diff --git a/AGENTS.md b/AGENTS.md index 7e8ae57d0..f41636f66 100644 --- a/AGENTS.md +++ b/AGENTS.md @@ -1,26 +1,38 @@ # Working on Matching One -Read `docs/RESEARCH-FRONTIER.md` and `docs/ROADMAP.md`, then the relevant live -Issue and its latest comments. `docs/STATUS.md` distinguishes current scope -from the verbatim historical ledger. Old acquisition instructions, opening -Issue bodies and publication-portfolio snapshots are not a current run queue. +Read `docs/WORK-NOW.md`, live Issue #650 and the relevant issue's latest +comments before scheduling new work. Issue #801 maps the owner's seven running +machines to actual jobs; do not infer a runtime inventory from issue titles. +`docs/RESEARCH-FRONTIER.md` and `docs/ROADMAP.md` retain dated research history. +Their older publication/acceptance focus and automatic width ladders are not +the current execution queue. The previous AGENTS version remains in Git. -Owner-delegated focus dated 2026-09-13, reconciled with the latest #650: -- One active acceptance package: consolidate and independently audit #735's - arbitrary-period root theorem, with #613/#736's sharp axial full-law boundary, - as one probability manuscript. Return the package to #735. -- Uniform oblique winding corridors are a possible strengthening within this - programme, not a new mandatory parallel project or duplicate census. -- #636 is a visible representation reserve, paused for new extension this round. - Reuse #708/#710/#733/#737 rather than re-commissioning their finite results. -- #275: original-U candidate maps before more sampling; theory-blocked P1, - with its original source/normalizer/moving-root contract retained. +Owner-delegated allocation, 2026-09-14: +- P0 question A: #768 identifies the first non-common sector insertion; #802 + predicts and computes a physical joint root-tangent / shape-normal response. + An intrinsic curve can erase a thermal-tangent correction that moves the root. +- P0 question B: #780 measures/proves the shared-label lineage merger bound + needed to transport the dilute cut model to a fixed-subcritical thermal window. + More exact moments of the limiting model are not a substitute for that bridge. +- P1 support: #613, #740, #760, #761, #769, #775, #776, #778. Support is bounded, + not permission to run all tasks in parallel. #769/#775 share raw rank tables; + finish the current smallest useful artifact, not an automatic L5--8 ladder. +- P2 topics remain legitimate mathematical interests but have no default new + shared-machine allocation. #787 -> #740, #788 -> #778, #784 -> #783 are merged + issue entrypoints; their original research text and results remain available. +- #275 keeps its original source, normalizer and moving-root contract. New rank + observers do not replace original U, and no frozen adjudication is changed. + +For running work, preserve checkpoints, completed shards and seeds before an +operator pauses at a safe batch boundary. Do not kill unknown processes, delete +raw data, change credentials, purchase resources or assume a priority label has +already stopped a machine. Unused capacity is preferable to duplicate production. Choose work by the mathematical or physical distinction it can settle, not by -how many PRs, scripts, new widths or compatibility scores it generates. A useful -new idea may replace this allocation, but state the new question and why its -answer would change the research. Do the small calculation now rather than -opening an issue asking someone else to repeat an already available result. +how many PRs, scripts, new widths or compatibility scores it generates. Pure +curiosity needs no publication rationale, but does not automatically expand the +shared compute queue. Do the small calculation directly instead of assigning +someone to repeat an already available result. Preserve existing data and frozen target history. Count one random block once. Name site versus bond, occupied graph versus matching complement, rank versus @@ -29,11 +41,10 @@ of sites, not the linear size. Distinguish actual operator Jordan structure from a derivative-jet lift and from an observed polynomial length factor. Open/closed, merged/unmerged and correct/incorrect are separate. Do not silently -merge another research PR, alter a source contract, or change a result's claim -level merely to tidy navigation. Use a narrow branch and a clear diff. Run -mathematical checks proportional to the change; report exactly what ran. -Do not claim parent CI for a child commit or full CI from local tests. +merge another research PR, alter a source contract, or promote a result merely +to tidy navigation. Use a narrow diff. Run mathematical checks proportional to +the change and report what actually ran; local checks are not full-repository CI. -`GOVERNANCE.md` keeps exploration lightweight. Do not add document-wording -tests, digest ceremonies, a new audit framework or a speculative production -scheduler. Keep proof, executable control, inference and novelty claims apart. +`GOVERNANCE.md` keeps exploration lightweight. Do not add wording tests, digest +ceremonies, another audit framework or an autonomous production scheduler. +Keep proof, executable control, inference and conjecture separate. diff --git a/docs/WORK-NOW.md b/docs/WORK-NOW.md new file mode 100644 index 000000000..a27a6fa6f --- /dev/null +++ b/docs/WORK-NOW.md @@ -0,0 +1,76 @@ +# 当前研究与算力分配：两处缺口，不再按公式数扩张 + +2026-09-14。所有者授权的优先级重排。本页与 #650 是当前分配入口，优先于旧 README、AGENTS、Issue 正文中的自动尺寸阶梯及发表/验收计划。数学兴趣不需要发表理由；共享算力需要一个尚缺的判断。 + +## 现在主攻什么 + +**问题 A：什么微观差异在拓扑消去之后仍然移动匹配根？** + +- **#768 / P0，深度理论与定向检索**：比较 scalar 8-arm、thermal-family spin ±4 及低阶竞争通道。区分旋转允许、匹配奇性、实际出现、差矩阵元非零。优先核对 c=0、h=5/8 的 level-2 奇异向量在实际模块里是否解耦，以及积分总导数、接触项和周期闭合。 +- **#802 / P0，分析后定向 CPU**：同一实际源下同时保留根位移和定 b 的形状变化。内禀 C_L(b) 会消掉纯热切向修正；不能仅凭它很小就排除一个根位移机制。先给两个不同预测，再计算一项现有表缺少的响应。 + +**问题 B：稀疏极限的漂亮过程是否真的运送到固定亚临界热窗口？** + +- **#780 / P0，共同标签理论与计算**：新的[双见证证明](manuscripts/geometric-balance/thermal-window-no-merger-20260914.md)已给出固定严格亚临界 p0 附近收缩参数窗口内的无宏观并合界，并控制 B_w/nu_- 指数下降。它是有明确输入的作者推导，不是小尺寸外推。接下来先处理强度时钟的可实现性、一致误差和实际标签接口；已有两参数谱系作业可完成一个最小产物用于核对或反例，不需要直接模拟指数长窗口反复等待并合。静态边缘、Laguerre 层级和更多极限核的矩不能替代该接口。 + +#650 是协调 P0；#801 是作业映射 P0。它们不是新增数学主线，也不是两项重计算。 + +## 算力边界：七台机器的真实作业还未核实 + +所有者报告七台机器在运行；本轮没有读取主机进程、当前资源使用或调度器。以下是给操作人员的指令，**不是已经停止/迁移作业的记录**。 + +先在 #801 为每台回填：机器别名、Issue、代码与输入、进度、检查点、下一最小独立产物、剩余工作的估计依据、CPU/RAM瓶颈。不要贴地址、令牌、密钥。未知内容写未知。 + +对无明确新问题的作业，在安全批次边界保存检查点、分片和种子，取消自动续跑。已经接近一个真正有用产物的，可以先完成该产物。不得杀未知进程或丢弃不可恢复工作。已经完成的结果照常提交，负结果也保留。空闲机器不是失败，不用一个新猜想填满每个槽位。 + +### 七个逻辑槽位（不是已识别的七台硬件） + +|槽位|建议用途|什么时候不继续占用| +|---|---|---| +|1|#802 一项源响应主算|候选尚无不同预测时不启动大算| +|2|同一响应的短交叉核对/必要分片|不是长期重复生产| +|3|#780 同标签两参数接口、时钟与有限误差|不重跑渐近无并合，也不退回更多静态矩| +|4|#761 独立质量区间|精度改善不能改变振幅/根判断时停| +|5|#769 或 #775 当前最小有用产物|两者共享原始 rank 表，不重复生成| +|6|#778 既有档案及便宜后处理|短作业按需用，不常驻大机器| +|7|预留|不为满载而开新梯度/宽度扫描| + +操作人员按实际内存与作业依赖重排。#801 回报之前，不对任何具体机器作“浪费/该杀”的断言。本页不启动付费资源、GPU、自动任务或远程凭据操作。 + +## 已经落地的优先级 + +P0：#768、#802、#780；协调 #650、作业映射 #801。 + +P1，有限支持而非全部并行：#613（已有几何论证的具体缺口）、#740（实际周期缝合，先比较界）、#760（质量局部性）、#761（独立质量区间）、#769（当前 L5 原始联合表封口）、#775（当前最小系数产物后停）、#776（复用自匹配控制，不开新阶梯）、#778（已有 paired 档案）。 + +P2，不默认分配新增共享大算力：#593、#758、#762、#767、#772、#774、#777、#779、#781、#783、#786、#789、#790。具体数学障碍或真正有区分力的反例可重新提升；这不禁止参与者自愿做理论探索。 + +已合并入口并关闭 duplicate：#787 → #740（周期留数）；#788 → #778（二次变差后处理）；#784 → #783（Fourier 有限接口）。原文与已有结果保留；关闭不等于证明。 + +#275 原始 U 保留其原源、正规化和 moving-root 合同。没有用新 rank 观察量完成旧候选识别，也没有修改冻结实验判决。未列入本轮主队列的历史标题不是七机自动运行指令。 + +## 当前重计算怎样瘦身 + +#775 不再要求首轮 L=5,6,7,8 全部完成。先复用 #769 可能已给出的 L5 rank-by-occupancy 表，计算原始 C[L,j,k] 与一个必要的根/形状对比。Fisher 曲率、面积、Bayes 距离、TP2、KK、源零点和高阶累积量都可以是后处理，但不能各自变成第二份原始枚举。 + +#769 保留 paired-flip 语义、位移原始表和代表见证，不再因新增一个描述符延长全部枚举。jump-two 的匹配偶性与 odd-sector 幅度是不同问题。 + +#593 的旧 width9/10/11 阶梯不再是当前授权。#774、#776 不因缺几行表就自动补齐宽度。相关在运行产物先交 #801 安全评估。 + +## 三项前序理论推进及本次跟进 + +前序推导见 [root-response-and-bridge-targets-20260914.md](root-response-and-bridge-targets-20260914.md)。 + +1. **切向盲点**：消去热坐标同时可能消去根位移机制。用根的切向响应 T 与形状法向响应 N 做联合比较。 +2. **条件模块筛选**：若 c=0、h=5/8 的 level-2 null 真正解耦，模去 L_-1 导数后 level 1..4 的维数为 0,0,1,1。C4 从这一热模块允许的首个非导数手征修正出现在 level4；这仍不证明它在实际扇区中非零，也不排除其他模块。 +3. **并合界本次已推进**：最终簇中的两个不交 essential 见证给 BK 平方界；统一体积尾处理大簇。固定 p0 下，宏观并合概率和 B_w/nu_- 的宽度对数上率均不超过 -kappa(p0)。连续正强度比 Lambda 的统一收敛仍是标记 Poisson 结论的明示条件；仿射热坐标、近临界统一性和所有路径/高阶矩不自动成立。 + +有限 p0 的 fixed-location 应使用固定纵向行的 essential 跨度覆盖/最近屏障。固定微观点可能为黑色或属于有限白簇，不能照搬 p→0 对照中异常概率趋零的取样约定。 + +这些结果决定下一步问什么；不是又一批为了精确而精确的宽度任务。 + +## 返回方式 + +理论返回一个被证明/反驳的具体箭头，或明确的最小缺口。计算返回原始新对象、成本、误差及它排除了什么；不能排除也如实记录。新文献只需要服务于实际模型、模块和扇区接口，不再按标题数量堆综述。 + +不修改 STATUS 的科学判决，不删除数据，不自动合并 PR。局部测试、数学正确性和作者猜想分别记录，但不用一套新的“灯”代替思考。 diff --git a/docs/manuscripts/geometric-balance/README.md b/docs/manuscripts/geometric-balance/README.md index 96a3ccd83..5e3e2aed1 100644 --- a/docs/manuscripts/geometric-balance/README.md +++ b/docs/manuscripts/geometric-balance/README.md @@ -27,3 +27,71 @@ contains the entire root and concentration arguments, and gives a bounded closest-source comparison. It is an author-supplied proof; no independent publication acceptance or originality certification is claimed. Existing proofs, data, frozen designs and research branches are retained unchanged. + +## 2026-09-14 integrated structural continuation + +The stacked continuation on PR #771 collects consequences that reduce several +previously separate analysis directions without changing the parent #739 +acceptance status. + +**Start review with [`round2-claim-ledger-20260914.md`](round2-claim-ledger-20260914.md).** +It is the current claim-level index and explicitly separates `EXACT`, +`AUTHOR-PROOF`, `CONDITIONAL`, `CONJECTURE / PROGRAMME`, and external-literature +boundary statements. It also records the shortcuts that were rejected or +corrected during self-audit. The thematic list below is an earlier compact +index; the ledger supersedes it when statuses differ or when newer continuation +files are not listed here. + +Since this compact index was first written, PR #771 has also added the +full-period first-exit torus bound, arbitrary-shape fixed-`p` homological free +energy, varying-direction centre theorem, all-direction quantitative mass +monotonicity, exact charge-neutral crossover coordinates, fixed-direction +Poisson--Gumbel extension, giant-white mean reward theorem, fixed-width charge +free energy, and a 2026 near-critical OZ/SITE-renewal boundary audit. Those +results are organized in the claim ledger rather than duplicated exhaustively +below. + +### Exact finite topology and birth structure + +- [`structural-consequences-20260914.md`](structural-consequences-20260914.md): persistent 4/8 birth reflection; dual-even/odd birth coordinates; exact same-parameter `(rank,K)` count reduction; alternating black/white barrier Palm identities; marked-Poisson transport; complementary Palm score constraints; convex loop/branch frontier; the `D^{-1}=partial_yy tau` consistency relation; and deterministic directional separation in exponential elongation. +- [`finite-reflection-dominance-20260914.md`](finite-reflection-dominance-20260914.md): graph inclusion plus persistent Alexander duality gives the exact finite stochastic order `1-T2 <=st T1`, hence `P2(1-p)<=P0(p)`, `M(p)+M(1-p)<=0`, `F(p)+F(1-p)<=1`, `Q(u)+Q(1-u)>=1`, and `int_0^1 M<=0` on every honest torus. +- [`rare-charge-balance-20260914.md`](rare-charge-balance-20260914.md): exact finite factorization of the matching-root slope into rare topological-charge susceptibility and conditional endpoint-odds slope, making `balance without concentration` algebraically explicit. +- [`neutral-gas-topological-charge-20260914.md`](neutral-gas-topological-charge-20260914.md): extensive winding counts live in one neutral `(1,1)` source direction while `D=W4-W8=r-1` is a bounded subextensive charge; includes the area-under-`M` reconstruction of the two sharp birth centres. +- [`projective-homology-gas-20260914.md`](projective-homology-gas-20260914.md): exact rank-one state `(slope,K)` and the resulting projective hard-core-gas language for multi-direction competition. + +### Alternating black/white geometry + +- [`poisson-tessellation-consequence-20260914.md`](poisson-tessellation-consequence-20260914.md): explicit total-variation/Markov-kernel derivation of the Exp component-Palm gap, Gamma(2,1) stationary-location gap, uniform relative position, and higher Gamma spacings. The coarse two-colour limit is a Poisson interval tessellation, not two independent Poisson clouds. +- [`supercritical-white-slab-bulk-20260914.md`](supercritical-white-slab-bulk-20260914.md): exact infinite-cluster boundary-density identity `beta(q)=(1-q)theta(q)/q` and a concrete bulk LLN/CLT target for the huge complementary white component. +- [`dual-surface-excess-slope-20260914.md`](dual-surface-excess-slope-20260914.md): exact finite dual identity `E_black[q n-p b]=E_white[q B-p N]=partial_z log nu`; at regular points the giant-white `O(exp(kappa w))` bulk terms cancel, leaving the `O(w)` excess `pq v w` that directly measures the Gumbel mass slope `v=-kappa'`. + +### Directional mass, first-exit bodies and large-d centres + +- [`matching-enhancement-mass-gap-20260914.md`](matching-enhancement-mass-gap-20260914.md): author-level two-terminal enhancement proof, using Grimmett--Li plus the corrected square-lattice rerouting theorem of Balister--Bollobas--Riordan, giving `kappa_8(p)1`. +- [`directional-enhancement-sandwich-20260914.md`](directional-enhancement-sandwich-20260914.md): one positive `p` sprinkling is beaten by full matching enhancement uniformly over endpoint direction, yielding `tau_8,p(e)<=tau_4,p+delta(e)` for all directions on compact parameter intervals. +- [`vector-first-exit-domain-20260914.md`](vector-first-exit-domain-20260914.md): repeated first-exit skeleton + site BK proves `B_S(t)<1` implies finite exponential susceptibility; certified domains are convex, certificates from different boxes may be convex-hulled, their large-box limit exhausts compact interiors of the Wulff/exponential-moment domain, and the note separates support functions from radial intercepts. +- [`dilute-directional-mass-centres-20260914.md`](dilute-directional-mass-centres-20260914.md): elementary axial dilute bounds `kappa_4=-log p+O(p)`, `kappa_8=-log(3p)+O(p)` and sharp `d->infinity` centre asymptotics. +- [`dilute-directional-geodesic-entropy-20260914.md`](dilute-directional-geodesic-entropy-20260914.md): fixed rational directions satisfy “graph-distance cost minus geodesic entropy”; gives explicit large-`d` tilted-centre asymptotics, e.g. diagonal `p4~(1/2)e^{-d/sqrt2}` versus `p8~e^{-sqrt2 d}`. +- [`p-regularity-audit-20260914.md`](p-regularity-audit-20260914.md): bond mass p-analyticity is classical, but no direct whole-subcritical square-SITE mass p-analyticity theorem is promoted; the at-most-countable exceptional `d` set in the SITE Gumbel theorem therefore remains. + +### Prefactor, sewing and finite-width locality + +- [`matrix-sewing-unit-residue-20260914.md`](matrix-sewing-unit-residue-20260914.md): finite-state matrix cyclic-sewing theorem. A simple Perron band with finite Markov memory gives exactly `exp(-kappa w)/sqrt(2 pi D w)` with unit logarithmic residue; finite local memory alone cannot explain an anomalous power or amplitude. +- [`sewing-amplitude-diagnostic-20260914.md`](sewing-amplitude-diagnostic-20260914.md): separates pure log-determinant residue, multiple soft-band multiplicity and genuine insertion/mark amplitudes; gives a falsification table for future `beta,zeta,D` certificates. +- [`periodic-mass-locality-mechanism-20260914.md`](periodic-mass-locality-mechanism-20260914.md): pure transverse periodization preserves the zero Fourier Perron mode exactly; any `gamma_w-kappa` comes from wrap-sensitive pieces/decorations. Exponential decoration locality would imply `gamma_w-kappa=O(e^{-cw})` by Perron perturbation. +- [`loop-branch-linear-tail-tests-20260914.md`](loop-branch-linear-tail-tests-20260914.md): because the variational bulge saturates at `r_*<1/2`, every `A>=1/2` is in the candidate's exactly linear branch regime; in particular #762's A=1,2 test predicts `-(1/w)log[P(L>=2w)/P(L>=w)]->kappa` if the #758 rate is correct. +- [`research-frontier-20260914.md`](research-frontier-20260914.md): the homological-free-energy variational picture, a small-`p` actual-SITE transfer/local-CLT programme for the complete-component `w^{-1/2}` prefactor, and the recommended order for the remaining common-window work. + +Finite controls for the persistent reflection are in +`scripts/persistent_alexander_birth_reflection.py`, with committed L=3 and L=4 +outputs in `results/geometric-consistency/`. The L=3 run exhausts all 512 +configurations and all 362,880 strict site orders; the L=4 run exhausts all +65,536 configurations and checks 20,000 fixed-seed site orders. A separate +zero-cost calculation of the constrained-Poisson/topology constants is in +`scripts/poisson_topology_constants.py`. + +These notes deliberately separate deterministic consequences, author-level +proofs requiring independent review, deductions that use existing #739 +probability inputs, and conjectural proof programmes. They do not identify a +continuum field, claim a full all-subcritical SITE sewing theorem, or turn the +near-critical crossover into an accepted square-site theorem. diff --git a/docs/manuscripts/geometric-balance/angular-irrep-improved-root-20260914.md b/docs/manuscripts/geometric-balance/angular-irrep-improved-root-20260914.md new file mode 100644 index 000000000..931255397 --- /dev/null +++ b/docs/manuscripts/geometric-balance/angular-irrep-improved-root-20260914.md @@ -0,0 +1,343 @@ +# Angular-irrep improvement of the matching charge root + +Date: 2026-09-14 + +Status: finite deterministic / exploratory transfer analysis plus a new finite-size-improvement conjecture. The transfer semantics are the committed safe lifted-gain semantics; local reimplementations were checked against committed roots to about `1e-11` or better. No new infinite-volume theorem is claimed. + +## 1. Motivation + +The oblique safe-transfer programme shows that the dominant finite-width matching-root displacement on the square lattice transforms as the real spin-four harmonic + +```text +H4(theta)=cos(4 theta). +``` + +The usual presentation is + +```text +p_ch(ell,theta)-pc + = -A4 H4(theta) ell^-4 + ... . +``` + +A stronger use of this fact is possible. Different orientations at the **same physical circumference** can be combined as an exact D4-irrep projector which removes the entire observed spin-four component without knowing `pc` or `A4` in advance. + +This turns angular dependence from a field-identification diagnostic into a finite-size-improvement mechanism. + +## 2. Two-orientation projector + +Suppose two short-period directions have the same physical circumference `ell`, with spin-four harmonics `h1!=h2` and charge roots `p1,p2`. + +At leading order + +```text +p_i = pc - A4 h_i ell^-4 + higher. +``` + +Then + +```text +A4_hat = ell^4 (p2-p1)/(h1-h2), + +pc_hat^(4-perp) + = (h1 p2 - h2 p1)/(h1-h2). +``` + +Neither formula uses an external threshold value. The second is the component of the two-root vector orthogonal to the `H4` irrep. + +More generally, write a D4 harmonic expansion + +```text +p_ch(ell,theta) + = pc + S0(ell) + + H4(theta) S4(ell) + + H8(theta) S8(ell) + + ... . +``` + +The two-angle projector removes **all** terms proportional to `H4(theta)`, not only the first `ell^-4` coefficient. If the axial `ell^-6` correction belongs to the same spin-four tower, it is removed as well. The remainder directly probes scalar and higher-irrep contamination. + +This is the finite-size analogue of a Symanzik/improved-observable projection. + +## 3. Exact equal-circumference pair at ell=5 + +The primitive direction + +```text +u=(3,4) +``` + +has Euclidean length `5`. Thus + +```text +axis: (1,0), n=5, ell=5, H4=1, +oblique: (3,4), n=1, ell=5, H4=-527/625. +``` + +The `n=1` oblique quotient is honest: the primitive period has length five, larger than the local NN/matching interaction range, and the transformed edges have strictly positive longitudinal displacement. The lifted-gain transfer has only + +```text +G4 states = 45, +G8 states = 147. +``` + +Roots: + +```text +p_axis5 = 0.5922358232050258, +p_34,1 = 0.5931582013380546. +``` + +The spin-four projector gives + +```text +pc_hat_ell5 = 0.5927362453692130, +A4_hat_ell5 = 0.3127638526162537. +``` + +For comparison only after the fact, the diagnostic high-precision threshold reference `0.59274605079` differs by + +```text +-9.8054e-6. +``` + +The individual raw roots are displaced by roughly `5e-4`; the irrep projection has already removed most of that finite-width error. + +## 4. Exact equal-circumference pair at ell=sqrt(65) + +The integer circle `a^2+b^2=65` contains two inequivalent primitive directions + +```text +u1=(1,8), +u2=(4,7), +``` + +with + +```text +H4(u1)= 3713/4225 = 0.8788165680..., +H4(u2)=-2047/4225 =-0.4844970414.... +``` + +Both use `n=1`, hence + +```text +ell=sqrt(65)=8.062257748... . +``` + +Safe-transfer state counts remain small: + +```text +(1,8): G4/G8 = 1394 / 2153, +(4,7): G4/G8 = 1134 / 6216. +``` + +The roots are + +```text +p_18 = 0.5926839084948207, +p_47 = 0.5927803979513810. +``` + +The exact harmonic projector is + +```text +pc_hat_sqrt65 + = (3713 p_47 + 2047 p_18)/5760 + = 0.5927461073406902, + +A4_hat_sqrt65 + = 0.2990272752626905. +``` + +Again the reference threshold is not used in the estimator. Its after-the-fact difference is + +```text ++5.66e-8. +``` + +This pair is especially useful because neither direction is an axis: the improvement is therefore not an axis-specific cancellation. + +## 5. Exact equal-circumference pair at ell=10 + +Use + +```text +axis: (1,0), n=10, ell=10, H4=1, +oblique: (3,4), n=2, ell=10, H4=-527/625. +``` + +The oblique transfer has + +```text +G4/G8 states = 7259 / 75541, +row memory = 4 / 7, +p_34,2 = 0.5927709164472407. +``` + +A dedicated C++ realization of the committed axial lifted-gain algorithm was checked at `w=9` against the repository root to below `1e-12`, and gives + +```text +w=10 safe states G4/G8 = 8953 / 8953, +p_axis10 = 0.5927163956300879. +``` + +Therefore + +```text +pc_hat_ell10 + = (625 p_34,2 + 527 p_axis10)/1152 + = 0.5927459750664773, + +A4_hat_ell10 + = 0.2957943638928858. +``` + +After the fit, comparison to the diagnostic reference gives + +```text +-7.57e-8. +``` + +The ell=5 and ell=10 projected errors differ in magnitude by about a factor `129`. With only two related scales this must **not** be promoted to an exponent fit. It does show that the algebraic angular cancellation removes far more than the raw `ell^-4` error. + +## 6. Strong constraint on a leading scalar x=21/4 explanation + +A scalar correction of the same root exponent four would enter `S0(ell)`, not `H4 S4`, and would survive the projector. + +The projected residuals at `ell≈8--10` are of order `1e-7`, while the observed spin-four raw correction at the same sizes is of order `3e-5`--`7e-5` in `p`. Therefore an orientation-independent exponent-four component large enough to explain the original root displacement is incompatible with these angular projections. + +This does not prove that every scalar `x=21/4` amplitude is exactly zero. It says that such a scalar cannot be the dominant mechanism behind the observed matching-root `ell^-4` shift. + +## 7. Thermal denominator is also orientation-scalar at equal ell + +Define the physical thermal-slope indicator + +```text +D(theta,ell) + = |u| d_p Theta_row(p_root) ell^(1/4) + = Q_p(p_root)/ell^(3/4), + +Q = n |u|^2 Theta_row. +``` + +For the exact `ell=10` pair, + +```text +D_axis10 = 3.39179932, +D_34,2 = 3.39187874, +relative difference = 2.34e-5. +``` + +For the equal `ell=sqrt(65)` pair, + +```text +D_18 = 3.40273574, +D_47 = 3.40296413, +``` + +again differing only at the `1e-4` level. + +Thus the root angular law is cleanly factorized at these sizes into + +```text +spin-four critical numerator / scalar leading thermal denominator. +``` + +## 8. Root/slope-normalized shape: the TANGENT_SPIN4 diagnostic + +For each orientation define + +```text +y = Q_p(p_root)(p-p_root), +Psi(y)=Q(p). +``` + +If the leading spin-four correction acts only by translating the thermal scaling coordinate, all orientation dependence should disappear from `Psi` to first order. + +At `ell=10`, axis versus `(3,4)` gives + +```text + axis10 (3,4),n2 difference + +y=-0.50 -0.4995465515 -0.4995187757 +2.78e-5 +y=-0.25 -0.2495311293 -0.2495230387 +8.09e-6 +y=+0.25 +0.2512046459 +0.2512132325 +8.59e-6 +y=+0.50 +0.5063352579 +0.5063680241 +3.28e-5 +``` + +Local normalized coefficients + +```text +Psi(y)=y+a2 y^2+a3 y^3+... +``` + +are + +```text +axis10: a2≈0.0133250, a3≈0.0235507, +(3,4),n2: a2≈0.0134625, a3≈0.0235648. +``` + +The equal `sqrt(65)` pair behaves similarly. Its slope-normalized curves differ by only `~1e-5--5e-5` on `|y|<=0.5`, despite roots lying on opposite sides of the threshold. + +This is strong finite-width evidence for + +```text +Q_{ell,theta}(X) + = F(X + delta X_4(theta,ell)) + + smaller normal correction, +``` + +rather than an orientation-dependent deformation of the full scaling-function shape at leading order. + +## 9. Angular-improvement hierarchy conjecture + +The natural stronger conjecture is + +```text +p_ch(ell,theta)-pc + = H4(theta) [a4 ell^-4+a6 ell^-6+...] + + H8(theta) [b8 ell^-8+...] + + S0(ell) + + ... . +``` + +If the known axial `4,6,...` correction ladder initially belongs to one spin-four family, the two-orientation projector removes the whole bracket, not just `a4 ell^-4`. The first surviving term could then be an `H8` channel or a genuinely scalar dual-odd correction. + +For the `(3,4)` angle, + +```text +H8=cos(8 theta)=164833/390625, +``` + +and the H4-cancelled axis/(3,4) combination carries effective + +```text +H8_eff = 429/625 = 0.6864. +``` + +The current ell=5 and 10 residuals are not sufficient to decide whether the next asymptotic term is scalar `ell^-6`, spin-eight `ell^-8`, a logarithmic collision term, or a mixture. Their main use is to demonstrate that the leading spin-four tower can be removed by symmetry before fitting the next mechanism. + +## 10. Practical consequence + +The angular projector provides two outputs from the same small transfer calculations: + +1. a mechanism test for the D4 representation content of the root correction; +2. an improved threshold estimator which can converge much faster than either orientation root separately. + +This suggests a new order of operations for future high-precision work: + +```text +first project out known lattice irreps, +then fit / identify the residual correction. +``` + +Do not spend larger widths to fit a correction that an exact symmetry projector can remove at smaller widths. + +## 11. Claim boundary + +- Integer directions, physical circumferences, D4 harmonics, safe-state counts and finite roots listed above are deterministic finite-transfer quantities. +- `pc_hat^(4-perp)` is a reference-free finite-size estimator conditional only on using the observed D4 harmonic as the projected component; comparison with the external threshold is diagnostic after the fact. +- The statement that the entire `ell^-6` axial correction belongs to the same spin-four tower is a conjecture. +- The apparent superconvergence of the projected estimator is not yet assigned an exponent. +- The functional `TANGENT_SPIN4` interpretation remains a scaling conjecture, albeit now supported by equal-circumference full-curve controls rather than root exponents alone. diff --git a/docs/manuscripts/geometric-balance/angular-irrep-radial-spectrum-research-map-20260914.md b/docs/manuscripts/geometric-balance/angular-irrep-radial-spectrum-research-map-20260914.md new file mode 100644 index 000000000..055d394fe --- /dev/null +++ b/docs/manuscripts/geometric-balance/angular-irrep-radial-spectrum-research-map-20260914.md @@ -0,0 +1,165 @@ +# Angular irrep first, radial spectrum second: a corrected research map + +Date: 2026-09-14 + +Status: synthesis after the reverse audit. This is a research-ordering note, not a theorem or publication plan. + +## 1. Why the order matters + +The recent programme repeatedly used one observed finite-size power to answer three different questions at once: + +```text +which square-lattice angular representation is present? +which continuum scaling dimension controls the radial decay? +which matching/pair-exchange source direction does the correction occupy? +``` + +These are independent axes. The current evidence and errata show that mixing them is a major source of over-interpretation. + +The corrected order is: + +```text +1. angular irrep / geometry, +2. normalized microscopic source direction, +3. radial exponent / logarithmic structure, +4. continuum module / field identification, +5. only then OPE/RG parity claims. +``` + +## 2. Angular layer: treat `H_(4n)=cos(4n theta)` as the primary basis + +For the square model at fixed physical circumference, write + +```text +p_ch(ell,theta) + = P0(ell) + + P4(ell) H4(theta) + + P8(ell) H8(theta) + + P12(ell) H12(theta) + + ... . +``` + +This is just a D4/reflection-compatible Fourier decomposition. + +The strongest current deterministic result is that the leading orientation-dependent coefficient is overwhelmingly H4-like. Independent oblique automata now validate the underlying safe-root calculations. + +The next task is not to fit another power to raw roots. It is to isolate `P0,P8,...` after H4 is removed. + +### Current gates + +- axis/(3,4) equal-ell pair: clean H4 projector at `ell=10`, but with nonzero higher-harmonic leakage; +- #808 N377 pair: exact H4 notch and near H8 notch, much stronger angular-scalar gate; +- N1105 four-angle tomography: attractive algebraically but current automaton hits a prohibitive state-space wall. + +Thus #808 is the present high-information experiment; N1105 is an algorithm-development target, not default production. + +## 3. Radial layer: each angular coefficient has its own expansion + +After angular separation, fit/derive each coefficient independently: + +```text +P4(ell) = a4 ell^-4 [1+c4,2 ell^-2+...], +P0(ell)-pc = a0 ell^-q0 [1+...], +P8(ell) = a8 ell^-q8 [1+...], +... +``` + +The old “4,6,8,... correction ladder” should not be treated as one operator series before this decomposition. + +In particular: + +- `ell^-6` inside `P4` can be a radial dressing of the leading H4 coefficient; +- an `ell^-7` signal in an H4-null/angular-scalar channel is a different object; +- equality of exponents across angular sectors would be a resonance/mixing clue, not proof of identical mechanism. + +## 4. Current `ell^-7` candidate should be typed conservatively + +The axis/(3,4) H4-null residual at `ell=5,10` is compatible with a reference-free `ell^-7` extrapolation. #808 is designed to test this with a geometry that strongly suppresses H8. + +A positive result should first be called + +```text +post-H4 angular-scalar ell^-7 correction. +``` + +Only after the angular leakage budget and radial law are stable should it be compared to + +```text +x = x_t + 7 = 33/4 +``` + +candidates such as `V_<1,4>` / its Q=1 logarithmic collision block. + +The historical #47 warning remains important: an `ell^-7` root correction is not the ordinary next thermal spin-four quasiprimary in the simple tower. + +## 5. Source layer is a separate coordinate + +A local perturbation should first be represented by its normalized likelihood score and quotiented by constant/thermal nuisance directions. + +Use the exact Bernoulli-chaos grading as a microscopic source basis, not as a continuum parity theorem. + +The relevant data object is a response matrix + +```text +angular channel × residualized source direction × width. +``` + +A continuum interpretation should explain this matrix, not merely one column/root sequence. + +## 6. A three-index correction notation + +To avoid future aliasing, label a finite-size correction schematically by + +```text +C[a, q, s] +``` + +where + +```text +a : angular harmonic index (0,4,8,...), +q : measured radial root exponent or correction exponent, +s : microscopic residualized source class / intervention label. +``` + +Only after theory identifies a stable continuum block attach representation/module names. + +Example current hypotheses: + +```text +C[4,4,uniform-root] : strong deterministic support; +C[0,7,H4-null] : candidate under #808; +C[8,? ,...] : nuisance/adversary until isolated; +``` + +This notation is deliberately phenomenological. It prevents “spin 4” from being inferred from exponent 4 or “matching odd” from one S/D observable. + +## 7. What would count as a real big step now + +A high-value result is one of: + +1. #808 demonstrates a robust angular-scalar residual whose higher-harmonic leakage is too small to explain it; +2. a residualized degree-3 microscopic source reveals a new response direction not aliased with thermal/pair sources; +3. a transfer/RG construction maps the finite pair-exchange source basis into a continuum tangent block; +4. a direct tagged-operator calculation assigns a distinct topological action to the #800 slow doublet; +5. an algorithm compresses large-memory oblique safe states enough to make 3+ same-circle angles feasible. + +A lower-value continuation is another root width or another derived moment that does not change one of these distinctions. + +## 8. Claim boundary + +Exact/factual inputs: + +- D4 angular harmonic algebra; +- independent oblique transfer verification; +- #802 source-normalization/alias erratum; +- N325/N1105 cost wall; +- exact H4-null leakage formulas. + +Research organisation: + +- angular-first/source-second/radial-third ordering; +- conservative naming of `ell^-7` residual; +- three-index phenomenological correction labels. + +The purpose is to make new computations answer one typed question at a time. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/audit-780-monotone-fragmentation-20260914.md b/docs/manuscripts/geometric-balance/audit-780-monotone-fragmentation-20260914.md new file mode 100644 index 000000000..9814d4dcb --- /dev/null +++ b/docs/manuscripts/geometric-balance/audit-780-monotone-fragmentation-20260914.md @@ -0,0 +1,235 @@ +# mono780 · 任务B：核心数学声称审计（9 条） + +格式：**主张 → 我的独立计算 → 裁定 → 最小修正措辞**。 +我的计算全部在云机用自己的脚本完成（`/workspace/mono780/work/audit.py`、`clock.py`），**不 import 包内模块**，只依赖 `fractions.Fraction`。 +强度标注：【精确恒等式】= 有理数可证；【有限枚举】= 有限穷尽/构造；【条件定理】= 结论依赖未验证假设；【猜想】。 + +--- + +## B1 `[z^0](z^-1+1+z)^4 = 19` + +**主张**：宽度 4 的最小黑 matching 屏障的纵向平移类有 19 个，且"按最低行锚定，不再除以 4"。 + +**我的独立计算**： +1. 卷积：`(z^-1+1+z)^4` 的 `z^0` 系数 = **19**。 +2. 暴力：`|{w∈{-1,0,1}^4 : Σw=0}|` = **19**。 +3. 显式：`Σ_{k=0..2} 4!/(k!k!(4-2k)!)` = `1+12+6` = **19**。 +4. 枚举构造 19 个点集：**每个都恰好每列 1 点**（列重数全为 (1,1,1,1)）； + 在 **matching(8 邻接)** 下 **19/19 全部绕行**。 +5. 单射性：`word → 归一化点集` 的像大小 = **19**（无碰撞）。 +6. 旋转：19 个里 **恰 1 个旋转不变**（全水平环），其余 18 个轨道长 4。 + +**裁定**：【精确恒等式 + 有限枚举】**成立，且论证比正文更强**。既然存在 1 个旋转不动点，`19/4` 根本不是整数，**除以 4 本身不自洽**——所以"不除 4"不只是"锚定技巧"，而是唯一自洽的计数。 + +**最小修正措辞**：把"保留列标号、按最低行锚定；不再除以 4"补一句量化理由： +>「19 个平移类中含 1 个旋转不变类，旋转轨道大小不齐，故商掉旋转会得到非整数 19/4；列标号在 `a_x` 逐列不同的模型里是真实的，不能商。」 + +**附带 caveat（我测到的，正文没提）**：这 19 个集合若搬到**黑为 NN(4 邻接)** 的模型里，**只有 1 个（全水平环）仍绕行**，其余 18 个在 NN 下断开。对本包 model B（黑=matching）无影响，但复用时要注意颜色-邻接配对。 + +--- + +## B2 4 点孔洞围栏（中心的四个轴邻点） + +**主张**：B 的最小查询围栏有 4 点，恰为中心的四个轴向邻点，每行 4 个；与 19 个 essential 模板不同类。 + +**我的独立计算**： +- 点数 = **4**；列重数 = **(1,1,2)**（中心列 2 点，另一列 0 点）→ 与"每列恰好一点"的 19 个模板**必然不同类**，且 `fence ∉ 19 模板` ✓。 +- 绕行判定：matching 下 **False**、NN 下 **False** → 可缩 ✓。 +- 隔离性：中心四邻全黑 ⇒ 中心在**白 NN 图**中孤立（四邻无白，故与任何白簇 4-连通断开）✓。 +- 最小性：非零绕行至少 w=4 步且等号强制每步横向正进 → 每列恰一点，故 4 点绕行集必为 19 类之一；围栏不绕行 → 两者互斥，4 点是"围住一个格点"的下界。 + +**对照表复核**： + +| 列场 a | A_B | A_H | r = A_H/A_B | 正文 | 裁定 | +|---|---:|---:|---:|---|---| +| w4 B 均匀 (1,1,1,1) | 19 | 4 | 4/19 | (19, 4/19) | 一致 | +| w4 B 变形 (2,1,1/2,1) | 19 | 25/4 | **25/76** | (19, 25/76) | 一致 | + +**裁定**：【精确恒等式 + 有限枚举】成立。**最小修正措辞**：无（"每行 4 个"宜写"每列的平移类 1 个、共 4 类"，与 B1 的"19 类"同口径，见 `interfaces.md`）。 + +--- + +## B3 `r_B = (a0/a2 + a2/a0 + a1/a3 + a3/a1)/19 >= 4/19` + +**主张**：保持 `Πa_x = 1` 就保持**整个领先 cuts 过程**不变（不只是某个平均值），而孔洞比 `r_B` 可被改大；等号 ⟺ `a0=a2 且 a1=a3`。 + +**我的独立计算**： +- `A_H = Σ_{x=0}^{3} a_x² a_{x-1} a_{x+1}`（4 个十字模板，逐个按重数 `a_x²·a_{x-1}·a_{x+1}` 构造）；两边除以 `A_B = 19 Πa_x` 得 + `r_B = (a0/a2 + a2/a0 + a1/a3 + a3/a1)/19`（逐项展开对账 4/4 吻合）。 +- 数值：均匀 `4/19`；变形 `(2,1,1/2,1)` → `(4 + 1/4 + 1 + 1)/19 = (25/4)/19 = 25/76` ✓；交替 `(u,u^{-1},u,u^{-1})` → `4/19` 且 `A_B=19` ✓。 +- 下界：`t + 1/t ≥ 2` ⇒ 两个倒数对之和 `≥ 4` ⇒ `r_B ≥ 4/19`；**在 `a0=a2, a1=a3` 时取等**（二维等号类）。 + +**对"只用了 `A_B` 不变"这一步的重点检查（我做了，结论比正文更强）**： +正文说"保持 leading cuts 不变"。我做的是**逐模板检查**：19 个屏障模板**每一个都恰好每列 1 点**，因此 +`Π_{v∈S} a_{x_v} = Π_{x} a_x` 与 `S` 无关 ⇒ `P(T_S ≤ t) = ε^b t^b Πa_x` 对**所有 19 个模板、所有空间位置**都相同。 +而模板的**空间位置与 `a` 无关**（`a` 只进 `T_S` 的时间坐标）。 +⇒ **整个领先 cut 点过程的律（不只是强度/均值）在固定 `Πa` 下逐点不变**。所以"不是只有一个平均值被保住"这句是**站得住的**；真正的机制是"每列恰好一点"。 + +**裁定**：【精确恒等式】成立；不变性论证成立且我给出更清晰的机制。**唯一范围限制**：这是 **ε↓0 的领先阶**结论；有限 ε 的高阶重叠修正 `b1,b2` 依赖 `a`（作者在 §6 末尾已声明"有限 epsilon 及更高阶图案仍能不同"）。 + +**最小修正措辞**：在 §6 加一句机制说明： +>「不变性来自每个最小屏障模板恰含每列一点，故 `Π_{v∈S}a` 与模板无关；这不是"只保住一个平均值"。」 + +--- + +## B4 `(19p⁴L, H) → (E,H)`，`H|E ~ Poi((4/19)E)`，`P(H=k)=(19/23)(4/23)^k`，`P(H=0)=19/23` + +**我的独立计算**： +- `H|E=e ~ Poi(r e)`，`E~Exp(1)` ⇒ `P(H=k)=∫₀^∞ e^{-e}(re)^k e^{-re}/k! de = r^k/(1+r)^{k+1}`。 +- 令 `r=4/19`：`r^k/(1+r)^{k+1} = (4/19)^k·(19/23)^{k+1} = (19/23)(4/23)^k` —— 逐项（k=0..7）**精确相等** ✓。 +- `P(H=0) = 1/(1+r) = 19/23` ✓；`E[H]=E[rE]=r=4/19` ✓；`E[H|E]=rE=(4/19)E` ✓；PGF `1/[1+s+r(1-z)]` ✓。 +- tagged（固定空间位置，`Y~Gamma(2,1)`）：`P(H_tag=k)=(k+1)r^k/(1+r)^{k+2}` 是负二项；`P(H_tag=0)=(1+r)^{-2}`： + `r=4/19 → (19/23)² = 361/529` ✓；`r=8 → 1/81` ✓。 + +**免责一致性检查**：§5 末"这些首先是有界变换收敛，所列均值方差属于**显式极限模型**；不由弱收敛单独推出实际有限 p 的高阶矩"，与 §4 末"这是极限过程的精确矩，若报告实际孔洞协方差的收敛还需相应**一致可积性**，不用弱收敛偷换"——**口径一致**，未见自相矛盾。 + +**裁定**：【精确恒等式（极限模型内）+ 明确声明的弱收敛免责】成立。 +**最小修正措辞**：§5 小标题「角色交换的静态预测现在是**定理后果**」偏强，建议改为： +>「角色交换的静态预测是**低密度极限（ε↓0，误差 O(ε)）的推论**；其矩按作者声明属于显式极限模型。」 + +--- + +## B5 w8 的 `r = 8` 是否只作输入引用 + +**主张**：w8 静态屏障/孔洞律是团队已有结果，本轮不重复计为新结果。 + +**我的核对**：README 第 15 行、正文开篇第 3 行、§5 第 109 行**三处**都把它标为输入/不计新。 +但**脚本仍然重算**了 w8：`profiles` 含 w8（`A_B=1, r=8`）、`motif_dependencies` 含 w8（`barrier=1, hole=8, b1=249, b2=12/13/14/15`）、测试 `test_column_control_eight` 断言 `r=45/2`、§6 (6.2) 推导 `r_A ≥ 8`。 + +**裁定**：声明**诚实**（文字上确实只作输入）。两点 nuance： +1. 脚本把 w8 当**对照/控制**重算（合理），不等于"计为新结果"； +2. 但 §6 的 **`r_A ≥ 8` 不等式与"等号仅常数列场"** 是**本轮新增**（AM-GM + Fourier），与 w8 的数值 `r=8` 是两件事，正文没显式分开。 + +**最小修正措辞**：§5 或 §6 注明「w8 的**数值** `r=8` 为既有输入；`r_A ≥ 8` 的不等式与等号刻划为本轮新增」。 + +--- + +## B6 四组 `executed_controls` 的真实性（216 / 610 / 49） + +**我的独立计算**： +- **216** = `k(3)×r(2)×z(3)×v(3)×s(2)×t(2)` = 216（`two_time` 网格）。 +- **610** = w4 的 `19+4=23` 个模板有序对 `23²=529` ＋ w8 的 `1+8=9` 个有序对 `9²=81`。 +- **49** = `7²`（`m,n ∈ 0..6`）。 +- 重跑脚本内对应代码：`physical_control`、`two_time_integral`、`motif_multitime_check`、`adjoint_moment` 全部自洽（`audit.out`）。 + +**裁定**：三个数**逐项复现**。但**证据强度有水分**： +- 216 与 49 是**参数网格大小**（不是 216/49 个独立数学事实）； +- 216 对应的"独立积分分解"`two_time_by_integrals` 与 `two_time` **代数恒等**（见 `issues-found.md` #2），**不构成独立证据**； +- 49 在 `report()` 里是**字面常量**（见 #3）。 +- 只有 **610**（`motif_multitime_check` 里真的遍历所有有序对并断言 `direct==separated`）是实质检查。 + +**最小修正措辞**：JSON/正文区分「枚举计数」与「参数网格大小」，并把 `two_time_integral_equalities` 改名为 `two_time_closed_form_grid_points` 之类，或换成真正的独立积分分解。 + +--- + +## B7 第二篇：log 时钟 Poisson 计数、两时刻孔洞律与矩 + +**主张**：`N_split ~ Poi(2log(τ₂/τ₁))`；两时刻联合变换 (4.1)；孔洞也有精确时间记忆，`Cov(H₁,H_k)=2r(1+r)/k`、`Cov(Y₁,Y_k)=2/k`、`Cov(Y₁,H_k)=2r/k`、`Corr=1/k=e^{-Δs}`。 + +**我的独立计算（从模型积分从零做，不经包内代数）**： +1. **转移核 (2.2)** 精确化：`P(X₂>u) = ∫_{u/k}^∞ e^{-x}e^{-(k-1)u/k}dx = e^{-u}` ⇒ **边缘恰为 Exp(1)**（数值核对 `max|P(X2>u)-e^{-u}| ≈ 1e-16`）。`E[X₂|X₁=x] = k(1-e^{-(k-1)x})/(k-1)`（解析积分，MC 旁证）。 +2. **两时刻联合 PGF**：我按"`x~Exp(1)`、`y=min(x,Z)`、`Z~Exp(k-1)`"，把指数 `-sx-tyk-rx(1-z)-ry(1-v)(z+k-1)` 对 `(x,Z)` 做两个积分，得 + `per-side = (A+k-1)/[A(A+k-1+B)]` → 平方即 (4.1)。**在 1620 个有理参数点上与 (4.1) 逐点相等，0 失配**。 +3. **孔洞 PGF (4.2)**：同一积分在 `s=t=0` 时给出 `(k+r(1-z))/[(1+r(1-z))(k(1+r)-rv(z+k-1))]` 平方，**96 点 0 失配**。 +4. **矩 (4.3)**：由 (4.1) 在 `z=v=1` 的 `M(s,t)=((s+k)/((1+s)(s+k+kt)))²` 做精确 2 阶 Taylor： + `E[Y₁]=E[Y_k]=2, Var(Y)=2, Cov(Y₁,Y_k)=2/k`（k=1,2,4,8 全中）。 +5. **孔洞协方差**：按**孔洞持久性**自算——旧孔在 `[0,y]` 被计两次、`(y,x]` 只计一次、新孔只计第二次： + `Cov(H₁,H_k|one side) = r·E[y] + r²k·Cov(x,y) = r/k + r²/k`，两侧独立 ⇒ `2r(1+r)/k`。 + **r=4/19**：`184/361, 92/361, 46/361`（k=1,2,4）✓；**r=8**：`144, 72, 36` ✓。`Corr = 1/k` ✓。 + `Cov(Q₁,Q_k)=2r/k` 与 `Cov(Q,Y)=0`（Q=H−rY）亦复现。 +6. **§3**：单侧无新 cut 概率 `= E_x[e^{-x(k-1)}] = 1/k = τ₁/τ₂` ⇒ 双侧 `(τ₁/τ₂)²` ✓；与 `Poi(2log k)` 的 `P(N=0)=1/k²` 一致 ✓。 + +**裁定**：【精确恒等式】全部成立，且我是**独立重建模型再积分**得到的，不是复用它的闭式。 +**"孔洞也有精确的时间记忆"对应的数学事实**：**孔洞的位置持久性**——cut 只删除落在它上面的孔洞，其余孔洞跨参数保留**同一位置**；因此 H 的两时刻协方差与长度**同阶衰减 `1/k`**（不是独立重抽 Poisson 会得到的 0 相关），且 `Q=H−rY` 与 `Y` 在全滞后不相关。这是该短语唯一自洽的数学对应物。 + +**最小修正措辞**：把"孔洞也有精确的时间记忆"补上机制一句话（"源于孔洞位置持久、cut 只删除其上的孔洞"），以免读者误解为"每时刻重抽 Poisson"。 + +--- + +## B8 逆向过程：确定性收缩 + 指数移入 + +**主张**：`A_rev f = -x f' + ∫₀^∞ e^{-z}[f(x+z)-f(x)]dz`；对所有多项式 `⟨x^m, A x^n⟩_Exp = ⟨A_rev x^m, x^n⟩_Exp`（本轮核对 m,n=0..6 共 49 个）。 + +**我的独立计算**： +- `A x^n = n x^n - n/(n+1) x^{n+1}` ⇒ `⟨x^m,A x^n⟩ = n(m+n)! - n(m+n+1)!/(n+1)`。 +- `A_rev x^m = -m x^m + Σ_{j=1}^m C(m,j) j! x^{m-j}` ⇒ `⟨A_rev x^m,x^n⟩ = (m+n+1)!/(n+1) - m(m+n)!`。 +- **两式相减恒为 0**（代数恒等式，对**所有** `m,n ≥ 0`）。数值核对 `m,n = 0..12`（169 个）**全对**。 +⇒ **49 不是特殊数**；如果它把 `range(7)` 扩到 `range(13)`，计数会是 169，结论不变。 +- 平稳性：`E_Exp[A x^n] = n·n! - n/(n+1)·(n+1)! = 0` 对**所有 n** 成立。 +- 均值：`E[X_s|X_0=x] = e^{-s}x + ∫₀^s e^{-(s-t)}·1dt = e^{-s}x + 1 - e^{-s} → 1`（与 `Exp(1)` 均值一致）；收缩率 1 与移入率 1×`E[E_j]=1` 精确平衡。 + +**裁定**：【精确恒等式】成立（且是**全参数**恒等式，不是"核对了 49 个"）。 + +**最小修正措辞**：把"本轮核对 m,n=0..6 的 49 个有理数等式"改为 +>「该对偶关系对**所有** `m,n ≥ 0` 是代数恒等式（`forward - reverse ≡ 0`）；脚本核对了 m,n=0..6 的 49 个特例。」 + +--- + +## B9 §7「固定亚临界 p 的热窗口是否由同一过程控制」 + +**主张（原文措辞）**：提出假设 `nu_w(p_w(x))/nu_0 → Λ(x)` 与条件 (a)(b)(c)，在此三条件下得 **条件定理**（时空 cuts 强度 `dz dΛ(x)`）；并明确写"**这里没有把 (a)–(c) 写成已检查完毕的实际固定 p 定理**"、"**可行推导**"、"**可行计算**"，以及"如果已有…能使 `p_w=p0+x/(vw)`、`Λ(x)=e^x`，那么…（7.2）"。 + +**我的核对**：**它自己定性为条件定理 + 未验证假设**，且**没有给出 (7.1)/(a)/(b)/(c) 的任何数值或解析验证**。文中给出的"可检验判据"只有一个**提案**： +- 用两/三个 p 的**共同标签**输出（较早屏障是否保留、是否合并、最早新增屏障位置）＋ tagged 白区间**两时刻 Laplace 变换**； +- 并明确警告不要把分别独立抽取的单参数表当联合数据。 + +**裁定**：**是"有条件的定理 + 未验证假设"，不是已证明的结果，也不是无判据的猜想**。判据（可检验量）**有提出但未执行**；没有给出任何数值支撑，因此**不能宣称热窗口已由同一过程控制**。 +**不确定度量级**：无（零数值支撑，不适用三点相容/支撑的区分）。 +**最小修正措辞**：把这句明确写成一句独立的 scope 声明： +>「(7.1) 与 (a)–(c) 均为**未验证假设**；本节只给出"若它们成立则…"的条件定理，(7.2) 亦为条件推论。文中未提供任何宽度/参数下的检验。」 + +--- + +## 小结 + +| 条目 | 裁定 | 强度 | +|---|---|---| +| B1 `[z^0]=19` + 19 模板 | 成立（且论证可加强） | 精确恒等式 + 有限枚举 | +| B2 4 点围栏 & 对照表 | 成立 | 精确恒等式 + 有限枚举 | +| B3 `r_B ≥ 4/19` 与 `25/76` | 成立；不变性机制我给出更强版本 | 精确恒等式（领先阶） | +| B4 `H|E~Poi((4/19)E)`、`(19/23)²`、`1/81` | 成立；免责一致 | 精确恒等式（极限模型） | +| B5 w8 只作输入 | 声明诚实；`r_A≥8` 为本轮新增 | 文本核对 | +| B6 216/610/49 | 数值成立；216/49 证据强度偏弱 | 部分为参数网格大小 | +| B7 两时刻律与矩 | **全部独立重建并成立** | 精确恒等式 | +| B8 逆向多项式恒等式 | 成立（对所有 m,n） | 精确恒等式 | +| B9 §7 热窗口 | 条件定理、假设未验证 | 条件定理 | + +--- + +## 任务 D：包**没想到**的 4 条（我另做的） + +### D1 `r_B ≥ 4/19` 的**可达性与全局形状** + +- **下界可达**：`r_B = 4/19` ⟺ `a0=a2 且 a1=a3`（**二维**等号类）。均匀场 `(1,1,1,1)` 与交替场 `(u,u^{-1},u,u^{-1})` 都取等。 +- **没有上界**：固定 `Πa=1` 时取 `a=(M,1,1/M,1)` ⇒ `r_B=(M²+M^{-2}+2)/19 → ∞`。 +- 所以"保持 leading cuts 不变"对孔洞噪声的作用是**单侧可调**：**只能放大、不能压到 4/19 以下**，且放大倍数无界。 + 这是我另算出来的量化结论（正文只说"可以不同"，未给区间）。 +- **可检验签名**（我建议加进正文，用来兑现"保持拓扑不变却改变孔洞噪声"）： + 变形 `(2,1,1/2,1)` 下 leading cuts **逐点不变**，但 + **component-Palm `P(H=0)`：`19/23` → `76/101`**； + **tagged `P(H=0)`：`361/529` → `5776/10201`**。 + 这是同一批 uniform 标签、同一屏障过程下**纯孔洞**的可观测量差异，直接可用 w4 稀疏模拟检验。 + +### D2 不变性的**真正机制**比正文的措辞更强 + +正文写"保持 `Πa_x=1` 就保持整个领先 cuts 过程不变，**不仅**保持某一个平均值"。 +我把机制找出来了：**19 个最小屏障模板每一个都恰含每列 1 点** ⇒ `Π_{v∈S} a_{x_v} = Π_x a_x` 与模板无关； +又因模板的**空间位置与 `a` 无关**（`a` 只进完成时间的 Laplace 因子）， +⇒ 在固定 `Πa` 下，**整个 cut 点过程的（位置,τ）联合律**逐点相同，**并不只是强度或一阶矩相同**。 +（这也解释了为什么 `b1,b2` 这类有限 ε 修正**会**变：它们依赖交集/并集的乘权重，而交集不再"每列一点"。） +**建议**：把这条机制写进 §6，比"不仅保持某一个平均值"更有说服力，也顺手说明高阶项为何例外。 + +### D3 对数时钟的"平稳"**不需要额外重整化条件** + +正文只说 `Exp(1)` 是平稳分布。我直接验证了**转移核的边缘**： +`P(X₂>u) = ∫_{u/k}^∞ e^{-x}e^{-(k-1)u/k}dx = e^{-u}`，**对任意 `k=τ₂/τ₁` 精确**（数值核对 `max|diff| ≈ 1e-16`）。 +**原因**：cut/hole 测度 `dz dτ` 在 `τ` 方向**平移不变**，所以 `τ0` 取多少都一样，无需 `τ0→∞` 的重整化。 +⇒ 这是**模型的精确性质**，而不是"施加了重整化才得到"的极限。正文在此处的措辞已经克制（"在任意固定 tau"），可以更明确地写出这一理由。 + +### D4 (界外话) 一个可加的**结构性**观察 + +w4 与 w8 的角色交换并不对称：w4 的极小化子是**二维**族 `{a0=a2, a1=a3}`（保留一个周期 2 模式）， +w8 的唯一极小点是**常数列场**（Fourier 特征值 `2+6cos(2πj/8)` 无零 ⇒ 无周期模式可取等）。 +⇒ 若要用"极小性"做判据，**w4 的判据必须是二维的**；只用"均匀 vs 非均匀"会在 `(u,u^{-1},u,u^{-1})` 上失效（它同样取等且 `A_B` 也不变）。 +**建议**：把 w4/w8 的这一差别写成一句对照，避免读者以为两处的极小化子是同一类型。 + diff --git a/docs/manuscripts/geometric-balance/audit-802-source-normalization-20260914.md b/docs/manuscripts/geometric-balance/audit-802-source-normalization-20260914.md new file mode 100644 index 000000000..aa72bc402 --- /dev/null +++ b/docs/manuscripts/geometric-balance/audit-802-source-normalization-20260914.md @@ -0,0 +1,239 @@ +# 独立核验：`#802` 交付「源归一化 / 源冗余」的纠正（对分析方 §7 的逐条判定） + +2026-09-14。代理 `verify802src`。机器 `DevEnvC_NePnUn`（**账号2**）。 +**本机 Mac 未做任何计算**；全部脚本在 `/workspace/v802src/scripts/`（`v1_exact.py`、`v2_c4.py`、`v3_evidence.py`）执行， +原始输出在 `/workspace/v802src/out/`，已下载回本机 `verify802src-out/`。 +**未做任何仓库写操作**（GitHub 只读）。 + +被核验的交付（本团队）：`sector802-out/note-root-response.md`（远端 `docs/manuscripts/geometric-balance/root-tangential-normal-response-20260914.md`）、 +`sector802-out/root-response.json`（远端 `results/research-dispatch/sector-root-response-20260914.json`）。 +纠正方：`two-observable-response-and-angular-alias-20260914.md` **§7** +（本机副本 `/tmp/concurrent-out/two-observable-response-and-angular-alias-20260914.md`）。 + +**证据分级**：【精确】=有理数/逐配置全枚举；【实测】=本团队 JSON 的满精度值（误差资格 = 该 JSON 自身的数值误差）；【结构】=读代码 + 线性代数事实。 + +--- + +## 0. 四句话结论 + +| 判据 | 判定 | 置信度 | 是否改变算力决策 | +|---|---|---|---| +| **C1** 黑对源与白对源是**同一概率族**（非两个独立源） | **成立**（我独立重证，且在本团队自己的数据上见 (12) 精确成立） | 极高 | **是** | +| **C2** 物理 safe 衰减率漏了 `f`，`N̂` 需 `+f_g` 修复 | **成立**（代码确用未正规化行权；精确恒等式 + 满精度复算逐格对上） | 高 | **是（最硬的一条）** | +| **C3** §5.2 的「`N=0 ⟺ T=−1`」及「`N≈0` 而 `T≠0` 不可能」 | **原更正成立**（本团队原陈述**错**） | 高 | **是（推翻了原先宣称的"不可能"）** | +| **C4** 奇周长环面一阶响应的**归因** | **原更正成立**（观察对、**归因错**） | 高 | 否（措辞/方法，无数字改变） | + +**没有一条是「不成立」。但 C1/C3 里本团队有一半是对的、一半是错的，下面逐条给出真值。** + +--- + +## 1. C1（§7.1）源冗余 —— **成立**【精确】 + +### 1.1 式 (10) `H_W = N − 2K + H_B` 逐配置恒成立（我独立全枚举） +`verify802src-out/v1_exact.json → C1a_identity`：真实环面 **3×3（512 配置）/ 4×4（65536）/ 3×4（4096）全部 0 违例**； +单行**全部** mask（w=3,4,5,6，含循环相邻）**0 违例**。等式在每一行内先成立，求和即得。 +一行证明（我自己的）：设该行黑占据数 `K`、黑游程数 `R`（`0 关于「rank1 的每行压力 = `f`，不能同时假设其未正规化概率趋 1」：这与上面的账一致—— +> 未正规化秩 1 权重 `≤ Σ_all w = Z_row^m`，故其每行压力**上限恰为 `f`**； +> 若再假定它（未正规化）趋于 1，就与「正规化 `P_1→1`」互斥。此点是同一本账的另一种说法，不是独立假设。 + +### 2.4 `N̂` 修复表 —— 我用满精度独立复算（`v1_exact.json → C2_recompute`） +`f = log Z_row` 的 `g` 导数用精确值 `w p_c²` / `w(1−p_c)²`，`p_c = 0.5927460507921`： + +| w | 原黑 `N̂` | 原白 `N̂` | `f_g`(黑) | `f_g`(白) | **修正黑 `N̂`** | **修正白 `N̂`** | 两条修正差 | `w·N̂`(修正) | +|---:|---:|---:|---:|---:|---:|---:|---:|---:| +| 4 | −1.3500497537668 | −0.6080813474300 | 1.40539152294 | 0.66342311663 | **0.05534176915** | **0.05534176915** | 0.0e+00 | 0.22137 | +| 5 | −1.7138942463384 | −0.7864337384174 | 1.75673940357 | 0.82927889568 | **0.04284515731** | **0.04284515731** | −1.1e−16 | 0.21423 | +| 6 | −2.0732497837478 | −0.9602971742426 | 2.10808728440 | 0.99513467494 | **0.03483750063** | **0.03483750063** | −7.8e−16 | 0.20903 | +| 7 | −2.4300006319441 | −1.1315559208547 | 2.45943516510 | 1.16099045400 | **0.02943453316** | **0.02943453316** | +8.9e−16 | 0.20604 | +| 8 | −2.7852763705546 | −1.3013395578810 | 2.81078304581 | 1.32684623319 | **0.02550667528** | **0.02550667528** | +2.2e−16 | 0.20405 | + +- 与**分析方给的修正表逐格比对**：最大绝对偏差 **3.3e−12**（差全部来自 `root-response.json` 打印位数 + `p_c` 位数）。 +- 分析方「两条修正序列差 ≤1.5e−15」——**我独立复算：最大差 8.9e−16，成立**（不是照抄）。 +- `w·N̂` 修正后 `0.22136708 / 0.21422579 / 0.20902500 / 0.20604173 / 0.20405340`，与分析方 `0.22137 / 0.21423 / 0.20903 / 0.20604 / 0.20405` 一致。 +- **另有一条我自己的独立一致性证据**：原始数据本身满足 `N̂_raw^W − N̂_raw^B = w(2p_c−1)` + （w=8：`1.4839368126735992` vs `1.4839368126736`，逐位到 1e−13）。 + 这正是 `N̂_phys^W = N̂_phys^B` **必须**蕴含的（由 C1 的 `N_W=N_B` + (14)）。⇒ 修复形式（差值 `= w p² − w(1−p)²`）**不是巧合**。 + +### 2.5 对本笔记结论的连带后果（**这才是改变决策的地方**） +`φ_g = (S'_w Δ_g)/(Δ'_w S_g)` 用修正后 `S_g^phys = S_g^raw + 2f_g` 重算（`v3_evidence.py`）： + +| w | `φ`(原, 黑) | `1/\|φ\|`(原) | `φ`(修正, 黑) | `1/\|φ\|`(修正) | +|---:|---:|---:|---:|---:| +| 4 | +1.95e−3 | 513 | −5.00e−2 | **20.0** | +| 5 | +9.72e−4 | 1029 | −4.05e−2 | 24.7 | +| 6 | +5.20e−4 | 1923 | −3.19e−2 | 31.3 | +| 7 | +3.02e−4 | 3307 | −2.56e−2 | 39.0 | +| 8 | **+1.88e−4** | **5330** | **−2.09e−2** | **47.8** | + +⇒ §5.1 那句「**法向读出比切向大约 1/φ ≈ 5×10³ 倍**」**作废**（修正后 ≈ **48 倍**，且**符号相反**）； +「`N̂` 两个源都显著非零、随 w 增长（`O(w)`）」**作废**（修正后 `N̂ = O(1/w)`，两源逐位相同）。 + +**C2 判定：成立，置信度高。** 唯一我没有做的是重跑 `w=4..8` 转移引擎；但修复内容是**每行正规化账**， +前提（未正规化行权）我已从代码逐字核实，且 `f` 恒等式与满精度复算都通过，**不需要新特征值求解**（与分析方一致）。 + +--- + +## 3. C3（§7.3）`N=0 ⇔ T=−1` 的**真值** —— **原更正成立；本团队 §5.2 陈述错** + +### 3.1 先固定本接口里的定义（逐字取自 `root-response.json → convention` + `s2_s3_response.py`） +``` +b = (1/2)log(P0/P2) ; d = log[P1/√(P0P2)] (rank 概率的坐标) +T_g = −b_g/b_p → −Δ_g/Δ_p +N_g = d_g − (d_p/b_p) b_g +``` +（`b_*`、`d_*` = 对源耦合 `γ` / 对 `p` 的导数。） + +### 3.2 正确的逻辑链（**恰好为真的那一条**） +``` +N = 0 ⟺ d_g − (d_p/b_p)b_g = 0 ⟺ (b_g,d_g) = (b_g/b_p)·(b_p,d_p) ⟺ (b_g,d_g) ∥ (b_p,d_p) + ⟺ ∂_γ 与 ∂_p 平行 ⟺ T = −a , a := ∂_γ 的等效 p 位移 (∂_γ = a·∂_p) +T = −1 ⟺ a = 1 ⟺ 源恰为**单位直接 p 平移**（一种归一化约定） +``` +⇒ 前两个等价（`N=0 ⟺ 平行`）**本团队写对了**；**最后一步 `⟺ T=−1` 写错了**（应为 `T=−a`）。 +⇒ 由此得到的「**`N≈0` 而 `T≠0` 在线性阶上不可能**」**不成立**。 + +### 3.3 我自己的反例（**真实环面，非构造性空谈**） +`v1_exact.json → C3_couterexample`（3×3 环面，源 `H = K`（占据数），`p₀ = 0.4`，全枚举 + 秩 + 中心差分）： +- 测得 **`T = −0.2399999999`（= −p(1−p) = −0.24），`N = 1.6e−10`（= 0）**，而 `T ≠ −1`。 +- **精确机理**：`p^K(1−p)^{N−K}e^{γK} = (p e^{γ})^K (1−p)^{N−K}` ⇒ 恰为**同一 Bernoulli 族在 `p_eff`**， + `logit(p_eff)=logit(p)+γ` ⇒ `∂_γ = p(1−p)∂_p`（`a = p(1−p)`）⇒ `N ≡ 0`、`T = −p(1−p)`。 + 我在脚本里用 `Fraction` 逐配置验证了「不同比值个数 = 1」（即 `exp(γK)` 精确等于 `p_eff` 的 Bernoulli 测度）。 +- 分析方举的 `p→p+a g`（`T=−a`）与 `exp(gK)`（`T=−p(1−p)`）**两个反例都正确**。 +- 另注：本团队自己 §5.1 的 `thermal` 行之所以 `T=−1`，是**定义**（脚本里硬写 `A_g := dI0_dx`）， + 即「单位 p 平移」这一**约定**，不是一般定理。本团队把一个约定当成了等价式。 + +**C3 判定：分析方的更正成立（本团队 §5.2 的最后一步与 V2 结论均错），置信度高。** +（连带：§6 V2 里「题面设想的切向盲点实例在一阶退化」一句也须撤回。） + +--- + +## 4. C4（§3.2 附带纠正）奇周长环面的归因 —— **原更正成立：观察对、归因错**【精确】 + +### 4.1 我独立全枚举的结果（`v2_c4.json`；`H_s = Σ_y s_y K_y`，`s_y=(+1)^y`） +| w×m | 周期 | `⟨H_s⟩` 实测 | `w p Σ_y s_y` | **`w_0` 平移不变性违例** | **接缝缺陷 `H_s(T cfg)+H_s(cfg) ≠ 0` 的配置数** | `b_g` | +|---|---|---|---:|---:|---:|---:| +| 3×3 | 奇 | 1.2000000000 | 1.2 | **0 / 512** | 448 / 512 | −5.448e−1 | +| 3×3 (p_c) | 奇 | 1.7782381524 | 1.7782381524 | **0 / 512** | 448 / 512 | −4.900e−1 | +| 4×4 | 偶 | 0.0000000000 | 0 | **0 / 65536** | **0 / 65536** | +6.3e−15 | +| 3×4 | 偶 | 0.0000000000 | 0 | **0 / 4096** | **0 / 4096** | −2.9e−16 | +| 4×3 | 奇 | 1.6000000000 | 1.6 | **0 / 4096** | 3840 / 4096 | −6.401e−1 | +| 3×5 | 奇 | 1.2000000000 | 1.2 | **0 / 32768** | 28672 / 32768 | −4.463e−1 | + +（我复现了本团队的观察：`m` 偶 → `b_g` 机器零；`m` 奇 → `b_g ~ −0.39…−0.64`。**观察对**。） + +### 4.2 正确的原因（**我自己的推导 + 上述精确枚举**） +- `w_0 = p^K(1−p)^{N−K}` 只依赖 `K` ⇒ 对**任意** `m`（含奇数）**逐配置平移不变**：上表 `w_0` 违例**恒为 0**。 + ⇒ **奇长度没有破坏均匀背景的平移不变性。本团队那句「周期方向上的平移不变性被奇长度破坏」是错的。** +- 真正的原因：`(−1)^y` 在**奇**长度循环上**不是周期相容的**：沿 `y` 平移 1 有 + `H_s(T cfg) = −H_s(cfg) + 2K_{m−1}`，多出一个**接缝项**；`m` 偶时 `(−1)^y` 周期相容，`H_s` **严格奇**（上表缺陷恒 0）。 + ⇒ 于是 + - `⟨H_s⟩_{w_0} = w p Σ_{y}(−1)^y = w p·(1 if m odd else 0)`（上表逐格精确吻合）； + - 对**平移不变**的秩指示 `1_{r}`：`m` 偶时 `H_s` 奇 ⇒ `⟨H_s 1_r⟩ = 0` ⇒ `b_g ≡ 0`； + `m` 奇时该对称性论证失效（`H_s` 既非奇也非不变）⇒ `b_g ≠ 0`。 +- ⇒ **正确表述**：奇周期上**交替源自身**带有**非零均值/接缝**（`Σ_y(−1)^y ≠ 0`）， + 这才是非零一阶信号的来源；**被破坏的不是均匀背景的平移不变性**。分析方 §7.3 末段**成立**。 +- 实践含义不变的部分：本团队「均值为零的行/列场在平移不变背景下一阶必为零、用它挑自旋必须推到二阶」—— + **在偶周期上仍然正确**（`H_s` 严格奇），但**不能**给出「奇周期破坏平移不变性」这个理由； + 在奇周期上该场**本来就不是均值为零、也不是奇场**，所以它**不能**被当作「均值为零的场」来用。 + +**C4 判定：原更正成立，置信度高（逐配置全枚举，0 违例）。** 影响为**措辞/方法**，无数字改变。 + +--- + +## 5. 净效应：哪几条改变算力决策 + +1. **C1（改决策）**：黑/白对源**不是两个独立非热源**，是同一概率族的参数重写（`H_W=N−2K+H_B`）。 + ⇒ 「用第二个源家族做对比/提供第二份独立证据」的预算**不必要**；§3.1 里「另设一个对照源 `H'` 作为第二个非热源」 + 与 §5.1「两个非热源给出明显不同的像 ⇒ 可区分」**须撤回**。 +2. **C2（改决策，最硬）**：物理安全衰减率含 `f`，`N̂_phys = N̂_raw + f_g`。 + ⇒ §5.1 的 `N̂` 两列数值、`φ≈2e-4`、`1/φ≈5×10³`、`O(w)` 法向值、V2 的「法向读出比切向大约三个半数量级」 + **全部作废**；正确值 `N̂ = 0.05534/0.04285/0.03484/0.02943/0.02551`（两源**逐位相同**）、`1/|φ| ≈ 20…48`。 + ⇒ 「再为黑/白各跑一排宽度以分离两个源」的后续算力**不必要**；真正剩下的自由度是分析方 §8 的「度量方向」（`Ibar/G` 比值）。 + **`Δ`、`Δ'`、`S`、`S'`、控制表、根位移表、尺度指数诊断（`−17/4`、`−1/4`、`−4`）不受影响**（`f` 与 `f_p` 在 `g=0` 处为 0）。 +3. **C3（改决策）**：删掉了一条**被当成定理的负面结论**。原 §5.2/§6 V2 宣称「`N≈0` 而 `T≠0` 一阶不可能」， + 使题面 V2 的搜寻目标**自证为空**。更正后该目标**重新有意义**：`N=0` 只要求源沿热向（`T=−a`，`a` 任意）。 + ⇒ 计划上不应再以「不可能」为由停止寻找 `N≈0, T≠0`。 +4. **C4（不改决策，改措辞）**：无数字改变；但「奇长度破坏平移不变性」这个**理由**不能再用， + 也不能拿「一阶为零」去**无条件**论证均值为零的场无信号（只在偶周期/严格奇场下成立）。 +5. **不改动的部分（须保留）**：S1 控制表逐位复现、`Δ_w/Δ'_w` 表与解析导数、`p*_w−p_c` 表、 + 尺度指数「相容未认证」的谨慎表述、以及全部「不能宣称什么」的自查（尤其 §7 第 3、4、6 条）。 + +**综合**：分析方 §7 的四条纠正**我一条都没能推翻**，其中 **C1、C2 会改变本案的算力分配**（C2 还会改数字）， +**C3 撤回一条被误用的"不可能性"**，**C4 只改措辞**。勘误文本见 `verify802src-out/erratum-draft.md`。 + +--- + +## 6. 复现与原始数字 +- 脚本：`/workspace/v802src/scripts/v1_exact.py`、`v2_c4.py`、`v3_evidence.py`（本机副本 `verify802src-scripts/`）。 +- 输出：`/workspace/v802src/out/v1_exact.json`、`v2_c4.json`、`evidence.json`（本机副本 `verify802src-out/`）。 +- 环境：`DevEnvC_NePnUn`（账号2，16 vCPU / 32 GiB，Python 3.9.9），仅用标准库（`Fraction`/`math`/`json`），**未装任何包**。 +- **本机 Mac 只做读写/上传下载/看日志**；**未做任何仓库写操作**。 diff --git a/docs/manuscripts/geometric-balance/barrier-cavity-resonance-20260914.md b/docs/manuscripts/geometric-balance/barrier-cavity-resonance-20260914.md new file mode 100644 index 000000000..eb45d1f8d --- /dev/null +++ b/docs/manuscripts/geometric-balance/barrier-cavity-resonance-20260914.md @@ -0,0 +1,213 @@ +# 当最小屏障与最小孔洞同阶：宽度八的两个 Poisson 过程和离散内部模式 + +2026-09-14。续接 #764 / #777，但改变极限：本记录先固定圆柱宽度 w=8，再令黑色 NN 概率 p↓0。白色仍为互补 matching site，q=1-p。它不是固定 p、w→∞ 的结论，也不借用那个极限下的二维 CLT。 + +## 1. 新结果 + +对一个按完整白色 essential 簇一次取样的 C，定义 + +- L=max_y C-min_y C+1； +- K=|C|；B=不同的外部黑站点数，包含跨度外相邻行； +- H(C)=跨度内不属于 C、也不属于其外边界的站点数，即真正没有被这个簇查询到的孔洞站点数。 + +固定 w=8，令 p↓0，则 + +\[ +\boxed{(p^8L,H(C))\Rightarrow(E,H),\quad +E\sim\operatorname{Exp}(1),\quad H\mid E\sim\operatorname{Poi}(8E).} +\tag{1} +\] + +联合变换为 + +\[ +\boxed{E[e^{-uE}z^H]=\frac1{1+u+8(1-z)},\quad u\ge0,\ 0\le z\le1.} +\tag{2} +\] + +所以 + +\[ +\boxed{P(H=k)=\frac19\left(\frac89\right)^k,\quad k=0,1,\ldots.} +\tag{3} +\] + +这是离散几何分布，不是高斯。极限分布的均值为 8、方差为 72，Corr(E,H)^2=8/9。这里先只宣称实际孔洞数的弱收敛和有界变换收敛；极限分布的矩不自动当作有限 p 矩的收敛证明。 + +还有一个能直接由旧 (L,K,B) 观察的量： + +\[ +\boxed{K+B-8L\Rightarrow16-H.} +\tag{4} +\] + +十六是上下两条黑色满行的外边界贡献，不应忘掉。孔洞数应首先按定义计算；不能对任何有限 p 都不加条件地用 16-(K+B-8L) 替代它。 + +若 S=K/q-B/p，并令 + +\[ +a_p=\sqrt{p q p^8/8}, +\] + +则进一步有联合弱极限 + +\[ +\boxed{(p^8L,H,a_pS)\Rightarrow(E,H,\sqrt E Z),} +\tag{5} +\] + +其中给定 E 后，H 与标准正态 Z 独立。三变量变换为 + +\[ +\boxed{E[e^{-uE}z^H e^{it\sqrt E Z}] +=\frac1{1+u+8(1-z)+t^2/2}.} +\tag{6} +\] + +因此同一个簇内，高斯的探索评分与离散的查询孔洞可以同时存在；宏观长度把它们耦合起来。评分的边缘仍为 Laplace，但剩余孔洞方向不服从高斯。没有与固定 p、w→∞ 的高斯结论矛盾。 + +## 2. 只有两种八黑点图案与本问题相关 + +用圆柱的真实 NN 位移保留绕行信息。 + +**拓扑屏障**：一条水平全黑行 R_y，共八个黑点。每行有一个候选，概率 p^8。 + +**局部孔洞**：一个站点的八个 matching 邻点全部黑色，中心自身不受要求。每行有八个候选，概率各为 p^8。 + +有限几何分类如下。 + +1. 黑色 NN essential 分量至少八点；恰好八点时只能是完整水平行。因为一个简单非零绕行圈至少走八步；等号时所有步都必须水平同向。 +2. 非essential 黑色 NN 分量若造成一个未查询位置，就必须含一条包围格点的可缩 NN 圈。长度至少八；等号只能是 3×3 的八点外围。 +3. 八点外围的中心若又是黑色，整个黑分量至少九点。这种额外情况在本极限的固定宏观窗口内消失。 + +第二条使用 4/8 的包围关系；最短圈分类也可由水平、竖直跨度至少为二及 NN 周长界证明。独立枚举了黑点数≤8 的全部 3792 个圆柱平移类：只有一个 essential 八点类和一个封闭孔洞八点类。其他八点动物虽同阶出现，但不制造本记录计数的屏障或孔洞。 + +## 3. 为什么可以暂时排除九点以上的黑簇 + +在最大度四图上，一个至少九点的根簇含一个九点连通动物。用其深度优先树遍历编码，可粗略界定这种动物数量为一个仅依赖九的有限常数。因此 + +\[ +P(|C_B(v)|\ge9)\le C p^9. +\] + +在长度不超过 T p^-8 的窗口内，联合界给 + +\[ +P(\text{有一个大小≥9的黑簇与窗口相交})\le C_T p. +\tag{7} +\] + +计数允许该簇伸出窗口；不是错误地只枚举完全包含于窗口的簇。 + +若条件于第零行为一个黑色满行，触及这条行的更大黑簇只需通过上下相邻的十六个站点检查，概率 O(p)。不接触它的九点动物仍使用独立未固定标签。因此 (7) 在这个满行 Palm 下也成立，另加 O(p) 端部项。 + +完整黑 essential 簇强度 ν_8(p)=p^8[1+O(p)]：孤立黑满行给下界 p^8q^16；所有非满行或带额外点的 essential 簇都进入九点动物误差。按簇取样与按满行取样的差异因而趋零。通过黑白 essential 配对，将结果传到完整白簇 Palm；不是把随机行所见的白簇当作 component Palm。 + +## 4. 两个局部图案过程联合趋于独立 Poisson + +把纵向位置乘 p^8。屏障图案每行一个；孔洞图案每行八个。各图案只依赖有限行、有限站点。 + +任何两个不同、相交的八点图案，并集至少十二点。孔洞–孔洞最小并集为十二；孔洞–满行的并集至少十三；不同满行的标签互不相交。故在 O(p^-8) 行里，邻近同时出现的总误差为 O(p^4)。不相交图案严格独立。 + +由有限依赖的阶乘矩展开（或标准局部依赖 Poisson 定理），得到独立的率一屏障过程与率八孔洞过程。这个独立性只属于这两种稀有局部黑图案；不是声称黑白 essential 锚点独立。 + +条件在零点有一条屏障时，只改变 O(1) 行的图案条件，孔洞靠近这个端点的概率仍趋零。下一道屏障的缩放距离 E 因此服从 Exp(1)，其间的孔洞数给定 E 后服从 Poi(8E)。 + +可先将下一屏障限制在 T p^-8 行，再用 (7) 完成真实簇的对应，最后令 T→∞。满行间隔本身是几何等待时间，尾部控制是直接的。这给出了 (1) 的完整路径，而不只在一个固定窗口内画出两种事件。 + +## 5. 真正的完整簇与查询数 + +在上述好事件上，两道相邻黑满行不带附着点。它们之间的白色外部连通分量是唯一的 essential 白簇，且每一中间行都有它的顶点；有限白分量只能是八点圈中的一个白色中心。 + +若两黑行相距 D，则 + +\[ +L=D-1,\qquad B_{\rm out}=16. +\] + +跨度内除了这些孔洞中心外，所有站点要么属于 C，要么是邻接 C 的黑点。因此 + +\[ +K+B=8L+16-H. +\tag{8} +\] + +这是好事件上的逐配置恒等式；它以概率趋一给 (4)。它也说明为什么只观察 K 或 B 的巨大共同主项，很容易漏掉这个 O(1) 的离散模式。 + +## 6. 联合高斯评分不要求先调用大宽度 bulk 定理 + +在长度 O(p^-8) 的未条件化窗口中，定义逐标签评分 (n_v-q)/(pq)。其每行均值为零、方差为 8/(pq)。乘 a_p 后，部分和趋于标准 Brownian 过程。 + +将行分成长 ℓ_p=⌊p^-2⌋ 的主体块，留三行缓冲。主体块相互独立；归一化块的最大值不超过 C p^(3/2)，故 Lindeberg 条件成立。一个块出现任意稀有图案的概率为 O(ℓ_p p^8)。对固定特征函数参数，块内“评分与稀有图案”混合项的绝对值不超过 + +\[ +C\,a_p\,\ell_p/p\times P(\text{块内稀有图案}). +\] + +在 O(p^-8/ℓ_p) 个块上求和为 O(p^(3/2))，趋零。缓冲占比为 O(p²)，评分方差和图案期望贡献都消失。这给出相互独立的 Brownian 评分过程、屏障 Poisson 过程、孔洞 Poisson 过程。 + +在满行 Palm 下，端点有限标签对评分的贡献至多 O(1/p)，乘 a_p 后消失。停在下一屏障处使用连续映射。好事件上实际完整簇评分与全中间行评分的差是 + +\[ +S_C-S_{\rm middle}=-H/q-16/p. +\] + +H 紧，故这个差归一化后趋零。由此得到 (5)、(6)。这证明弱极限，不预先承诺所有高阶混合矩收敛。 + +## 7. 再闭成环面：孔洞怎样分给宏观白簇 + +若 m p^8→t>0，两个独立过程在圆周上给出 + +\[ +J\sim\operatorname{Poi}(t),\qquad Z\sim\operatorname{Poi}(8t),\qquad J\perp Z. +\] + +J 是黑色横屏障数，Z 是总孔洞数。J=0 时存在一个白色 cross；J=j≥1 时有 j 个宏观白簇，间距比例服从 Dirichlet(1,…,1)。 + +条件于 J=j≥1、Z=h，孔洞对 j 个白簇的分配是 Dirichlet–multinomial(1,…,1)。更简单地说： + +\[ +\boxed{P(H_1=h_1,\ldots,H_j=h_j\mid J=j,Z=h) +=\binom{h+j-1}{j-1}^{-1},\quad \sum h_i=h.} +\tag{9} +\] + +所有有序非负整数分配等概率。它不是声称排列后的分区也等概率。J=0 或 J=1 都只有一个宏观白簇，但拓扑类型不同。 + +## 8. 一个局部孔洞场可以改变宏观长度 + +在已求出的极限模型中，用 z^H 重新加权，z≥0。若 z<9/8，则 + +\[ +E z^H=(9-8z)^{-1}. +\] + +正规化后 E 的分布变成 Exp(9-8z)，且给定长度时孔洞率从 8 变成 8z。因此偏好局部孔洞会选择更长的宏观白簇。 + +z=9/8 是该极限模型的生成函数极点。不能仅由弱收敛就宣布实际有限 p 生成函数的极点也趋向 9/8；z>1 的统一指数矩仍是一个明确的后续问题。这不是新的体相渗流临界点。 + +## 9. 更一般的猜想：最小成本共振 + +设某个窄周期模型的最小拓扑屏障需要 b 个稀有站点，每纵向单位有强度 a_b p^b；最小局部查询缺陷需要 h 个稀有站点，强度 a_h p^h。一个屏障间隔的预期局部缺陷数由 + +\[ +\Lambda_p=(a_h/a_b)p^{h-b} +\] + +控制。在稀有图案不成不可忽略的团簇、且它们确实对应完整几何对象的条件下，预期有： + +- h>b：多数间隔没有缺陷； +- h=b：Poisson 缺陷放在指数间隔中，产生离散几何模式； +- h8 时，\(w p^{8-w}\)→∞；要把真实孔洞数的中心极限定理做出来，必须保留更大孔洞对均值的修正，不能只拿裸八点图案的均值居中。 + +这是比“第二模式小所以可以忽略”更细的图景：它可以真实存在，却在某个极限中消失；也可以在共振处保持离散；再在另一个极限中高斯化。 + +## 10. 执行范围与来源 + +本轮脚本精确检查围栏、最小黑动物、图案重叠和截断布尔系数；另给一个明确标成独立块 toy 的有理等待时间控制。没有用它冒充真实稀有孔洞的长圆柱模拟。 + +外部背景：Mertens–Ziff arXiv:1603.07289v2 §II 的匹配包围／绕行分类；Chen–Xia arXiv:math/0410169 的 Palm／局部依赖 Poisson 方法（本轮读取摘要，仅作方法定位）。本记录所需的有限范围图案率、并集阶数、块独立性及真实簇对应均已在上面写明，不从摘要借用一个未核对的 site 定理。 + +内部输入：黑白 essential 配对的精确定义见 `black-white-gap-law.md`；没有借用其固定 p、w→∞ 的概率结论。本轮所有高阶矩与生成函数极点的未完成范围已在各节单独标出。 diff --git a/docs/manuscripts/geometric-balance/bernoulli-chaos-source-grading-20260914.md b/docs/manuscripts/geometric-balance/bernoulli-chaos-source-grading-20260914.md new file mode 100644 index 000000000..177b9a941 --- /dev/null +++ b/docs/manuscripts/geometric-balance/bernoulli-chaos-source-grading-20260914.md @@ -0,0 +1,185 @@ +# Bernoulli-chaos source grading: an exact microscopic pair-exchange tangent basis + +Date: 2026-09-14 + +Status: exact finite product-measure/source algebra. It is proposed as a microscopic basis for #61/#802; it does **not** by itself establish RG/OPE parity of continuum fields. + +## 1. Exact paired standardized variables + +For the primal Bernoulli site law with parameter `p`, define + +```text +xi_v = (n_v-p)/sqrt(pq), q=1-p. +``` + +Under the matching/complement pairing + +```text +n_hat_v=1-n_v, +q=1-p, +``` + +the standardized variable of the paired model is + +```text +xi_hat_v + = (n_hat_v-q)/sqrt(qp) + = -xi_v. +``` + +Therefore for every finite site set `A`, + +```text +Psi_A = product_(v in A) xi_v +``` + +obeys the configurationwise exact relation + +```text +boxed: Psi_A -> (-1)^|A| Psi_A. +``` + +For a finite Bernoulli product space the `Psi_A` form the usual Hoeffding/Walsh orthonormal basis. Thus any square-integrable microscopic source has a unique decomposition by chaos degree. + +This is a grading of **source functions under the exact complement pairing**. It is not a theorem that the RG scaling fields reached by these sources carry the same scalar parity. + +## 2. Degree 0 and degree 1 are nuisance directions for the present question + +- degree 0: constant normalization; +- uniform degree 1: thermal/logit score; +- nonuniform degree 1: spatial one-site probability field, whose first-order response may vanish by translation/symmetry but is still a one-site measure deformation. + +For a source-identification problem where the thermal coordinate is profiled out, the relevant source space should first be quotiented by + +```text +span{1, K} +``` + +(or the exactly corresponding normalized thermal score). This is the same quotient already present in #773 source calculus. + +## 3. #802 black/white pair-source redundancy is a degree-2 identity + +For one adjacent pair `(i,j)`, write `x_i=n_i-p=sqrt(pq) xi_i`. Then + +```text +n_i n_j + = p^2 + + p sqrt(pq)(xi_i+xi_j) + + pq xi_i xi_j, +``` + +while + +```text +(1-n_i)(1-n_j) + = q^2 + - q sqrt(pq)(xi_i+xi_j) + + pq xi_i xi_j. +``` + +Their genuine degree-2 component is **identical**. Summing over a regular row/cycle gives exactly the observed relation + +```text +H_W = const - 2K + H_B. +``` + +Hence, after quotienting constant + thermal directions, black-pair and white-pair sources are the same vector. Their apparent first-order distinguishability before normalization was necessarily spurious. + +This turns the #802 erratum into a general source-design rule rather than a one-off bug. + +## 4. The first exact exchange-odd nonthermal source occurs at odd degree >=3 + +After removing degree-1 thermal motion, an odd chaos source of degree three is the smallest local source carrying an exact minus sign under complement without being merely thermal. + +A generic translationally symmetrized example is + +```text +H_3 = sum_x sum_a c_a + xi_x xi_(x+r_a) xi_(x+s_a), +``` + +with coefficients chosen to form a declared spatial/D4 representation. + +Under the exact primal/matching complement pairing: + +```text +H_3 -> -H_3. +``` + +This gives a genuine lattice-side `D` source direction. It should be compared with an even degree-2 control and the degree-1 thermal control. + +Again, the conclusion is only about microscopic source exchange. An RG-compatible continuum tangent map must still be demonstrated before assigning a local scaling field a parity label. + +## 5. A concrete finite-width tangent experiment + +Choose a small, preregistered source basis, for example: + +```text +H1 : uniform degree-1 thermal score; +H2 : one D4-symmetrized degree-2 pair/motif source; +H3 : one D4-symmetrized degree-3 motif source. +``` + +For each safe phase and width compute physical normalized Perron score responses + +```text +mu_G,a = partial_(g_a) log lambda_G^physical, +``` + +or equivalently the normalization-safe free-energy derivatives. + +Then record + +```text +D_a(w) = mu_4,a - paired(mu_8,a), +S_a(w) = mu_4,a + paired(mu_8,a), +``` + +with the exact microscopic exchange grade carried separately. + +The useful question is not “does H3 prove an odd CFT field?” but + +> does the large-width response matrix become block-compatible with the exact microscopic complement grading, and which continuum scaling powers live in each block? + +This is an operational version of the RG-tangent programme in #61. + +## 6. Why degree alone will not identify spin four + +Square-lattice spin zero and spin four both lie in the trivial `C4` representation. A D4-invariant degree-3 source therefore does not by itself separate a scalar continuum field from spin four. + +Angular geometry / same-circle projectors remain the correct analyzer of the H0/H4/... decomposition. The source-chaos grading solves a different ambiguity: normalized pair-exchange direction versus thermal/gauge redundancy. + +The two analyses should be crossed only after each is separately typed: + +```text +angular irrep : H0/H4/H8/... +source exchange : microscopic chaos grade / S-D response +radial exponent : measured only after the first two are separated. +``` + +## 7. Relation to original-U + +Any candidate physical source for #275 should first be decomposed in the same way. If two proposed source implementations differ only by degree 0/thermal directions, no amount of extra sampling can distinguish them after nuisance profiling. + +Conversely, a candidate forward map that predicts different higher-chaos components supplies a real new observable direction. + +This is a finite probability statement and does not replace the original-U continuum normalization contract. + +## 8. Claim boundary + +Exact: + +```text +xi_hat=-xi, +Psi_A -> (-1)^|A| Psi_A, +black/white pair sources share the same degree-2 component, +source quotient modulo normalization/thermal directions. +``` + +Programme/hypothesis: + +- large-width RG tangent map preserves or asymptotically respects the microscopic grading; +- particular continuum fields can be assigned to the observed response blocks; +- a chosen degree-3 source has useful overlap with the sector correction of interest. + +The main correction is conceptual: construct parity first on the **source space where it is exact**, then test whether RG transports it. Do not infer the reverse direction from a finite matching identity. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/black-white-gap-law.md b/docs/manuscripts/geometric-balance/black-white-gap-law.md new file mode 100644 index 000000000..67198b86d --- /dev/null +++ b/docs/manuscripts/geometric-balance/black-white-gap-law.md @@ -0,0 +1,498 @@ +# Rare black barriers and giant white components on a cylinder + +2026-09-14. A continuation of the mathematical question in #764/#739. + +The point of this note is a connection, not a publication plan: a rare, short +component of one colour determines the longitudinal scale of a very large +component of the other colour. All colour probabilities below use the SAME +Bernoulli labels. No independent recolouring is introduced. + +## 1. Objects and results + +Use the cylinder `(Z/wZ) x Z`, with its physical horizontal edge gains. Black +sites have probability p and NN connectivity; white sites have probability +q=1-p and matching (NN+NNN) connectivity. Take w>=3. The width-two controls +retain the two lifted parallel horizontal edges explicitly. + +An essential component contains a cycle of nonzero horizontal winding. Its +complete span is `L=max y-min y+1`. A component is counted ONCE, with anchor +its lowest occupied row (the least column in that row is only a tie-breaking +vertex label). Write nu_B,nu_W for the numbers of essential components per +vertical row, and E_B,E_W for their component-Palm averages. Neither average +is a random-vertex average. At any fixed w and 0infinity: + + nu E_B L^k = O(nu w^k) -> 0, every fixed k; + nu L_white => Exp(1), with all fixed positive moments; + nu^k E_W L^k -> Gamma(k+1), k>0; + nu E_W L -> 1, Var_W(L)/(E_W L)^2 -> 1. (1.5) + +Here `Gamma(k+1)` is a moment, not a Gamma-distributed component length. +The white parameter is q=1-p, on the supercritical side of the plane +matching graph. A Brownian-span assertion for a FIXED SUBCRITICAL white +parameter would address a different experiment. + +Under a uniformly chosen longitudinal position, the containing interval, +and asymptotically the unique white essential component spanning that row, +has instead + + nu L_seen_from_row => Gamma(shape=2,rate=1). (1.6) + +This is length bias. It is not a second physical phase transition. + +The full black and white ANCHOR point processes, scaled vertically by nu, +converge jointly to the SAME rate-one Poisson point process. They are not two +independent Poisson processes. Consecutive white spans under component Palm +converge jointly to independent exponential spacings; this independence is +an asymptotic consequence, not a finite-width assumption. + +The proofs below use only elementary cylinder topology, the plane site +volume tail [AV], the exploration coupling, and local-dependence Poisson +approximation [AGG]. They do NOT use the earlier mass-inversion theorem, +Gumbel affine-window theorem, p-differentiability, OZ sewing, or a unit residue. + +## 2. The alternating chain is a topological fact + +Use the usual complementary regular-neighbourhood representation of the +4/8 digital matching pair. One construction thickens occupied NN vertices +and edges and fills elementary all-black faces. Complementary regions have +the connectivity and essential topology of the white matching graph. It is +enough to check the sixteen face patterns. Crossing white diagonals in an +all-white face are connected, not two crossing distinct components. This is +the same local boundary-connectivity fact underlying [MZ, Section II]. + +A compact connected essential neighbourhood on the open annulus has exactly +two essential boundary curves. Additional boundary curves enclose discs. +The region immediately on its upper boundary belongs to ONE opposite-colour +component, because the boundary is connected and the local matching rule +connects its surrounding opposite-colour sites. This opposite component is +itself essential. The same argument applies below. Two disjoint connected +essential components are ordered by which end of the annulus lies on their +side of an essential separating curve. A contractible component cannot be +inserted into this order: it does not separate the two ends. + +Starting from any essential component and following its upper neighbour +therefore alternates the colours. There is no omitted essential component +between neighbours, since they meet along the same separating boundary. +The full monochromatic rows ensure that the chain extends to both ends and +exhausts all essential components. Local finiteness excludes accumulation +inside a bounded set of rows. This proves (1.1). + +### Height order and contact bounds + +If D is above C, choose a simple essential curve inside C. A lattice vertex +of D at or below the lowest row of C could be joined downwards to the bottom +end without intersecting this curve: below that row there are no vertices +or edge segments of C, and the proposed vertical ray at the vertex's column +cannot pass through C's distinct lattice vertex. Such a point belongs to +the lower, not upper, region. Hence `min y(D)>min y(C)`. The analogous upward +ray gives `max y(D)>max y(C)`. The argument works also for the planarized +matching curve: its added face points stay between their endpoint heights. + +Neighbouring opposite-colour components have boundary-adjacent sites whose +vertical coordinates differ by at most one. Thus, for B_i,W_i,B_(i+1), + + a_i < min W_i < a_(i+1), + min W_i <= max B_i+1 = a_i+L_i, + a_(i+1)-1 <= max W_i < max B_(i+1). + +Writing `L(W_i)=max W_i-min W_i+1` gives (1.3). Branches, holes and multiple +essential cycles in a component do not change this argument. The height is +the complete component's height, not a chosen loop's height. + +There is another useful version. Let N_B(y),N_W(y) count essential components +whose vertical projections include row y. Their active indices form a +contiguous segment of the alternating chain, because both endpoints of the +spans are ordered. This segment is nonempty: neighbouring spans have no +integer-row gap by the contact bound. Consequently + + N_B(y)+N_W(y)>=1, |N_B(y)-N_W(y)|<=1. (2.1) + +In particular, if no black essential component spans y, exactly one white +essential component does. Campbell counting gives `E N_B(0)=nu E_B L`. + +## 3. Stationarity, gaps and the mean law + +Pair B_i with its upper neighbour W_i. This pairing is bijective and +commutes with vertical translations. Counting matched components in a long +interval, or applying the mass-transport identity, proves (1.2). No renewal +property is needed. + +For the black anchor process, assign integer rows +`a_i,...,a_(i+1)-1` to anchor a_i. They partition Z. Therefore + + nu E_B D_i=1. (3.1) + +Moving to the next black anchor preserves the component-Palm law, so +`E_B L_(i+1)=E_B L_i`. The bijection also transports the W_i mark to white +component Palm without size bias. Averaging (1.3) proves (1.4). + +Equivalently, from (2.1), + + 1-nu E_B L <= nu E_W L <= 1+nu E_B L. (3.2) + +Thus the entire mean relation is topological plus stationary. Poisson +approximation is only needed to determine the DISTRIBUTION around this mean. + +## 4. Uniform black volume control and rarity + +[AV, Theorems 2--3] gives, for fixed p0 such that the +plane occupied root cluster has `Pr(|C(0)|>=n)<=C exp(-cn)`. + +To apply it uniformly in the cylinder width, use the following coupling. +Explore a cylinder cluster using a fixed queue. At the first query of a +quotient vertex, choose a neighbouring lift of its already assigned parent +and read that as-yet unqueried plane label. Distinct quotient vertices have +distinct lifts. A repeated quotient vertex uses its old answer and does not +query a new copy. The answers thus have exactly the independent cylinder +law, and the discovered occupied TREE injects into the independent plane +root cluster. Cycles need not lift as cycles; size, which is all this step +uses, is preserved. Hence the same C,c give the cylinder volume bound. + +A winding component contains at least w distinct vertices. An isolated full +black row, with white rows immediately above and below, is one complete +essential component. Thus + + [p(1-p)^2]^w <= nu <= C w exp(-cw). (4.1) + +For component Palm, selecting its anchored lowest-row site gives + + nu Pr_B(K>=n) <= C w exp(-cn). (4.2) + +Split the moment integral at a sufficiently large constant times w and use +the lower bound in (4.1). For every fixed r>0 this proves + + E_B K^r=O(w^r), E_B L^r=O(w^r). (4.3) + +In particular epsilon_w=nu E_B L tends to zero exponentially up to a +polynomial. The mean conclusion `nu E_W L->1` already follows, without any +point-process limit. Equation (2.1) also proves that the probability a fixed +row is not spanned by exactly one white essential component tends to zero. +This is row COVERAGE, not probability that a uniformly chosen site belongs +to the white component. + +## 5. A self-contained rare-anchor Poisson argument + +This section shortens the needed probabilistic input. It does not assume +that the cylinder intensity's exponent has already been identified with the +plane inverse correlation length. + +Let H=w^2. Let I_y^H indicate a COMPLETE black essential component with +lowest row y and span at most H. There is at most one such component. +The event can be decided from the rows y-1,...,y+H, including the rows that +certify no connection outside. Its probability is nu_H and + + 0<=nu-nu_H<=Cw exp(-cH)=:eta_H. (5.1) + +Consider n<=T/nu anchor rows, T fixed. Indicators at distances greater than +H+1 have disjoint supports. Thus, in the notation of [AGG], b3=0 and + + b1<=n(2H+3)nu^2. (5.2) + +Here is the essential b2 step. Two distinct complete anchored components +have vertex-disjoint occupied winding witnesses. Each witness is contained +in a common slab of at most 3H+4 rows. If E is the increasing event of ANY +occupied essential cycle in this slab, then + + {I_y^H=I_z^H=1} subset E disjoint-occurrence E. + +We use site BK on E, NOT on the nonmonotone complete-anchor events. +An occurrence of E either belongs to a full component of span <=H, whose +anchor lies in an enlarged interval of <=4H+4 rows, or a vertex in the slab +belongs to a component of size >H. The intensity and volume bounds give + + Pr(E)<= (4H+4)nu + Cw(3H+4) exp(-cH)=:a_H. + +Hence + + b2<=n(2H+3)a_H^2. (5.3) + +Equations (4.1),(5.1)--(5.3) give b1+b2=o(1): the leading term is +`O_T(H^3 nu)`, and every remaining term contains exp(-c w^2) divided by at +most an exponential in w. The probability of omitting any true anchor in +the observation interval is at most n eta_H=o(1). + +[AGG, Theorem 2], with our total variation convention `sup_A|P(A)-Q(A)|`, +then gives the joint count approximation on any finite partition. Their +printed norm is twice this convention; the bound we use is 2(b1+b2) when +b3=0. On rescaling row coordinates by nu, the independent Poisson intensity +measure converges to Lebesgue measure. The black anchors therefore converge +to a stationary rate-one Poisson point process on R. + +### Uniform gap tails, not just bounded-window convergence + +Choose a block of length `ceil(1/nu)+2H+2`. Restrict local anchors to the +interior so their H+2 supporting rows are wholly inside the block. By the +just proved Poisson limit, the probability of at least one such anchor is +bounded below by a fixed a>0 for all sufficiently large w. These events on +disjoint blocks are independent. Therefore the stationary void probability +V_w(n) satisfies, with constants independent of sufficiently large w, + + V_w(n)<=C0 exp(-c0 nu n). (5.4) + +The event used here is a complete local anchor, not just a path belonging +to a possible component rooted outside the block. + +## 6. From voids to component-Palm spacings + +For a stationary integer anchor process with gaps D, direct counting yields + + V_w(n):=Pr(no anchor in rows 1,...,n) + =nu E_B[(D-n)_+]. (6.1) + +Each gap of length D contains exactly `(D-n)_+` starting positions with no +anchor among the next n rows. This identity is valid without independent +gaps. At n=0 it includes (3.1). + +Linearly interpolate (6.1) and put X_w=nu D. The Poisson limit gives + + E[(X_w-t)_+] -> exp(-t), t>=0. (6.2) + +These are convex functions of t, with a differentiable limit. Their +one-sided derivatives bracket `-Pr(X_w>t)`. Difference quotients and then +a shrinking increment imply + + X_w => Exp(1). + +There is no unjustified differentiation of an arbitrary asymptotic error. +For t>=1, `Pr(X_w>t)<=E[(X_w-(t-1))_+]`; (5.4) gives a uniform exponential +tail. All fixed positive moments of X_w therefore converge. + +The same point-process limit also gives finite vectors of successive Palm +gaps: the limit is a vector of independent Exp(1) variables. One way to +justify the Palm step explicitly is Campbell's formula on a bounded +rescaled interval. The summand is bounded by its anchor count. The latter +has uniformly bounded second moment: distant truncated indicators are +independent, nearby pair contributions are bounded by (5.3), and the full +versus truncated second-moment discrepancy is bounded by a polynomial in +n times exp(-cH). This supplies uniform integrability. Local continuous +Palm tests pass to the Poisson Palm law; truncate very long gaps using +(5.4) before removing the truncation. + +## 7. The white law, its moments, and the coupled point processes + +Under the black-to-white pairing, (1.3) implies + + |L(W_i)-D_i| <= max(L_i,L_(i+1)). (7.1) + +Together with (4.3), nu times this difference tends to zero in every fixed +L^r space. Equations (6.2),(7.1) prove (1.5), including all fixed moments and +finite vectors of consecutive white spans. In particular the limiting +white span coefficient of variation squared is ONE, not the Brownian-range +constant pi/3-1 applicable to a different, subcritical component law. + +For the lower anchor c_i of W_i the sharper geometric relation is + + 1<=c_i-a_i<=L_i. (7.2) + +On a bounded rescaled interval, the expected number of paired anchors whose +rescaled displacement exceeds epsilon is bounded by a constant times +`Pr_B(nu L_i>epsilon)`, plus negligible endpoint terms. This tends to zero. +Thus the white anchor process and black anchor process collapse onto the +SAME Poisson process. For disjoint bounded intervals J_l, + + (N_B(J_l/nu),N_W(J_l/nu))_l => (Z_l,Z_l)_l, + Z_l independently Poisson(|J_l|). (7.3) + +This does not violate the torus prohibition on having no wrapping component +of either colour. Here N counts ANCHORS IN A WINDOW. Both counts can vanish +while one white component anchored outside spans the entire window. + +### A row-selected component is length biased + +Let D^dagger be the black gap containing a uniformly chosen row. Exactly, + + E f(nu D^dagger)=nu E_B[D f(nu D)]. (7.4) + +Consequently `nu D^dagger=>Gamma(2,1)`, with all fixed moments. Its mean +is asymptotically 2, whereas the component-Palm gap mean is 1. + +There is also an exact incidence-Palm identity for white spans: selecting +one pair (row, white component whose projection includes that row) gives + + E_inc f(nu L)= E_W[L f(nu L)]/E_W L. (7.5) + +It has the same Gamma(2,1) limit, by the already proved white moments. +The symmetric difference of the paired intervals +`[a_i,a_(i+1)-1]` and `[min W_i,max W_i]` has length at most +`L_i+L_(i+1)`. The proportion of rows affected is at most 2 epsilon_w. +Moreover (2.1) says there is a unique white projected component with +probability at least 1-epsilon_w. Thus ordinary row observation and (7.5) +agree asymptotically. Higher moment convergence follows by the incidence +moment formula and uniform integrability at one higher order. + +## 8. Closing the long cylinder into a torus changes the zero-count cell + +Take an axial torus with `m nu_w(p)->t in (0,infinity)`, p fixed subcritical. +The same truncated-anchor proof works with periodic dependency neighbourhoods. +The chance of any component with size >H is at most `wm C exp(-cH)=o(1)`. +Hence vertical wrapping is absent with probability tending to one, and the +number K of black horizontal components tends to Poisson(t). + +The finite matching topology [MZ] then determines the WHITE global count: + + (W_black,W_white) => (K, K+1{K=0}). (8.1) + +If K=0, there is one white rank-two component. If K>=1, there are K white +rank-one components. Consequently, for 0<=z,y<=1, + + E[z^W_black y^W_white] + -> exp(t(zy-1))+exp(-t)(y-1). (8.2) + +The limit is NOT independent Poisson and is NOT simply two identical +Poisson counts. The one exceptional zero-count term is the global closure. +The black rank probabilities tend to `(exp(-t),1-exp(-t),0)`. + +Conditionally on K=k>=1, the cyclic macroscopic gaps between anchors are +times t a Dirichlet(1,...,1) spacing vector, after a uniformly selected anchor +fixes the cyclic origin. White components occupy those gaps up to vanishing +scaled boundary spans. At k=0 the white cross is a different topology, not +an artificial additional anchor from an independent process. + +## 9. Exact all-height controls and a new bulk-volume question + +The attached script reuses UNMODIFIED tagged_winding_span.py, Git blob +`52f3611990ce2b1331d9e5296e0262f5e402e0d7`. It computes both colours at the +complementary parameters p=1/4 and q=3/4, using exact rational arithmetic. +It does not use a cutoff histogram to estimate a long white mean. + +| w | nu | E_B L | E_W L | nu E_W L | CV_W^2 | +|---|---:|---:|---:|---:|---:| +| 2 | .05679086538 | 2.07733008 | 16.46886447 | .9352810651 | .8865951229 | +| 4 | .004429300955 | 3.13710294 | 224.88167088 | .9960685995 | .9846477960 | +| 6 | .0004219779152 | 3.93801838 | 2369.27391382 | .9997812668 | .9978072200 | + +The JSON stores fractions, not only these rounded displays. The errors in +`nu E_W L-1` satisfy (1.4) exactly. It also evaluates the rational transform +`E[(1+s nu)^(-L)]`; its limiting target is 1/(1+s), without rounding exp(-s nu). + +### Exact volume identities + +Let K be the number of sites in a COMPLETE white essential component. Define + + theta_w=Pr(a uniform site belongs to an essential white component), + chi_w=E[|C(0)| 1{C(0) is white and essential}]. + +Two direct mass transports give + + theta_w=(nu/w) E_W K, + chi_w=(nu/w) E_W K^2. (9.1) + +Thus + + nu chi_w/[2 w theta_w^2] = (1+CV_W(K)^2)/2. (9.2) + +The independent direct activity transfer verifies these finite identities. +For w=2,3,4 at q=3/4 it gives `Corr_W(L,K)^2` approximately +`.99362833, .99849621, .99964572`. These are finite exact-rational results, +not proof of a large-width correlation limit. + +### Bulk-filling conjecture (NOT proved here) + +Write theta_8(q)=Pr_plane(0 lies in the infinite matching cluster). The new +question is whether a giant white slab has asymptotically deterministic bulk +density theta_8(q): + + (nu L, nu K/w) => (E,theta_8(q) E), E~Exp(1), (9.3) + +with at least second moments. If true, it gives + + theta_w -> theta_8(q), + chi_w ~ 2 w theta_8(q)^2/nu, + Corr_W(L,K)^2 -> 1. (9.4) + +This is a supercritical-dual statement. It does not contradict the separate +subcritical hypothesis of decorrelation between a diffusive shape and a +centred additive occupation fluctuation. The missing input is a bulk-density +law in the randomly delimited, exponentially long slab, under COMPLETE +component Palm rather than vertex Palm. Exponentially rare boundary selection +must not be divided out without a conditional argument. + +## 10. What this does not yet say about the ultimate white pole + +It is tempting to infer that the fixed-width far-tail exponent gamma_white +satisfies gamma_white/nu->1. Equation (1.5), even with convergence of every +fixed moment, does NOT by itself establish that claim: the tail limit takes +height to infinity before width. + +An explicit inference counterexample uses nu_n=1/n and a mixture of positive +geometric variables: weight 1-2^(-n^2) at success parameter 1/n, weight +2^(-n^2) at parameter 1/n^2. Then L/n tends to Exp(1) with all fixed moments, +but the ultimate tail rate is `-log(1-1/n^2)`, whose ratio to nu_n tends to +zero. This is not asserted to occur in site percolation. + +A useful further conjecture is the genuine site metastable relation +`gamma_white/nu_black->1`. Proving it requires uniform long-time control, +not another confirmation of the first few moments. It is recorded here as +a separate question, not silently included in (1.5). + +## 11. Executed scope + +The script checks 41,984 finite annular configurations with black endpoint +rings, using lifted BFS and an independently coded displacement DSU for both +colours. It checks alternation, strict height order, contact bounds, gap +sandwiches and row coverage. Those are finite geometric controls, not a +substitute for Sections 2--7. + +Every nonempty periodic point pattern of periods 2 through 11 is used to +verify the discrete stationary gap identities, including the length bias. +All-height complementary moments use widths 2 through 6; joint activity +controls use widths 2 through 4. No Monte Carlo, extra heavy-width build, +external machine, or full repository suite is run. The mathematical proofs +and the explicit bulk-filling conjecture are kept separate. + +## Sources actually used + +[MZ] S. Mertens and R. M. Ziff, *Percolation in Finite Matching Lattices*, +arXiv:1603.07289v2, Section II (surrounding-component connectivity and the +single/spiral/cross count classification). HTML relevant section read. +https://arxiv.org/html/1603.07289v2 + +[AV] T. Antunovic and I. Veselic, *Sharpness of the phase transition and +exponential decay of the subcritical cluster size for percolation on +quasi-transitive graphs*, arXiv:0707.1089v3, Theorems 2--3 and Section 3 +site/BK conventions. Relevant theorem statements and model definitions read. +https://arxiv.org/html/0707.1089v3 + +[AGG] R. Arratia, L. Goldstein and L. Gordon, *Two Moments Suffice for Poisson +Approximations: The Chen--Stein Method*, Ann. Probab. 17 (1989), 9--25, +Theorem 2, printed p.11. The PDF statement and total-variation convention +were visually read, not reconstructed from the parsed formula alone. +https://dornsife.usc.edu/larry-goldstein/wp-content/uploads/sites/221/2023/06/AGG-1.pdf + +[Palm context] G. Nieuwenhuis, *Bridging the gap between a stationary point +process and its Palm distribution*, Statistica Neerlandica (1994). +Abstract only; not used as the proof of an inversion formula. Equations +(3.1),(6.1),(7.4),(7.5) are derived by counting in this note. +https://doi.org/10.1111/j.1467-9574.1994.tb01430.x diff --git a/docs/manuscripts/geometric-balance/bounded-euler-defect-information-bottleneck-20260914.md b/docs/manuscripts/geometric-balance/bounded-euler-defect-information-bottleneck-20260914.md new file mode 100644 index 000000000..4c071ef7d --- /dev/null +++ b/docs/manuscripts/geometric-balance/bounded-euler-defect-information-bottleneck-20260914.md @@ -0,0 +1,198 @@ +# The bounded Euler defect as the root-information bottleneck + +Date: 2026-09-14 + +Status: exact consequence of the finite matching Euler identity, plus research interpretation. No new continuum field is identified. + +## 1. An extensive-looking expression is exactly three-valued + +For every honest square-torus site configuration, + +```text +X := r4-1 + = k4-k8-K+E-F0 + in {-1,0,+1}. +``` + +Each term on the right can be extensive: + +```text +k4, k8, K, E, F0 = O(L^2) +``` + +for typical configurations. Their combination is nevertheless bounded **configuration by configuration**, not only after expectation. + +This is much stronger than an average cancellation. + +## 2. Why typical-object asymptotics can miss the root + +The Matching-One root solves + +```text +E_p X = 0. +``` + +Suppose a coarse/effective theory approximates separately + +```text +k4/L^2, +k8/L^2, +K/L^2, +E/L^2, +F0/L^2 +``` + +to small absolute error. Unless those approximations preserve the exact correlated Euler cancellation, their reconstructed error in `X` can be much larger than the true `O(1)` signal. + +Thus very accurate bulk laws for cluster densities, component sizes, boundary fractions or local motifs do not automatically determine the root. + +The missing information is precisely the bounded topological Euler defect left after cancellation. + +This is a concrete microscopic realization of the research-compass warning that “a solved typical limit can discard the quantity that fixes the root.” + +## 3. Charged-sector susceptibility is the variance of the Euler defect + +Because + +```text +X=-1,0,+1, +``` + +we have exactly + +```text +E X = P2-P0 = M, +E X^2 = P0+P2 = chi, +Var X = chi-M^2. +``` + +At the balance root `M=0`, + +```text +boxed: chi = Var(X). +``` + +So the “charged-sector susceptibility” is literally the variance of the bounded Euler defect. + +The topological source generating function is + +```text +Z(h)=E e^{hX}=P1+P2 e^h+P0 e^-h. +``` + +At the root, odd source cumulants vanish and even cumulants are fixed by `chi`; for example + +```text +kappa_2(X)=chi, +kappa_4(X)=chi-3chi^2. +``` + +No separate high-dimensional cluster model is required to know the source law once `(M,chi)` are known. + +## 4. The dangerous direction is reconstruction, not direct evaluation + +There are two numerically/theoretically different strategies: + +### Direct topological evaluation + +Compute `r` or `X` from lifted homology / rank classification. The result is bounded and stable. + +### Reconstructed Euler evaluation + +Compute the large pieces `k4,k8,K,E,F0` and subtract. This is exact only if all pieces are obtained on the **same configuration** with mutually consistent conventions. + +Independent marginal approximations, separately fitted asymptotics, or source derivatives with missing normalization can destroy the cancellation. + +This gives a general warning for the project: + +> Whenever a proposed “explanation” of the root passes through several extensive pieces, verify the configurationwise identity before interpreting residuals as new physics. + +The #802 source-normalization failure is an example of the same structural hazard: an omitted common normalization turned an exact/gauge cancellation into an apparent large response. + +## 5. Local insertion form + +Let a white site `v` be switched to black. Write + +```text +c_b = number of distinct black NN neighbour components touched by v, +d_b = number of occupied NN neighbours of v, +f_b = number of elementary faces completed black by inserting v, +t_w = number of nonempty white-matching components produced when v is removed from its old white component. +``` + +Then + +```text +Delta k4 = 1-c_b, +Delta k8 = t_w-1, +Delta K = 1, +Delta E = d_b, +Delta F0 = f_b. +``` + +Hence the exact Euler source gives + +```text +boxed: +Delta_v X + = 1 - c_b - t_w + d_b - f_b. (5.1) +``` + +The left side is the rank insertion jump in `{0,1,2}`. + +Equation (5.1) is a local-connectivity certificate for topological birth. It connects the pivotal language of #768/#769 to the cluster/Euler language without naming an arm field. + +In particular, a direct `0->2` birth is not characterized by a bare local degree alone; it is a mismatch between black-component merging, white-component splitting and the local edge/face Euler terms. + +## 6. A revised microscopic question for the leading correction + +Instead of starting from + +```text +“is the L^-4 correction an 8-arm field or a spin-four field?” +``` + +one can ask a source-defined question: + +> which finite-size/angular correction first biases the distribution of the exact Euler defect `X` away from its continuum balanced law? + +The answer can then be decomposed empirically into + +```text +angular irrep (H0/H4/H8/...), +radial exponent, +microscopic source / pivotal channel, +``` + +before assigning a continuum operator. + +The existing oblique data say the first visible root bias is overwhelmingly H4-like. #808 asks what survives after that irrep is projected out. Equation (5.1) supplies a lattice-side target for classifying the corresponding insertion events. + +## 7. Consequence for effective models + +A component-Palm/Poisson/fragmentation model may correctly describe most geometry while not preserving the Euler defect. To claim it explains Matching One, it must additionally reproduce at least one of: + +```text +E X, +the sourced law Z(h), +or an equivalent sector free-energy difference. +``` + +Matching only component density, mean span or neutral-count law is insufficient unless a theorem transports the Euler defect through the approximation. + +This criterion is stricter than “the effective model fits several observables,” but directly aligned with the root. + +## 8. Claim boundary + +Exact: + +- configurationwise bounded Euler identity; +- `chi=Var X` at the root; +- local insertion formula (5.1). + +Interpretive/programmatic: + +- use Euler-defect preservation as a criterion for whether an effective model explains the root; +- classify leading angular/radial corrections through the distribution of `X` before CFT field naming. + +The main message is that the root lives in an exact correlated cancellation, not in any one of its extensive ingredients. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/c3-character-vs-embedded-spin4-20260914.md b/docs/manuscripts/geometric-balance/c3-character-vs-embedded-spin4-20260914.md new file mode 100644 index 000000000..c84206a5a --- /dev/null +++ b/docs/manuscripts/geometric-balance/c3-character-vs-embedded-spin4-20260914.md @@ -0,0 +1,230 @@ +# C3 homology character versus embedded physical spin four away from the hexagonal modulus + +2026-09-14. Typing clarification for the existing #156 primitive-sector pilot. No previous finite numbers are erased or rescored here. + +The three-line contrast used in #156 is a valid **homology-representation character**. The new projective-slope analysis shows that it equals the real part of the physically embedded spin-four harmonic only at the exact hexagonal modulus. Away from that fixed point, the two are distinct observables and should be named separately. + +## 1. The existing three-line character + +In the positive-`rho` convention of #156, the distinguished primitive lines are + +\[ +\ell_0=(1,0),\qquad +\ell_1=(0,1),\qquad +\ell_2=(1,-1). \tag{1.1} +\] + +The 60-degree homology action cycles these three lines at the Eisenstein/hexagonal point. With probabilities `P_0,P_1,P_2`, the real nontrivial C3 character used by #156 is + +\[ +\boxed{C_{C3}=P_0-\frac{P_1+P_2}{2}.} \tag{1.2} +\] + +This is the real part of the character with weights + +\[ +1,e^{2\pi i/3},e^{-2\pi i/3}. \tag{1.3} +\] + +It is an exact representation-theoretic coordinate for the declared three-line orbit. + +## 2. Physical spin-four weights depend on the actual modulus + +For an embedded torus with + +\[ +\tau=\frac12+i y, +\] + +the projective physical spin-four readout is + +\[ +Z_4(u,v)=\left(\frac{u+v\tau}{|u+v\tau|}\right)^4. \tag{2.1} +\] + +On the three distinguished lines, + +\[ +Z_4(1,0)=1, \tag{2.2} +\] + +while the other two are complex conjugates. Reflection symmetry gives `P_1=P_2`, so their imaginary parts cancel and the three-line physical contribution is + +\[ +\boxed{ +A_4^{(3)}(\tau) +=P_0+c_4(y)(P_1+P_2),} \tag{2.3} +\] + +where + +\[ +c_4(y)=\Re\left[ +\left(\frac{1/2+i y}{\sqrt{1/4+y^2}}\right)^4\right]. \tag{2.4} +\] + +The C3 contrast (1.2) corresponds instead to the fixed coefficient `-1/2`. + +Therefore + +\[ +\boxed{ +A_4^{(3)}(\tau)-C_{C3} +=\left(c_4(y)+\frac12\right)(P_1+P_2).} \tag{2.5} +\] + +The two coordinates coincide **if and only if** the relevant geometric coefficient is `c_4=-1/2` (apart from a trivial zero sector probability). + +## 3. Exact coincidence at the hexagonal fixed point + +For + +\[ +y=\sqrt3/2, +\qquad +\tau=e^{i\pi/3}, \tag{3.1} +\] + +the two non-axis physical directions make angles `+/- pi/3`. Hence + +\[ +c_4=\cos(4\pi/3)=-\frac12. \tag{3.2} +\] + +Thus + +\[ +\boxed{A_4^{(3)}(\tau_{hex})=C_{C3}.} \tag{3.3} +\] + +This is the geometric reason the three-line C3 real character can be described as a spin-four angular contrast **at the exact Eisenstein modulus**. + +The full primitive-sector harmonic nevertheless vanishes there after including the entire sixfold orbit structure, as proved by the 60-degree modular automorphism in `projective-slope-modular-covariance-20260914.md`. + +## 4. N30 and N56 are not at the exact fixed point + +The PR #213 baselines use + +\[ +\tau_{30}=\frac12+\frac56 i, +\qquad +\tau_{56}=\frac12+\frac78 i. \tag{4.1} +\] + +For these moduli, + +\[ +\boxed{c_4(5/6)=-0.5570934256055362\ldots,} \tag{4.2} +\] + +\[ +\boxed{c_4(7/8)=-0.4844970414201184\ldots.} \tag{4.3} +\] + +They are close to, but not equal to, `-1/2`. + +Using the already-merged PR #213 primitive probabilities: + +### N30 + +\[ +P_0=0.1107291776903850\ldots, +\qquad +P_1=P_2=0.1272155037499346\ldots. \tag{4.4} +\] + +The three-line C3 character is + +\[ +C_{C3}=-0.01648632605954963\ldots, \tag{4.5} +\] + +whereas the physically embedded three-line spin-four contribution is + +\[ +\boxed{A_4^{(3)}=-0.0310126638579850\ldots.} \tag{4.6} +\] + +The full all-primitive-sector continuum harmonic computed in the 2026-09-14 control is + +\[ +A_4^{full}=-0.03047986387219586\ldots. \tag{4.7} +\] + +Thus the `rank1_other` sectors supply a small but nonzero correction of about `+5.33e-4` to the embedded physical harmonic. + +### N56 + +\[ +P_0=0.1247005965715636\ldots, +\qquad +P_1=P_2=0.1201521157922650\ldots. \tag{4.8} +\] + +Here + +\[ +C_{C3}=+0.004548480779298590\ldots, \tag{4.9} +\] + +while + +\[ +\boxed{A_4^{(3)}=+0.00827390732812383\ldots,} \tag{4.10} +\] + +and the full primitive-sector value is + +\[ +A_4^{full}=+0.008136416970836356\ldots. \tag{4.11} +\] + +The other primitive sectors again shift the physical harmonic slightly. + +## 5. Interpretation of the old pilot + +The continuum-subtracted N30/N56 pilot in #156 remains a valid measurement of the **frozen C3 homology character** `C_{C3}` and its finite-size residual. This note does not reinterpret or invalidate that measurement. + +What changes is the allowed wording: + +- `C_{C3}` may be called the physical embedded spin-four three-line contrast at the exact hexagonal modulus; +- away from that modulus, including N30/N56, it is a fixed homology-character coordinate, not literally `Re e^{i4 theta}` in the physical embedding; +- the full physical spin-four harmonic additionally uses modulus-dependent weights and all primitive rank-one lines. + +Therefore a sign/radial transport statement about `C_{C3}` is a statement about that representation coordinate. It cannot be promoted to an embedded-H4 amplitude without the modulus-dependent map (2.3) and the `rank1_other` contribution. + +## 6. Why the distinction matters for H4/H8 alias discussions + +On the three exact Eisenstein orbit lines, real C3 characters cannot distinguish certain spin-four/spin-eight aliases because both restrict to the same finite character pattern up to the declared convention. The existing #156 note already recognized this. + +The projective embedded harmonic supplies the missing extra structure: + +1. away from the hexagonal point, the physical angle weights move continuously with `tau`; +2. other primitive slopes have distinct `e^{i4 theta}` and `e^{i8 theta}` values; +3. the full Pinson--Arguin sum therefore produces genuinely different spin-four and spin-eight modular-covariant functions. + +For example at `tau=i`, + +\[ +H_4^{cont}=0.7791813140\ldots, +\qquad +H_8^{cont}=0.9992417096\ldots. \tag{6.1} +\] + +while both vanish at the hexagonal fixed point by the appropriate automorphism selection rules. + +This gives a representation-faithful route to break the three-line alias **without** inventing an untyped microscopic source. + +## 7. Recommended naming discipline + +Use three distinct labels in future notes/data: + +```text +C_C3 fixed three-line homology character, +A4_three(tau) physical spin-4 projection on those three lines, +A4_full(tau) physical spin-4 sum over every primitive rank-one line. +``` + +At `tau_hex`, `C_C3=A4_three`, but `A4_full` still contains the remaining primitive orbit structure and is constrained by the full torus automorphism. + +This naming prevents a correct representation proxy from silently acquiring a stronger physical-spin interpretation than its definition supports. diff --git a/docs/manuscripts/geometric-balance/canonical-potts-topological-coordinates-20260914.md b/docs/manuscripts/geometric-balance/canonical-potts-topological-coordinates-20260914.md new file mode 100644 index 000000000..56851d85a --- /dev/null +++ b/docs/manuscripts/geometric-balance/canonical-potts-topological-coordinates-20260914.md @@ -0,0 +1,305 @@ +# Canonical Potts topological coordinates: loop fugacity, thermal self-duality and rank charge + +Date: 2026-09-14 + +Status: exact algebra for a toroidal FK model with the bounded rank source. It reorganizes the generic-Q source/tangent programme and makes the graph-polynomial source a symmetry point rather than an ad hoc closure deformation. + +## 1. Start from the sourced FK weight + +For a cellular torus FK subgraph `A`, + +```text +W(Q,v,h;A) + = Q^{k(A)} v^{|A|} exp[h X(A)], +X(A)=r(A)-1. +``` + +The Euler identity gives + +```text +X(A)=k(A)-k(A*)+|A|-|V|. +``` + +## 2. Define three canonical coordinates + +Introduce + +```text +boxed: +x = v/sqrt(Q), +eta = h + (1/2) log Q. +``` + +Equivalently, + +```text +v=x sqrt(Q), +h=eta-(1/2)log Q. +``` + +Substitute into the original weight and use the Euler identity. All powers of `Q` simplify: + +```text +W(Q,x,eta;A) + = Q^{|V|/2} + (sqrt Q)^{k(A)+k(A*)} + x^{|A|} + exp[eta X(A)]. (2.1) +``` + +The prefactor `Q^{|V|/2}` is global. + +Using + +```text +k+k*=b+1_(r!=1), +``` + +we obtain the loop/topology form + +```text +boxed: +W ∝ + (sqrt Q)^{b(A)+1_(r(A)!=1)} + x^{|A|} + e^{eta(r(A)-1)}. (2.2) +``` + +This is an exact finite configuration identity. + +## 3. Interpretation of the three coordinates + +Equation (2.2) cleanly separates three roles. + +### `Q`: common loop fugacity + +Every medial boundary loop carries the usual factor `sqrt(Q)`, together with the fixed torus extreme-sector bonus already derived. + +### `x=v/sqrt(Q)`: thermal / self-duality coordinate + +The self-dual bond-FK point is simply + +```text +x=1. +``` + +The logarithm `t=log x` is the natural duality-odd thermal coordinate at fixed `Q`. + +### `eta`: primal/dual topological charge + +The only remaining rank source is + +```text +e^{eta(r-1)}. +``` + +Thus `eta`, not the raw closure source `h`, is the canonical generic-Q topological chemical potential after the intrinsic FK cluster asymmetry has been removed. + +## 4. The graph-polynomial source is exactly `eta=0` + +The graph-polynomial balance source is + +```text +h_Q=-1/2 log Q. +``` + +Therefore + +```text +boxed: +eta(h_Q,Q)=0. +``` + +At this source the explicit topological charge factor disappears from (2.2). The only topology dependence left is the universal torus sector bonus already carried by the medial-loop count. + +This is why the primal and dual cluster fugacities become equal in the Euler-localized representation. + +Hence the graph-polynomial section is not best thought of as “turn on a special topological field”. In canonical coordinates it is the **zero topological-field section**. + +## 5. The square self-dual graph-polynomial point is `(x,eta)=(1,0)` + +At the square-lattice self-dual Potts/FK coupling + +```text +v=sqrt(Q), +``` + +we also have + +```text +x=1. +``` + +Therefore the distinguished point is + +```text +boxed: +(log x, eta)=(0,0). +``` + +It is simultaneously + +```text +thermal self-dual, +primal/dual topological-source symmetric. +``` + +This gives a compact structural explanation for the unusually strong cancellation exploited by the graph-polynomial/eigenvalue method. + +## 6. Exact duality action + +Let `A*` be the complementary dual state. Then + +```text +|A*|=|E|-|A|, +X(A*)=-X(A), +b(A*)=b(A), +1_(r(A*)!=1)=1_(r(A)!=1). +``` + +Therefore the weight (2.2) obeys, up to the global factor `x^{|E|}` and replacement by the dual lattice, + +```text +boxed: +(log x, eta) -> (-log x, -eta). (6.1) +``` + +On a self-dual lattice the partition function has the corresponding two-coordinate reflection relation. + +Thus thermal duality and topological charge conjugation are simply the two odd coordinates around the same fixed point. + +## 7. Reinterpretation of the generic-Q shifted rank coordinate + +The existing canonical rank coordinate is + +```text +b = (1/2) log(Z0/Z2). +``` + +The graph-polynomial balance occurs at + +```text +b=-1/2 log Q. +``` + +Hence the shifted odd variable + +```text +b_tilde=b+(1/2)log Q +``` + +is precisely the rank-law counterpart of the canonical source coordinate `eta`. + +Both vanish on the graph-polynomial symmetric section. + +This makes the earlier algebraic shift much less mysterious: it removes the built-in primal FK cluster fugacity and centers the theory on equal primal/dual cluster weight. + +## 8. Reinterpretation of the graded Q tangent + +Differentiate at fixed `eta=0`. + +Since + +```text +eta=h+(1/2)log Q, +``` + +keeping `eta=0` forces + +```text +dh/d log Q = -1/2. +``` + +This is exactly the previously identified kinematic topological-source term in the generic-Q tangent. + +Therefore the `-1/2` piece has a geometric meaning: + +> it is the compensating source motion required to stay on the primal--dual symmetric `eta=0` surface while Q changes. + +After this subtraction, the remaining Q tangent probes the common loop fugacity `sqrt(Q)`, any Q-dependence of the thermal coordinate `x`, and representation/module data. It is not contaminated by a trivial topological imbalance. + +This is the natural coordinate system for #746. + +## 9. Near-critical/source scaling should use `(t,eta)` + +At fixed Q define + +```text +t=log x. +``` + +The exact duality action is + +```text +(t,eta)->(-t,-eta). +``` + +Therefore a near-critical topology-resolved scaling function should be organized around these two odd coordinates, not raw `(v,h)`. + +At Q=1, + +```text +eta=h, +x=v, +``` + +so the distinction disappears and the Matching-One charge source is already canonical. + +At generic Q it is essential. + +## 10. Possible loop/height interpretation + +Equation (2.2) suggests a continuum dictionary: + +```text +sqrt(Q) : loop fugacity / Coulomb-gas background parameter, +t : thermal mass perturbation, +eta : electric/topological face-charge perturbation. +``` + +At `eta=0` the primal/dual faces are unbiased. Moving `eta` changes their relative cluster fugacities while leaving the product fixed. + +An oriented-loop/height representation should therefore encode `eta` as an electric/background-charge-like deformation rather than as a change of the unoriented bulk loop fugacity. + +This is a targeted representation conjecture, not yet a normalized Coulomb-gas charge assignment. + +## 11. Consequence for the massive torus programme + +The object sought in #782 can be restated as a finite-volume scaling function in + +```text +(Q,t,eta,alpha,tau), +``` + +where `alpha` is the already-identified rank-one neutral seam. + +The graph-polynomial / critical UV anchor is the slice + +```text +t=0, +eta=0. +``` + +The Matching-One rank response is the `eta` derivative (at Q=1), while the thermal response is the `t` derivative. + +This provides a cleaner target for a massive modified trace than separate ad hoc projectors for rank0, rank1 and rank2. + +## 12. Claim boundary + +Exact finite algebra: + +```text +x=v/sqrtQ, +eta=h+1/2 logQ, +W ∝ (sqrtQ)^{b+1_(r!=1)} x^{|A|} e^{eta(r-1)}, +duality: (log x,eta)->(-log x,-eta), +graph-polynomial source: eta=0. +``` + +Conjectural/interface: + +- identification of `eta` with a particular electric/height operator in the continuum; +- local periodic-TL implementation for arbitrary eta; +- massive/TBA scaling functions in these coordinates. + +The main structural payoff is that generic-Q thermal and topological asymmetries are now centered on one explicit duality fixed point. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/cavity-field-98-negative-20260914.md b/docs/manuscripts/geometric-balance/cavity-field-98-negative-20260914.md new file mode 100644 index 000000000..b8b607082 --- /dev/null +++ b/docs/manuscripts/geometric-balance/cavity-field-98-negative-20260914.md @@ -0,0 +1,224 @@ +# 任务 A —— #779 的 9/8 极点：正活动矩阵的谱、`z_c(p)`、以及一个 w-依赖的结论 + +机器：华为云 `DevEnvC_HZsCM6`（全新，16 vCPU / 32 GiB），工作目录 `/workspace/cavity779/`。 +引擎：`/workspace/cavity779/in/tagged_winding_span.py`，**未改动**， +sha256 `9621acf490dbc4b7dba28f2d2f0f4c2e0f1ee9e42d3987d1f7815bd1c9d5c9a3` +（= 仓库 pinned blob `52f3611990ce2b1331d9e5296e0262f5e402e0d7`）。 +`w=7,8` 用的是**自建副本**（单行放宽宽度上限 + 一处 O(n²)→O(n) 性能改写， +逐位一致性已验），**原件未动**（`issues-found.md` I7/I10）。全部计算在云机上执行。 + +--- + +## 0. 结论（四句话，含一处我自己的自我修正） + +1. **出发点通过自证**：`H = wL−K−B+B_out`、`0 ≤ B_out ≤ 2w` 在 7 个圆柱算例的**穷举**上 + 5 项几何检查 **0 违例**（§1）。活动矩阵的逐行标签 `(k,b)` 也与朴素几何定义 + **0/60734 转移不符**（§2），故每行 z 指数 = 该行孔洞数是**已核实**的。 +2. **`z_c(p)` 的答案强烈依赖宽度 `w`**，且**在 `w=8` 上确实逼近 9/8**： + + | `w` | `z_c(1/8)` | `z_c(1/16)` | `z_c(1/32)` | 结论 | + |---|---|---|---|---| + | 3 | ∞ | ∞ | ∞ | 精确无极点（`H≡0`） | + | 4 | 64.47 | 234.29 | 869.08 | 远离 9/8，`p↓0` 时**变大** | + | 5 | 8.429 | 15.580 | 32.395 | 同上 | + | 6 | 3.880 | 6.336 | — | 同上 | + | 7 | 2.387 | 3.422 | 5.322 | 同上 | + | **8** | **1.1788** | 见 §6（计算中） | — | **与 9/8 = 1.125 只差 4.8%** | + +3. **规范形式 `ρ = 1 + p^w[w(z−1) − 1]` 是 w-依赖的**：`w ≤ 6` 上它**定量错误** + （实测 z 斜率比 `w` 小 3000–7000 倍）；但 `w=8` 上实测斜率 **8.5 ≈ w=8**， + 截距 1.53（而非 1）⇒ `z_c = 1.18`。也就是说**这条形式在 `w=8` 上基本对（斜率对、常数差 53%）**。 +4. **我自己第一版的"结构性否定"过头了**：`connected=True` 确实排除了同色孤岛（`8p^8`）， + 但我当时断言"剩下的反色屏蔽孔洞被 `p^{w+1}` 压死 ⇒ `H→0`"是**错的**—— + 在 `q^K p^B` 这个测度里，一个孔洞让该行的**边界计数 `b` 减 1**，等于给了一个 `1/p` 的回补， + 所以孔洞**不是被压死，而是每孔代价 `≈ z/p`**。这正是 `w=8` 上斜率达到 `≈w` 的来源。 + 修正记录留在 §4。 + +--- + +## 1. 出发点自证：`H` 恒等式（穷举，精确） + +脚本 `ta1_h_identity.py`。**穷举**圆柱 `w×R` 上全部 `2^{wR}` 个组态；对每个完整绕行白簇 +（8 邻域、x 周期、y 开放；绕行用位势 DSU 的位移和 ≠ 0 判定）按**纯几何**定义算： + +| 记号 | 定义（不引用任何公式） | +|---|---| +| `L` | `y1−y0+1`，并检查该簇确实出现在跨度的每一行 | +| `K` | `|C|` | +| `B_in` | 跨度内、不属于 `C`、与 `C` 8-邻接的不同站点数 | +| `B_out` | 行 `y0−1`/`y1+1` 内（存在者）与 `C` 8-邻接的不同站点数 | +| `H` | 跨度内既不属于 `C` 也不属于其外部邻接集合的站点数 | + +检查：`I1: H = wL−K−B_in`；`I2: 0 ≤ B_out ≤ 2w`；`I3: 跨度每行都有该簇的点`； +`I4: B_in ∪ B_out = 该簇全部外部邻接站点（两条独立代码路径给同一集合）`； +`I5: Σ_{行∈跨度}(w − k_行 − b_行) = H`。 + +| `w` | `R` | 组态数 | 完整绕行簇数 | `H_min` | I1 | I2 | I3 | I4 | I5 | `k+b>w` 的行 | +|---|---|---|---|---|---|---|---|---|---|---| +| 3 | 4 | 4 096 | 3 020 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | +| 3 | 5 | 32 768 | 27 828 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | +| 3 | 6 | 262 144 | 248 668 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | +| 3 | 7 | 2 097 152 | 2 183 716 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | +| 4 | 4 | 65 536 | 41 456 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | +| 4 | 5 | 1 048 576 | 775 456 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | +| 5 | 4 | 1 048 576 | 579 780 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | + +**结论：6.1e6 个组态、3.87e6 个完整绕行簇，全部检查 0 违例。** 另有一个**非平凡**推论： +`w ≤ 3` 时 `H ≡ 0`（宽度 3 上任意两列环向距离 ≤1 ⇒ 跨度内每个非簇站点都与簇邻接）。 + +> 第一版脚本的 `I4` 写错过（`len(far)==0` 不是不变量），已修正并重跑， +> 两次运行的 I1/I2/I3/I5 完全一致（`issues-found.md` I6）。 + +## 2. 活动矩阵的标签语义（独立核对） + +脚本 `ta0_label_check.py`。用**朴素几何**独立定义：「当前行的一个站点是边界站点 ⟺ +它不是当前行的簇站点，且在 `上/本/下` 三行中存在一个环向距离 ≤1 的簇站点」，`k` = 当前行簇站点数； +与 `T.activity_transfer(w, matching=True)` 的 `(k,b)` 逐转移比较： + +| `w` | 2 | 3 | 4 | 5 | +|---|---|---|---|---| +| 比较转移数 | 42 | 560 | 5 650 | 54 482 | +| 不符 | **0** | **0** | **0** | **0** | + +且 `max(k+b) = w` 对所有已建宽度成立（w=8 也是 8）。**因此 +`M(p,z)_{i→j} = z^{w−k−b} q^k p^b` 的 z 指数确实等于该行的孔洞数。** + +## 3. 算法与 `z_c(p)`、`ρ` 的结果 + +``` +W_p(z) = Ψ_w(z^w, q/z, p/z)/ν = Σ_C z^{wL−K−B} q^K p^B / ν +H_p(z) = Σ_C z^{H} q^K p^B / ν （H = wL−K−B+B_out） +z_c(p) = inf{ z ≥ 1 : ρ(M(p,z)) ≥ 1 }, M(p,z)_{i→j} = z^{w−k−b} q^k p^b +``` + +`w−k−b ≥ 0` ⇒ `M` 每个元素对 z 非降 ⇒ `ρ(z)` 单调 ⇒ 二分；`ρ` 用幂迭代 + +**Collatz–Wielandt 括号**（`min_i(Mx)_i/x_i ≤ ρ ≤ max_i(Mx)_i/x_i` 对任意正 `x` 严格成立）， +所以下面的 `ρ` 是**有双侧界**的，不是三点拟合。**半径只依赖矩阵**，与源/退出向量无关。 + +### 3.1 `z_c(p)` 表 + +| `w` | 活动状态数 / 约简行 | 构建 | 峰值 RSS | p=1/8 | p=1/16 | p=1/32 | 扫描上限 | +|---|---|---|---|---|---|---|---| +| 3 | 80 / 5 | 0.02 s | — | ∞ | ∞ | ∞ | 1e7 | +| 4 | 386 / 12 | 0.17 s | 13 MB | 64.4708 | 234.287 | 869.082 | 1e7 | +| 5 | 1 832 / 23 | 1.9 s | 18 MB | 8.42866 | 15.5805 | 32.3951 | 1e7 | +| 6 | 8 774 / 54 | 19.9 s | 56 MB | 3.88009 | 6.33572 | — | 1e7 | +| 7 | 42 296 / 107 | 215.9 s | 393 MB | 2.38683 | 3.42202 | 5.32189 | 1e5 | +| 8 | **204 994 / 250** | **2 410 s** | **3 572 MB** | **1.17878** | §6 计算中 | — | 3.0 | + +（`w=3` 是**精确**结论：每行 `k+b=w` 恒成立 ⇒ `M(z)` 与 z 无关 ⇒ `ρ = 1−p^w`， +实测 `w=3,p=1/8`：`ρ = 0.998046875 = 1−(1/8)^3` 到机器精度，`w=3,p=1/32`： +`0.999969482421875 = 1−(1/32)^3`。） + +**裁定（Q1）**： +- `w ≤ 7`：`z_c(p)` **随 `p↓0` 变大**（放大指数：`w=4` 为 `p^{−1.88}`，`w=5` 为 `p^{−1.0}`， + `w=6` 为 `p^{−0.71}`，`w=7` 为 `p^{−0.52}`），**远离 9/8**。 +- `w = 8`：**`z_c(1/8) = 1.17878`，与 `9/8 = 1.125` 只差 4.8%**。这是"接近但还不等"。 + 由于 `w≤7` 的趋势是 `p↓0` 时 `z_c` 变大，**`w=8` 的 `p` 趋势是判定 Q1 的决定性一步**， + 正在用一次性构建 + 多 `p` 复用的脚本（`ta6_w8_trend.py`，含 pickle）跑； + §6 会如实写它是否跑完。**本报告不用 `w≤7` 的趋势替 `w=8` 表态。** + +### 3.2 `ρ(p,z)` 的 z 依赖：**是 w-依赖的，不能一刀切** + +定义 `S(w,p) := d[(ρ−1)/p^w]/dz`（在 `p^w` 单位下的 z 斜率），实测： + +| `w` | p | `(ρ−1)/p^w` 在若干 z 上的值 | `S` | 规范形式预测的 `S` | +|---|---|---|---|---| +| 4 | 1/8 | z=1.0001: −1.048360 → z=1.2: −1.048226 | 6.7e−4 | 4 | +| 5 | 1/8 | z=1.0001: −1.122772 → z=1.1: −1.121794 | 9.8e−3 | 5 | +| 6 | 1/8 | z=1.0001: −1.225116 → z=1.1: −1.215238 | 9.9e−2 | 6 | +| 7 | 1/16 | z=1.0001: −1.097843 → z=1.1: −1.053448 | 4.4e−1 | 7 | +| **8** | **1/8** | **z=1.02: −1.3558, z=1.05: −1.1008, z=1.10: −0.6746, z=1.125: −0.4609** | **8.5** | **8** | + +- `w ≤ 6`：`S ≪ w`（差 3–4 个数量级）⇒ 规范形式**定量错误**。 +- `w = 7`：`S ≈ 0.44`，仍比 `7` 小约 16 倍。 +- **`w = 8`：`S ≈ 8.5 ≈ w`** ⇒ 规范形式的**斜率对**了；但截距是 ~1.53 而不是 1 + （由 z=1.02 与 z=1.10 两点反推：`−1.3558 + 8.5×0.02 = −1.53`； + 由 z=1.10 与 z=1.125 反推也是 `−1.53`）⇒ 极点 `1 + 1.53/8.5 = 1.18` ✓ 与二分结果自洽。 +- `S` 每个宽度步约放大 **一个数量级～15 倍**；`w=8` 恰好落在 `S ≈ w`。 + **这个 w-依赖是本轮最重要的发现：同一句话在 `w=6` 上是错的，在 `w=8` 上基本对。** + +### 3.3 PGF（Q2），`w=8, p=1/8` + +`H_p(z)`（用**拆开终态标签**的正确版本，见 I9）：`1.0 / 1.01875 / 1.03822 / 1.05845 / +1.07948 / 1.10137 / 1.11837 / 1.12416 / 1.12883`（z = 1.0 … 1.124）。 +即 `E[H] ≈ (1.1288−1)/0.124 ≈ 1.04`，而 `1/(9−8z)` 在 `z=1.124` 给 **125**。 +**裁定（Q2）**：即使在最有利的 `w=8` 上，`Z_p(z)` 也**没有**达到 `1/(9−8z)`： +实测 `E[H] ≈ 1`（应为 8），`z=1.1` 处实测 1.10（预测 5.0）。方向上 `H` 确实在动 +（`w ≤ 6` 时 `E[H] ≈ 1e−4…1e−2` 且几乎全在 `{0,1}`），但幅度差一个量级以上。 + +## 4. 机制：一处自我修正 + 目前最自洽的图像 + +**先记住已核实的事实**：`activity_transfer` 每一步都调 `step(..., connected=True)`， +`step` 里 `if connected and (roots-kept or not kept): return None, -1` ⇒ +**跨度内任何不与标签分量相连的*占据*站点都被丢弃**（`issues-found.md` I3）。 +对「白 matching 簇」，这**整类删掉了同色孤岛**（被 8 个黑点围住的白孤立单点， +`q·p^8 × w` 位置 = `8p^8`/行）——也就是派单里"孔洞每行率 8p^8"的那条机制。 + +**我第一版的错**：我接着断言"剩下的反色（黑）屏蔽孔洞代价 `p^{w+1}` ⇒ `H → 0`"。这个推理 +忽略了一件事：我们用的测度是 `Σ_C z^{wL−K−B} q^K p^B = Σ_C Π_行 (z^{w−k−b} q^k p^b)`， +**一个孔洞让该行的边界计数 `b` 减 1**（孔洞不是边界），于是该行权重从 `q^k p^{b}` 变成 +`q^k p^{b−1} z`，**相对增益是 `z/p`**。在 `p = 1/8` 上 `z/p` 一次就是 `≈ 8z`—— +孔洞在 `w=8` 上不但不被压死，反而是**被奖励**的。这正好解释了： +- `w=8` 的实测 z 斜率 `S ≈ 8.5 ≈ w`（每行约 `w` 个可放孔洞的位置）； +- 而 `w ≤ 6` 时同样机制被 `p^{w}` 之外的结构压住，`S` 小 3–4 个数量级 + （宽度太窄，可屏蔽的孔洞几何受限）。 + +**目前最自洽的图像（标记为"与数据相容的解释"，不是已证命题）**： +在正活动矩阵（exclusive）里，孔洞是**反色屏蔽**位点，每行约 `w` 个候选位置， +每孔带来 `z/p` 的相对增益、以及该行少一个 `p`（少一个边界站点）的代价， +净效应让领先块写成 `ρ ≈ 1 + p^w[w(z−1) − c]`，`w=8` 上实测 `c ≈ 1.53`。 +若 `c → 1`（`p↓0`），则 `z_c → 1 + 1/w = 9/8` ✓ 与 #779 的断言一致； +若 `c → 常数 > 1`，则 `z_c → 1 + c/w > 9/8`。**`p↓0` 时 `c` 的去向就是 Q1 的全部内容**， +`ta6_w8_trend.py` 直接测它。 + +## 5. 相关对照：标记引擎自己的跨度尾率（前提检查） + +脚本 `ta4_span_tail.py`（引擎自己的精确律，`matching=True`，`bins=24`）： + +| `w` | p | `ν` | 平均跨度 | `d_24/d_23` | 每行衰减率 | `p^w` | +|---|---|---|---|---|---|---| +| 6 | 1/8 | 3.384e−4 | 4.060 | 0.3742 | 0.983 | 3.81e−6 | +| 6 | 1/16 | 6.730e−6 | 3.218 | 0.1925 | 1.648 | 5.96e−8 | +| 8 | 1/8 | 3.792e−5 | 4.712 | 0.3782 | 0.973 | 5.96e−8 | +| 8 | 1/16 | 1.974e−7 | 3.707 | 0.1931 | 1.644 | 2.33e−10 | + +`p` 减半 ⇒ 衰减率**变大**、平均跨度**变小**，与「`p^8 L ⇒ Exp(1)`」方向相反。 +**这不是否证**：该前提要求 `L ~ p^{−8}`（`p=1/8` 时 `1.7e7` 行），远超可算窗口； +它只说明在可算 `p` 上该前提不可见。（另注：这里的「跨度」是**标记引擎的完整绕行簇**跨度， +与 §3 的活动矩阵不是同一个对象——见 I1。） + +## 6. `w=8` 的可算性（派单要求报告：状态数 / 内存 / 时间） + +| 阶段 | 数值 | +|---|---| +| 活动状态数 | **204 994**（`state_cap=100000` 时**撞上限**：`RuntimeError: activity state cap`，如实记录） | +| 约简行数 | 250 | +| 构建时间 | 2 410 s（40 min，单核） | +| 峰值 RSS | 3 572 MB | +| 转移规模 | 204 994 × 256 | + +两个引擎层面的实测坑（`issues-found.md`）： +- `blocks.index(i)` 让约简表组装变成 `O(n²)`（`w=8` 时仅此一处 ~80 min）→ + 自建 `activity_fast.py` 改成一次 O(n) 扫描，在 `w=3,4,5` 上与引擎**逐位一致**后使用； +- 默认 `state_cap=100000` 对 `w=8` 不够。 + +**`w=8` 的 `p` 趋势（`ta6_w8_trend.py`，一次性构建 + pickle + 多 `p` 复用）**： +结果见 `cavity-98-w8-trend.json` / `ta6.log`。 +**若未在本轮预算内跑完，本报告不替它表态**：此时 Q1 的裁定是 +「`w ≤ 7` 上 `z_c(p)` 随 `p↓0` 变大、远离 9/8；`w=8` 在 `p=1/8` 上已到 1.179（与 9/8 差 4.8%）， +但 `p↓0` 的方向未知」。 + +## 7. 不能宣称的 + +1. **不能**说 9/8 被否证或被打实。本轮在 `w=8, p=1/8` 上得到 `z_c = 1.1788`， + 与 `9/8` 差 4.8%，且规范形式的**斜率**（8.5 vs 8）已对、**截距**（1.53 vs 1）未对； + `p↓0` 的走向未定（§6）。 +2. **不能**把 `w ≤ 7` 的"规范形式错"外推到 `w=8`：数据明确显示 `S(w)` 每个宽度步放大 + 一到两个数量级，`w=8` 恰好进入 `S ≈ w` 的区域。 +3. **不能**用本轮的 `p ≥ 1/32` 数据评价 `p^wL = O(1)` 的渐近前提（§5）。 +4. **不能**把 `w ≤ 3` 外推：那里 `H ≡ 0` 是**精确**结论（每行 `k+b=w`）。 +5. **不能**说 `ν` 唯一：活动矩阵的 `ν` 与引擎标准配方的 `ν` 差一个 z 无关因子 + （`w=3,p=1/4,matching`：`17305/8802304` vs `467235/8802304`，比值恰 `1/27`，I1）。 +6. **不能**说 §4 的机制解释是定理：它是"与数据相容"的解释，`O(p^2)` 级的修正未被证明。 +7. 未做：长度 `p^{−8}` 的大圆柱模拟、全高度年龄表、#741 六个密度重算、`w=9`。 diff --git a/docs/manuscripts/geometric-balance/charge-fugacity-reflection-20260914.md b/docs/manuscripts/geometric-balance/charge-fugacity-reflection-20260914.md new file mode 100644 index 000000000..aa9e0ee2b --- /dev/null +++ b/docs/manuscripts/geometric-balance/charge-fugacity-reflection-20260914.md @@ -0,0 +1,146 @@ +# Exact 4/8 reflection law for the charge fugacity + +2026-09-14. Short exact consequence of digital Alexander duality, graph inclusion and the charge-neutral coordinates. + +## 1. Separate graph coordinates + +For graph `G4` or `G8` at occupation density `p`, define + +\[ +\chi_G(p)=P_G(r=0)+P_G(r=2), \tag{1.1} +\] + +\[ +\theta_G(p)=\log\frac{P_G(r=2)}{P_G(r=0)}. \tag{1.2} +\] + +In the rank-one sector let `H_{G,p}(z)` be the conditional essential-component count PGF and let `L_G` denote the projective slope mark. + +## 2. Digital Alexander is an exact involution on the full tuple + +Configurationwise, + +\[ +r_4(\omega)+r_8(\omega^c)=2. \tag{2.1} +\] + +At black density `p`, the complement has matching density `1-p`. Therefore + +\[ +P_{8,p}(r=0)=P_{4,1-p}(r=2), \tag{2.2} +\] + +\[ +P_{8,p}(r=2)=P_{4,1-p}(r=0), \tag{2.3} +\] + +and rank one is preserved. + +Hence exactly + +\[ +\boxed{\chi_8(p)=\chi_4(1-p),} \tag{2.4} +\] + +\[ +\boxed{\theta_8(p)=-\theta_4(1-p).} \tag{2.5} +\] + +The stronger rank-one component theorem gives preservation of count and slope under complement, so + +\[ +\boxed{H_{8,p}(z)=H_{4,1-p}(z)} \tag{2.6} +\] + +and the full slope-marked neutral law is transported unchanged. + +Thus the exact charge-neutral involution is + +\[ +\boxed{(\chi,\theta,H,L)_8(p) +=(\chi,-\theta,H,L)_4(1-p).} \tag{2.7} +\] + +## 3. Same-occupied-set graph inclusion orders the fugacity + +For a fixed occupied set, the matching graph contains the NN graph. Therefore its ambient rank is at least the NN rank. + +At the probability level, + +\[ +P_{8,p}(r=2)\ge P_{4,p}(r=2), \tag{3.1} +\] + +\[ +P_{8,p}(r=0)\le P_{4,p}(r=0). \tag{3.2} +\] + +Consequently + +\[ +\boxed{\theta_8(p)\ge\theta_4(p).} \tag{3.3} +\] + +For ordinary square tori where the extra matching diagonals produce a strict enhancement with positive probability, the inequality is strict in the interior. + +## 4. Exact reflection dominance for the NN charge field + +Combine (2.5) and (3.3): + +\[ +-\theta_4(1-p)\ge\theta_4(p). \tag{4.1} +\] + +Therefore + +\[ +\boxed{\theta_4(p)+\theta_4(1-p)\le0.} \tag{4.2} +\] + +This is the charge-field version of the earlier probability inequalities + +\[ +P_2(1-p)\le P_0(p), \tag{4.3} +\] + +\[ +M(p)+M(1-p)\le0. \tag{4.4} +\] + +It is stronger conceptually because `theta` is the coordinate whose zero is **exactly** the matching root. + +At `p=1/2`, + +\[ +\boxed{\theta_4(1/2)\le0.} \tag{4.5} +\] + +Since `theta_4` is strictly increasing, the finite NN balance root obeys + +\[ +\boxed{p_*\ge1/2,} \tag{4.6} +\] + +with strict inequality under strict matching enhancement. + +## 5. Constraint on any scaling crossover + +Suppose a family of finite tori admits a charge-field scaling limit after a thermal reparameterization `x=x_w(p)`. If complement reflection acts asymptotically as `x -> x^c` under `p->1-p`, then the two graph scaling functions must satisfy + +\[ +\boxed{\Theta_8(x)= -\Theta_4(x^c).} \tag{5.1} +\] + +Graph inclusion additionally requires the appropriately aligned inequality + +\[ +\Theta_8(x)\ge\Theta_4(x). \tag{5.2} +\] + +A proposed common-window ansatz that violates these two relations is incompatible with finite topology regardless of how well it fits marginal count data. + +For a self-matching control model, the involution would reduce to an oddness condition around the self-dual parameter. The square-site NN/matching pair is not self-matching, so no such extra identification is assumed here. + +## 6. Boundary + +All identities through Section 4 are exact finite-lattice statements. Section 5 is only the induced constraint on any future scaling limit. No critical scaling function is asserted. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/charge-neutral-crossover-coordinates-20260914.md b/docs/manuscripts/geometric-balance/charge-neutral-crossover-coordinates-20260914.md new file mode 100644 index 000000000..09d724230 --- /dev/null +++ b/docs/manuscripts/geometric-balance/charge-neutral-crossover-coordinates-20260914.md @@ -0,0 +1,350 @@ +# Exact charge-neutral coordinates for the common black/white crossover + +2026-09-14. Exact finite reparameterization of the same-parameter joint law. This is a direct answer to the “minimal topology-compatible family” part of #767 before any near-critical scaling hypothesis is introduced. + +## 1. Exact support + +At one common parameter, with black NN and complementary white matching on the same labels, + +\[ +(W_4,W_8)\in\{(0,1),(1,0),(K,K):K\ge1\}. \tag{1.1} +\] + +Equivalently define the bounded topological charge + +\[ +D=W_4-W_8=r_4-1\in\{-1,0,1\}. \tag{1.2} +\] + +The charged sectors `D=-1,+1` have `K=0`; the neutral sector `D=0` has `K>=1`. + +## 2. Charged susceptibility and charge fugacity + +Write + +\[ +\boxed{\chi(p)=P_0(p)+P_2(p)=1-P_1(p)} \tag{2.1} +\] + +for the total charged-sector probability, and + +\[ +\boxed{\theta(p)=\log\frac{P_2(p)}{P_0(p)}} \tag{2.2} +\] + +for the log charge odds. + +For every interior `00` + +The two birth windows stay separated. In the plateau between them, + +\[ +\chi\to0, \tag{9.1} +\] + +while `theta` still passes through zero at the finite matching root. The common-parameter law is overwhelmingly neutral. + +### Fixed macroscopic aspect ratio + +There is no exponentially long rank-one plateau forced by geometry. `chi` may remain `O(1)` through the critical crossover, and the neutral count law need not be a rare-component Poisson law. + +### Intermediate `rho->infinity`, `log rho=o(w)` + +This is the natural merging regime for a nontrivial crossover: `chi`, `theta`, and `H_p` may all have nondegenerate scaling limits. Any proposed scaling function should be stated for these three objects (or the slope-marked extension), not for two unconstrained colour intensities. + +## 10. A concrete crossover target + +A minimal conditional programme is therefore: + +\[ +\chi_w(p)\to\Chi(x,\rho), +\qquad +\theta_w(p)\to\Theta(x,\rho), +\qquad +H_{w,p}(z)\to\mathcal H_{x,\rho}(z), \tag{10.1} +\] + +under a declared near-critical scaling variable `x` and aspect parameter `rho`. + +The exact constraints are + +\[ +0\le\Chi\le1, +\qquad +\Theta\text{ monotone in the thermal parameter}, \tag{10.2} +\] + +`H` is a PGF on positive integers, and the separated-window limits are (8.3)--(8.4). + +The matching root in the scaling theory is simply + +\[ +\boxed{\Theta=0.} \tag{10.3} +\] + +This is the smallest state space in which a common-window crossover can live without violating finite topology. + +## 11. Claim boundary + +Sections 1--9 are exact finite identities plus already-proved separated-window boundary conditions. Section 10 is a conditional scaling programme. No square-site near-critical universality function is asserted, and no fixed-p prefactor is analytically continued to criticality here. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/charge-sector-relaxation-gap-20260914.md b/docs/manuscripts/geometric-balance/charge-sector-relaxation-gap-20260914.md new file mode 100644 index 000000000..f941497b8 --- /dev/null +++ b/docs/manuscripts/geometric-balance/charge-sector-relaxation-gap-20260914.md @@ -0,0 +1,144 @@ +# The first charge-sector relaxation gap appears to be 2 pi / w + +2026-09-14. Small-width spectral control for the corrected finite-aspect charge crossover. + +## 1. Observable + +At the fixed-width charge-coexistence point `p_w^ch`, let + +\[ +\lambda_{1,G,w}>|\lambda_{2,G,w}|\ge\cdots \tag{1.1} +\] + +be the leading spectral moduli of the transparent safe-homology transfer kernel for graph `G`. Define the dimensionless relaxation gap + +\[ +\boxed{ +g_{G,w} +=w\log\frac{\lambda_{1,G,w}}{|\lambda_{2,G,w}|}.} \tag{1.2} +\] + +If critical transfer scaling applies, `g_{G,w}` should converge to `2 pi Delta x`, where `Delta x` is the first scaling-dimension gap inside the corresponding magnetic/topological sector. + +## 2. Direct spectrum + +Sparse Arnoldi evaluation of the same safe kernels used for the charge roots gives + +| `w` | `g_4,w` | `g_8,w` | +|---:|---:|---:| +| 4 | 7.1077281 | 5.5172789 | +| 5 | 6.7138550 | 5.9713148 | +| 6 | 6.5406250 | 6.1426213 | +| 7 | 6.4512010 | 6.2165936 | +| 8 | 6.4004937 | 6.2516436 | + +The two sequences approach from opposite sides and strongly bracket + +\[ +\boxed{2\pi=6.283185307179586\ldots}. \tag{2.1} +\] + +The simplest interpretation is + +\[ +\boxed{\Delta x=1.} \tag{2.2} +\] + +That is exactly the spacing of a level-one conformal descendant above a primary magnetic state on the cylinder. + +This is a much less speculative identification than the separate Kac-(4,2) hypothesis for the **sector-odd irrelevant correction**: the present gap concerns ordinary relaxation *within* each magnetic sector, not the tiny difference between the two sectors. + +## 3. Consequence for finite aspect + +For a periodic transfer trace, subleading contamination at aspect ratio + +\[ +\rho=m/w \tag{3.1} +\] + +is therefore expected to scale as + +\[ +\boxed{e^{-2\pi\rho}} \tag{3.2} +\] + +up to amplitudes and further gaps. + +Combining this with the charge thermal slope + +\[ +\Theta'_w\asymp w^{-1/4} \tag{3.3} +\] + +and the derivative factor `m`, the corrected finite-length root displacement has scale + +\[ +\boxed{ +|p^*_{w,m}-p_w^{ch}| +\sim \frac{e^{-2\pi\rho}}{\rho\,w^{3/4}}} \tag{3.4} +\] + +unless an additional symmetry cancels the first descendant contribution from the **difference** of the two periodic traces. + +Equation (3.4) is a falsifiable target, not a theorem of the present safe kernel alone. + +## 4. When is the intrinsic w^-4 shift visible? + +The semi-infinite pseudo-critical displacement is + +\[ +p_c-p_w^{ch}\asymp w^{-4}. \tag{4.1} +\] + +Demanding the finite-length term (3.4) be asymptotically smaller gives + +\[ +\frac{e^{-2\pi\rho}}\rho\ll w^{-13/4}. \tag{4.2} +\] + +If + +\[ +\rho=C\log w, \tag{4.3} +\] + +then a sufficient leading-power condition is + +\[ +\boxed{C>\frac{13}{8\pi}=0.517253\ldots.} \tag{4.4} +\] + +So only logarithmic aspect growth may be needed to expose the semi-infinite `w^-4` correction on periodic tori. + +This is qualitatively different from a generic open-boundary amplitude mismatch, which would give a `1/m` correction without the exponential factor. + +## 5. Possible extra cancellation + +The primal and complementary matching sectors approach the same continuum magnetic module. It is therefore possible that not only their leading primary eigenvalues but also the first descendant trace amplitudes agree in the **sector difference** at criticality. If so, the actual periodic-root correction would begin at `e^{-4 pi rho}` or another higher gap. + +This should be tested directly by constructing the exact periodic topological transfer blocks or by comparing finite-`m` roots at fixed `w`. The current safe-kernel spectrum shows that a `2 pi/w` relaxation mode exists; it does not by itself prove that this mode survives subtraction of the two torus traces. + +## 6. New hierarchy of spectral questions + +The charge programme now separates three distinct spectra. + +1. **Primary magnetic gap:** + \[ + w I^0_G(p_c)\to2\pi(5/48). + \] +2. **Within-sector relaxation:** + \[ + w\log(\lambda_1/|\lambda_2|)\to2\pi. + \] +3. **Primal/dual sector-odd mismatch:** + conjecturally + \[ + I^0_4(p_c)-I^0_8(1-p_c)\asymp w^{-17/4}, + \] + with the tentative `(4,2)` LCFT interpretation recorded separately. + +Confusing these three is exactly what makes the fast `w^-4` pseudo-critical convergence look mysterious. + +## 7. Claim boundary + +The finite spectral numbers in section 2 are direct outputs of the transparent safe transfer. The limit `2 pi`, the descendant interpretation, and the use of that gap in the exact periodic topological trace difference are scaling conjectures to be checked at larger widths / with the precise periodic sector transfer. No claim is made that the safe-kernel second eigenvalue is already a certified eigenvalue of Jacobsen's reduced TL block, although the matching leading eigenvalue and root sequence strongly support the shared sector semantics. diff --git a/docs/manuscripts/geometric-balance/charge-sector-scaling-hierarchy-20260914.md b/docs/manuscripts/geometric-balance/charge-sector-scaling-hierarchy-20260914.md new file mode 100644 index 000000000..84e65b2aa --- /dev/null +++ b/docs/manuscripts/geometric-balance/charge-sector-scaling-hierarchy-20260914.md @@ -0,0 +1,195 @@ +# Five scaling layers in the fixed-width charge transfer + +2026-09-14. Integration note. The purpose is to keep several numerically visible exponents from being assigned to the same physical mechanism. + +The first and fourth layers use standard percolation CFT/thermal scaling. The second and third are strong transfer diagnostics. The fifth is now a **rotational sector-odd anisotropy conjecture**; the earlier scalar Kac-(4,2) assignment was demoted after comparing square and kagome symmetry selection. + +## 1. Magnetic primary: x_m=5/48 + +At criticality the NN-safe and complementary-matching-safe sectors both propagate the magnetic/topological excitation. Thus + +\[ +I^0_{G,w}(p_c)\sim\frac{2\pi x_m}{w}, +\qquad x_m=5/48. \tag{1.1} +\] + +The transparent transfer gives + +| `w` | `w I4` | `w I8` | +|---:|---:|---:| +| 4 | 0.6808677452 | 0.6677642268 | +| 5 | 0.6702211440 | 0.6643381257 | +| 6 | 0.6651785592 | 0.6620431213 | +| 7 | 0.6622847725 | 0.6604282152 | +| 8 | 0.6604615126 | 0.6592750557 | + +against + +\[ +2\pi x_m=0.6544984694978736\ldots. \tag{1.2} +\] + +This is the common primary sector and is the reason the eigenvalue identity can converge unusually fast. + +## 2. Common sector-even irrelevant correction: candidate x=4 + +Average the two magnetic gaps: + +\[ +\bar I_w=\frac12\left[I^0_{4,w}(p_c)+I^0_{8,w}(1-p_c)\right].\tag{2.1} +\] + +The diagnostic + +\[ +w^2\left[w\bar I_w-2\pi x_m\right] \tag{2.2} +\] + +has values + +```text +w=4 0.31708 +w=5 0.31953 +w=6 0.32805 +w=7 0.33604 +w=8 0.34367 +``` + +so a correction + +\[ +I_w=\frac{2\pi x_m}{w}+C_{even}w^{-3}+\cdots \tag{2.3} +\] + +is natural. In CFT language a per-row correction `w^(1-x)` with exponent `-3` corresponds to an irrelevant dimension + +\[ +\boxed{x_{even}=4.} \tag{2.4} +\] + +A standard identity-family / lattice-anisotropy scalar combination is a natural source. The crucial matching-method point is not the exact operator name: this lower-dimensional correction appears **common to both magnetic sectors** and therefore cancels strongly from their difference. + +The `x=4` interpretation is a scaling diagnosis, not a matrix-element proof. + +## 3. Within-sector relaxation: Delta x=1 + +The second eigenvalue inside each safe magnetic kernel gives + +\[ +g_{G,w}=w\log(\lambda_1/|\lambda_2|). \tag{3.1} +\] + +For `w=4,...,8`, the NN values decrease toward `2 pi` and the matching values increase toward `2 pi`. This suggests + +\[ +\boxed{\Delta x_{relax}=1,} \tag{3.2} +\] + +consistent with a level-one descendant in the same magnetic module. + +This gap controls convergence in the **longitudinal aspect ratio** and should not be confused with the irrelevant correction that shifts the pseudo-critical width sequence. + +## 4. Thermal response: x_t=5/4 + +The charge-sector difference is + +\[ +\Theta_w(p)=I^0_{4,w}(p)-I^0_{8,w}(1-p). \tag{4.1} +\] + +Differentiation in the thermal parameter couples to the percolation thermal field + +\[ +x_t=5/4,\qquad y_t=2-x_t=3/4. \tag{4.2} +\] + +Hence an excitation energy per row has derivative + +\[ +\boxed{\Theta'_w\asymp w^{y_t-1}=w^{-1/4}.} \tag{4.3} +\] + +The same power follows from the exact safe-sector pivotal-density identity plus the four-arm scaling mechanism. + +## 5. Sector-odd rotational correction: conjectural spin 4, x=21/4 + +The difference at the critical point is far smaller than the average correction: + +\[ +\Theta_w(p_c)\asymp w^{-17/4}. \tag{5.1} +\] + +A per-row correction `w^(1-x)` therefore points to + +\[ +\boxed{x_{odd}=21/4=x_t+4.} \tag{5.2} +\] + +The current leading interpretation is **not** a scalar Kac-(4,2) field. Jacobsen finds exponent four on the square lattice but the corresponding amplitude vanishes on kagome, where the leading correction is exponent six; he explicitly suggests rotational symmetry as the reason. + +This strongly favors a lattice-anisotropy selection rule. The concrete conjecture is: + +\[ +\boxed{ +\text{square: first primal/dual-odd correction is spin }4 +\text{ in the thermal family, }x=x_t+4=21/4.} \tag{5.3} +\] + +Square `C4` permits the real spin `+/-4` combination. Kagome/triangular `C3/C6` does not; with inversion/reflection the next allowed rotational thermal-family correction is spin six, + +\[ +x=x_t+6=29/4, \tag{5.4} +\] + +which yields pseudo-critical exponent six. + +Dividing the square sector-odd mismatch by the thermal response gives + +\[ +\frac{w^{-17/4}}{w^{-1/4}}=w^{-4}. \tag{5.5} +\] + +This rotational interpretation is recorded in `sector-odd-spin4-anisotropy-20260914.md`. The old scalar Kac-(4,2) note now explicitly records its demotion. + +## 6. Why lower irrelevant dimensions do not contradict the n^-4 root shift + +A common source of confusion is to say: if an `x=4` irrelevant field exists, why does the pseudo-critical root not shift with exponent + +\[ +x=4\quad\Rightarrow\quad 4-x_t=11/4 ? \tag{6.1} +\] + +The answer is that the root uses the **difference** of the primal and dual magnetic-sector energies. A sector-even correction can be large in each energy and still disappear from + +\[ +I^0_4-I^0_8. \tag{6.2} +\] + +The transfer data show exactly this hierarchy: + +```text +individual/common correction >> sector difference. +``` + +Thus the relevant question for the matching root is not “what is the leading irrelevant field of percolation?” but + +> what is the lowest-dimensional lattice-symmetry-allowed field whose matrix-element difference is ODD under exchange of the two topological magnetic sectors? + +The current answer-candidate is a thermal-family spin-four anisotropy on the square lattice, upgraded to spin six on a six-fold lattice. + +## 7. Summary table + +| phenomenon | scale in per-row transfer | proposed dimension/spacing | role | +|---|---:|---:|---| +| magnetic primary | `w^-1` | `x_m=5/48` | common leading sector | +| common irrelevant | `w^-3` | `x≈4` | sector-even; cancels in difference | +| longitudinal relaxation | gap `2pi/w` | `Delta x=1` | aspect convergence | +| thermal derivative | `w^-1/4` | `x_t=5/4` | root susceptibility / pivotal density | +| square sector-odd mismatch | `w^-17/4` | conjectural spin 4, `x=x_t+4=21/4` | produces `w^-4` root shift | +| six-fold sector-odd mismatch | `w^-25/4` | conjectural spin 6, `x=x_t+6=29/4` | produces `w^-6` root shift | + +Assigning all of these scales to one generic “irrelevant exponent” would erase the mechanism. + +## 8. Claim boundary + +The finite transfer numbers are direct controls. The primary magnetic interpretation is the same CFT argument already used by Jacobsen. `x≈4` and `Delta x=1` are strong scaling diagnoses. The sector-odd spin-four/spin-six thermal-family assignment is a targeted conjecture motivated by the square/kagome symmetry contrast; it still requires an operator/sector matrix-element proof. diff --git a/docs/manuscripts/geometric-balance/charge-slope-pivotal-density-20260914.md b/docs/manuscripts/geometric-balance/charge-slope-pivotal-density-20260914.md new file mode 100644 index 000000000..454f80b41 --- /dev/null +++ b/docs/manuscripts/geometric-balance/charge-slope-pivotal-density-20260914.md @@ -0,0 +1,172 @@ +# Charge-free-energy slope as a safe-sector pivotal density + +2026-09-14. Exact finite-width identity plus a critical-scaling interpretation. + +This note connects the fixed-width charge transfer to the repository's existing pivotal/arm-analysis machinery. The algebraic identity is exact. The final `w^-1/4` arm interpretation uses critical scaling and is not promoted as a rigorous square-site exponent theorem. + +## 1. Finite strip event and Russo derivative + +For graph `G=G4` or `G8`, circumference `w`, and open strip height `m`, let + +\[ +A_{w,m}^G=\{\text{no horizontally essential occupied component}\}, +\qquad +Q_{w,m}^G(p)=P_p(A_{w,m}^G). \tag{1.1} +\] + +This is a decreasing event of the `wm` independent site variables. Russo's formula therefore gives + +\[ +\boxed{ +\frac{d}{dp}Q_{w,m}^G(p) +=-\sum_{v}P_p(v\text{ is pivotal for }A_{w,m}^G).} \tag{1.2} +\] + +Consequently + +\[ +-\frac1m\frac{d}{dp}\log Q_{w,m}^G(p) += rac1m\sum_v +\frac{P_p(v\text{ pivotal})}{Q_{w,m}^G(p)}. \tag{1.3} +\] + +The finite-state Perron representation makes the `m->infinity` limit analytic, so + +\[ +\boxed{ +(I^0_{G,w})'(p) +=\lim_{m\to\infty} +\frac1m\sum_v +\frac{P_p(v\text{ pivotal})}{Q_{w,m}^G(p)}.} \tag{1.4} +\] + +Thus the derivative of the topological void free energy is literally a pivotal intensity per transfer row, normalized by survival in the safe sector. + +## 2. Safe-conditioned form + +The pivotal event for site `v` depends only on the other site variables. Since `A` is decreasing, on a pivotal outside configuration the event `A` holds exactly when `v` is closed. Therefore + +\[ +P_p(A\cap\{v\text{ pivotal}\}) +=(1-p)P_p(v\text{ pivotal}). \tag{2.1} +\] + +Writing `R^G_{w,m}` for the number of pivotal sites in the strip, + +\[ +\boxed{ +-\partial_p\log Q_{w,m}^G(p) +=\frac1{1-p}E_p[R^G_{w,m}\mid A_{w,m}^G].} \tag{2.2} +\] + +Hence the infinite-strip safe/Q-process has a well-defined pivotal density per row + +\[ +r^0_{G,w}(p) +:=\lim_{m\to\infty}\frac1mE[R^G_{w,m}\mid A_{w,m}^G], \tag{2.3} +\] + +and + +\[ +\boxed{(I^0_{G,w})'(p)=\frac{r^0_{G,w}(p)}{1-p}.} \tag{2.4} +\] + +This is the pivotal version of the Perron-eigenvector derivative. + +## 3. Equivalence with the safe-row occupation deficit + +Let `bar K^0_{G,w}(p)` be the mean number of occupied sites in one added row under the Perron/Doob safe phase. Differentiating Bernoulli row weights gives independently + +\[ +(I^0_{G,w})'(p) +=\frac{wp-\bar K^0_{G,w}(p)}{p(1-p)}. \tag{3.1} +\] + +Equating (2.4) and (3.1) yields the exact identity + +\[ +\boxed{ +wp-\bar K^0_{G,w}(p)=p\,r^0_{G,w}(p).} \tag{3.2} +\] + +So the conditioned safe phase is depleted relative to an ordinary Bernoulli row by exactly `p` times its topological pivotal density. + +This gives a useful semantic check on the left/right Perron vectors: their occupation bias is not an arbitrary spectral statistic; it measures how many sites per row could destroy safe homology if opened. + +## 4. Charge slope at coexistence + +At black density `p` and complementary matching density `q=1-p`, + +\[ +\Theta_w(p)=I^0_{4,w}(p)-I^0_{8,w}(q). \tag{4.1} +\] + +Differentiating in black `p`, + +\[ +\Theta'_w(p) +=(I^0_{4,w})'(p)+(I^0_{8,w})'(q). \tag{4.2} +\] + +Using (2.4) separately in the two safe phases, + +\[ +\boxed{ +\Theta'_w(p) +=\frac{r^0_{4,w}(p)}{q} ++\frac{r^0_{8,w}(q)}{p}.} \tag{4.3} +\] + +Equivalently, from (3.2), + +\[ +\boxed{ +pq\,\Theta'_w(p) +=p\,r^0_{4,w}(p)+q\,r^0_{8,w}(q) +=w-\bar K^0_{4,w}(p)-\bar K^0_{8,w}(q).} \tag{4.4} +\] + +At the charge root this is the exact thermal response that divides the critical sector mismatch to produce the pseudo-critical shift. + +## 5. Critical arm interpretation + +A site that is pivotal for creation of horizontal homology must connect macroscopically distinct occupied/vacant topological channels. In an ordinary critical bulk picture this is a four-arm-type local event. The standard percolation four-arm exponent is + +\[ +x_4=5/4. \tag{5.1} +\] + +A row contains `w` possible pivotal locations, suggesting + +\[ +r^0_{G,w}(p_c)\asymp w\,\pi_4(w) +\asymp w^{1-5/4}=w^{-1/4}. \tag{5.2} +\] + +This is exactly the power observed in the safe-transfer control, + +\[ +\Theta'_w(p_w^{ch})w^{1/4} +=3.4851,3.4479,3.4262,3.4124,3.4031 \tag{5.3} +\] + +for `w=4,...,8`. + +Equation (5.2) should be read as a scaling mechanism, not as a completed square-site proof. The conditioning on survival in the magnetic/topological safe sector must be shown not to change the bulk thermal exponent; in CFT language this is precisely the statement that differentiating the magnetic-sector energy couples to the ordinary thermal field. + +## 6. Practical analysis route + +This identity suggests a much cheaper way to interrogate the charge slope than numerical finite differences of eigenvalues. + +1. Work in the Perron/Doob safe chain at the charge root. +2. Mark whether each site of the new row is topologically pivotal for safe survival. +3. Measure the two conditional pivotal densities `r^0_4,r^0_8`. +4. Verify (4.3) against the eigenvalue derivative. +5. Study the local multi-arm geometry of those pivotal samples. + +The repository already contains several C4/pivotal analysis tools; the missing step is to retarget them to the **safe-sector Q-process** rather than an unrelated unconditioned local source. + +## 7. Claim boundary + +Equations (1.2)--(4.4) are exact consequences of Russo's formula, Bernoulli score differentiation, and the fixed-width Perron limit. The identification of the asymptotic power with a four-arm exponent is a critical-scaling hypothesis for square-site percolation, albeit one that matches both the transfer data and the standard thermal exponent `y_t=3/4`. diff --git a/docs/manuscripts/geometric-balance/charged-susceptibility-and-flat-root-20260914.md b/docs/manuscripts/geometric-balance/charged-susceptibility-and-flat-root-20260914.md new file mode 100644 index 000000000..51ad4b15a --- /dev/null +++ b/docs/manuscripts/geometric-balance/charged-susceptibility-and-flat-root-20260914.md @@ -0,0 +1,317 @@ +# Magnetic charged susceptibility and the quantitative flattening of the balance CDF + +2026-09-14. Integration of exact charge coordinates, fixed-width transfer, and the already-merged Pinson--Arguin primitive-sector continuum formula. + +The key phenomenon is a product of two opposite effects: + +- the **conditional charge odds** cross zero increasingly sharply with width/aspect; +- the total probability of being in either charged endpoint sector becomes exponentially small in aspect. + +Their product is the ordinary CDF slope at the balance root. This gives a quantitative mechanism for “balance without concentration.” + +## 1. Exact finite identity + +Recall + +\[ +\chi=P_0+P_2, +\qquad +\theta=\log(P_2/P_0), \tag{1.1} +\] + +and + +\[ +M=\chi\tanh(\theta/2), +\qquad +F=(1+M)/2. \tag{1.2} +\] + +At the matching root `theta=0`, + +\[ +\boxed{ +F'(p_*)=\frac14\chi(p_*)\theta'(p_*).} \tag{1.3} +\] + +Thus a root can be uniquely selected by a huge conditional odds derivative even while the unconditional CDF is nearly flat, if `chi` is sufficiently small. + +## 2. Continuum critical charged susceptibility on a long rectangle + +Take the critical continuum torus + +\[ +\tau=i\rho, +\qquad \rho\to\infty. \tag{2.1} +\] + +The merged Pinson--Arguin primitive-sector evaluator gives for the horizontal primitive rank-one sector `{1,0}` + +\[ +\pi_{i\rho}(\{1,0\}) +=\frac{ +\theta_3(i\rho/6)-\theta_3(3i\rho/2)-2\theta_2(3i\rho/2) +}{2|\eta(i\rho)|^2}. \tag{2.2} +\] + +Use + +\[ +|\eta(i\rho)|^2 +=e^{-\pi\rho/6}[1+O(e^{-2\pi\rho})], \tag{2.3} +\] + +\[ +\theta_3(i\rho/6) +=1+2e^{-\pi\rho/6}+O(e^{-2\pi\rho/3}), \tag{2.4} +\] + +\[ +\theta_3(3i\rho/2)=1+O(e^{-3\pi\rho/2}), \tag{2.5} +\] + +\[ +\theta_2(3i\rho/2) +=2e^{-3\pi\rho/8}[1+O(e^{-3\pi\rho})]. \tag{2.6} +\] + +Then + +\[ +\boxed{ +\pi_{i\rho}(\{1,0\}) +=1-2e^{-5\pi\rho/24}+O(e^{-\pi\rho/2}).} \tag{2.7} +\] + +All nonhorizontal primitive rank-one sectors are `O(poly(rho)e^{-pi rho/2})` or smaller. Therefore the total rank-one probability satisfies + +\[ +P_1^{cont}(i\rho) +=1-2e^{-5\pi\rho/24}+O(\operatorname{poly}(\rho)e^{-\pi\rho/2}).\tag{2.8} +\] + +At `Q=1`, the trivial and cross topological sector probabilities are equal. Hence + +\[ +P_0^{cont}=P_2^{cont}=\frac{1-P_1^{cont}}2, \tag{2.9} +\] + +and the charged susceptibility is + +\[ +\boxed{ +\chi_{cont}(\rho) +=P_0^{cont}+P_2^{cont} +=2e^{-5\pi\rho/24}[1+o(1)].} \tag{2.10} +\] + +The exponent is + +\[ +\frac{5\pi}{24} +=2\pi\frac5{48} +=2\pi x_m, \tag{2.11} +\] + +exactly the magnetic cylinder gap used in the Jacobsen sector argument. + +The leading coefficient `2` is fixed by the continuum wrapping formula; it is not a fitted transfer amplitude. + +## 3. Fixed-width transfer version + +At fixed `w`, let `p_w^{ch}` be the semi-infinite charge root and + +\[ +I_w^{ch} +=I^0_{4,w}(p_w^{ch}) +=I^0_{8,w}(1-p_w^{ch}). \tag{3.1} +\] + +For the periodic topological trace, simple dominant eigenvalues give at large longitudinal length `m` + +\[ +P_0\sim e^{-mI_w^{ch}}, +\qquad +P_2\sim e^{-mI_w^{ch}}, \tag{3.2} +\] + +up to subleading trace eigenvalues. Thus + +\[ +\boxed{ +\chi(p_w^{ch}) +\sim2e^{-mI_w^{ch}}.} \tag{3.3} +\] + +Similarly + +\[ +\theta'(p_w^{ch}) +\sim m\Theta'_w(p_w^{ch}). \tag{3.4} +\] + +Insert these into the exact factorization (1.3): + +\[ +\boxed{ +F'(p_w^{ch}) +\sim\frac12\,m\Theta'_w(p_w^{ch}) +\,e^{-mI_w^{ch}}.} \tag{3.5} +\] + +This is the fixed-width large-aspect quantitative version of balance without concentration. + +The trace-amplitude coefficient one in (3.2) is the same periodic-sector issue discussed in `finite-aspect-charge-root-crossover-20260914.md`; if an exact finite rank transfer introduces a different sector normalization, (3.3)--(3.5) should be rechecked at that interface. + +## 4. Continuum fixed-aspect scaling + +Critical finite-size scaling gives + +\[ +I_w^{ch} +=\frac{2\pi x_m}{w}+o(w^{-1}), \tag{4.1} +\] + +and + +\[ +\Theta'_w(p_w^{ch}) +\sim A_\theta w^{-1/4}. \tag{4.2} +\] + +Take + +\[ +m=\rho w \tag{4.3} +\] + +with `rho` fixed while `w->infinity`, and only afterwards let `rho` become large. Equations (3.5), (4.1), (4.2) give + +\[ +\boxed{ +w^{-3/4}F'(p_*) +\sim\frac{A_\theta}{2}\, +\rho e^{-2\pi x_m\rho}.} \tag{4.4} +\] + +For square-site percolation the current safe-transfer controls suggest + +\[ +A_\theta\approx3.4 \tag{4.5} +\] + +in the raw Bernoulli `p` coordinate, so the prefactor in (4.4) is roughly `1.7` before final wide-width extrapolation. + +In logit coordinate `h`, the corresponding first derivative amplitude is about `0.82` in the present widths. + +## 5. Why odds sharpen while the CDF flattens + +At large `rho`, + +\[ +\theta'(p_*) +\asymp \rho w^{3/4}, \tag{5.1} +\] + +so the CONDITIONAL odds between the two charged sectors rotate through the root ever more steeply. + +But + +\[ +\chi(p_*) +\asymp e^{-2\pi x_m\rho}, \tag{5.2} +\] + +so almost no configurations lie in either charged sector at all. Therefore + +\[ +F'(p_*) +\asymp \rho w^{3/4}e^{-2\pi x_m\rho}, \tag{5.3} +\] + +which decays exponentially in aspect after removing the ordinary thermal `w^(3/4)` factor. + +This is the precise sense in which + +```text +charge odds: sharp +unconditioned CDF: flat +``` + +can occur simultaneously. + +## 6. Aspect of maximal normalized CDF slope + +The large-`rho` envelope + +\[ +\rho e^{-2\pi x_m\rho} \tag{6.1} +\] + +has its maximum at + +\[ +\boxed{ +\rho_{max}=\frac1{2\pi x_m} +=\frac{24}{5\pi} +\approx1.5279.} \tag{6.2} +\] + +The asymptotic formula itself is not expected to be quantitatively accurate all the way down to `rho≈1.5`, but (6.2) highlights that increasing aspect beyond a modest value can already reduce the median density even though the conditional charge sign becomes steeper. + +## 7. Important nonuniformity for exponential aspect + +Do **not** insert an exponentially growing + +\[ +\rho=e^{dw+o(w)}/w \tag{7.1} +\] + +directly into the continuum leading term (4.1). The correction + +\[ +I_w^{ch} +=\frac{2\pi x_m}{w}+Cw^{-3}+\cdots \tag{7.2} +\] + +is tiny per row but is multiplied by `m`; when `rho` is exponential, it changes the exponent by an enormous amount. + +For exponential aspect the correct statement is the fixed-width one (3.5): + +\[ +\boxed{ +F'(p_*)\sim\frac12m\Theta'_w e^{-mI_w^{ch}},} \tag{7.3} +\] + +using the actual finite-`w` charge gap. The continuum formula (4.4) is a fixed-/moderately-growing-aspect scaling statement, not a uniform exponential-aspect approximation. + +This nonuniformity is another example of why tiny finite-width free-energy corrections cannot be multiplied by an exponentially long direction without audit. + +## 8. Interface to #767 + +The common-window charge coordinates now have explicit large-aspect boundary behavior: + +\[ +\chi(0,\rho)\sim2e^{-2\pi x_m\rho}, \tag{8.1} +\] + +while the dual-odd thermal scaling note proposes + +\[ +\theta(X,\rho)\approx\rho\mathcal F(X). \tag{8.2} +\] + +Together, + +\[ +M(X,\rho) +=\chi(X,\rho)\tanh[\theta(X,\rho)/2] \tag{8.3} +\] + +is a topology-compatible two-variable crossover representation with both its neutral plateau weight and its charge sign mechanism explicitly separated. + +This is much more constrained than a generic bivariate Poisson/copula ansatz. + +## 9. Claim boundary + +The Pinson--Arguin expansion (2.7)--(2.10) is an analytic consequence of the already-merged continuum primitive-sector formula. The exact finite factorization (1.3) is rigorous. The square-site simultaneous finite-size scaling in sections 4--5 uses standard CFT/thermal assumptions and the safe-transfer amplitude sequence. Section 7 explicitly states the failure of uniformity for exponential aspect. diff --git a/docs/manuscripts/geometric-balance/circular-coalescent-and-topological-clock-20260914.md b/docs/manuscripts/geometric-balance/circular-coalescent-and-topological-clock-20260914.md new file mode 100644 index 000000000..40c057b3e --- /dev/null +++ b/docs/manuscripts/geometric-balance/circular-coalescent-and-topological-clock-20260914.md @@ -0,0 +1,124 @@ +# 白簇的圆周谱系：Kingman 跳链、孔洞剪枝与最后一个拓扑时钟 + +2026-09-14。接 `monotone-cut-mark-filtration-20260914.md`，不重算前轮最小屏障模板。本文把同一个切分过程从“跟随一个位置”推进到“所有终端白簇的祖先树”。一般 Dirichlet 切分/合并对偶已有 Bertoin–Goldschmidt 的明确结果；这里给出周期拓扑、取样方式、时间和孔洞标记的完整对应。 + +## 1. 三层对象，以及可以使用的实际点渗流极限 + +显式过程是在周长为 L0 的圆周上，放置强度 dz dτ 的 Poisson cuts，以及独立强度 r dz dτ 的孔洞标记。cuts 切开圆周；标记随所在区间分配，不能每个时刻重新生成。 + +前轮的最小模板论证给出两个实际独立 SITE 来源： + +- 固定 w=8，黑色 NN、白色 matching，p=εt，纵坐标 z=ε^8 y，τ=t^8，r=8。 +- 固定 w=4，黑色 matching、白色 NN，p=εt，z=ε^4 y，τ=19t^4，r=4/19。 + +取环面长度 m，使 mε^w→L0，先在任何 0 **provenance 声明（重要）** +> 产出该结果的子代理**在写出笔记前被中断**（运行 3m40s 后被用户取消回合）。 +> 其**脚本已跑完**（墙钟 463.9 s），原始输出 `-raw.json` 与 telemetry 日志 +> 由**操作者原样归档**（文件名带 `raw` / `telemetry`）。 +> **本文由操作者（主代理）依据该原始输出撰写**，只陈述原始输出中已有的字段值； +> **未新增任何计算、未重新拟合**。凡本文的解释性句子都标为【解释】。 + +--- + +## 0. 四句话结论 + +1. **前置命题被证否（精确）**：`Δ_w` 只由 **Perron 根**构成，因此**闭合幅度 `A_r` 无法进入它**。 + 原始输出字段 `identical_Delta_after_amplitude_removal = true`。 + ⇒ **原定的 `H1`（"幅度遮住了 `−17/4`"）与 `H2`（"`−17/4` 被排除"）这个二分本身是设错了**： + 幅度不可能移动 `Δ_w` 的幂次。【精确恒等式 + 数值验证】 +2. **验证方式**：比较 `Δ_w` 与有限 `m` 的迹估计 + `D(m) = −(1/m)log Tr(R_4^m) + (1/m)log Tr(R_8^m)`。 + 原始输出显示 `D(m) − Δ_w → 0` 如 `e^{−mΔ}/m`： + `m=1` 时 `D_m = 0.082922`（差 `0.086198`），`m=32` 时 `D_m = 0.0032758796085423925` + 与 `|Δ_w| = 0.0032758796085428643` 一致到 ~1e-13 。【数值验证】 +3. **幅度解剖（`A_r` 的各分量确实算出来了，只是它进不去 `Δ_w`）**： + `l^T r = 1` 归一化**精确成立**（`lTr_after_normalisation = 1.0`）； + 迹系数 `Tr(R^m)/ρ^m → 1`；开边界/**contact** 幅度 + `A_open = (1^T r)(l^T e_empty)`：`A_open(w=4) = 0.617686 → A_open(w=8) = 0.872497`（NN 扇区）， + `A_open8 = 2.506993 → 15.174843`（matching 扇区）。【有限宽度表】 +4. **外推指数（5 个宽度，`1/w` 外推）**：`Δ_w ~ w^{−4.035}`（rms 0.0112）、 + `Δ'_w ~ w^{−0.2311}`（rms 3.9e-4）、根偏移 `~ w^{−3.803}`（rms 0.0109）。 + ⇒ 外推值 **4.035 更靠近整数 4**，而**不靠近 `17/4 = 4.25`**。 + ⚠️ **这是 5 个宽度的外推，不是指数测量。**【有限宽度拟合】 + +--- + +## 1. 为什么原二分设错了（这一条是本交付的核心） + +前一位子代理（`sector802`）在 `V4` 列出的缺口是 + +> 「`P_r` 的闭合幅度 `A_r`（contact / seam / 归一化）从未计算，它把根与形状移动 `O(1/m)`。」 + +本轮直接检验这句话能否用来移动 `Δ_w`。 + +**结论是不能。** `Δ_w = I⁰_{4,w}(p) − I⁰_{8,w}(1−p)`，而 `I⁰ = −log ρ`、`ρ` 是 safe 子随机转移块的 +**最大特征值（Perron 根）**。**幅度（特征向量的归一化、`contact` 项、`seam` 系数）不出现在 `ρ` 里。** + +原始输出的直接证据： + +``` +"identical_Delta_after_amplitude_removal": true +``` + +以及迹估计的收敛（`trace_dm` 表，节选 w=4）： + +| `m` | `D(m)` | `D(m) − Δ_w` | `coef4 = Tr(R_4^m)/ρ_4^m` | `coef8` | +|---|---|---|---|---| +| 1 | 0.082922313 | 0.086198192 | 1.315555301 | 1.424620257 | +| 2 | 0.052508014 | 0.055783893 | 1.058605217 | 1.168144365 | +| 4 | 0.005043893 | 0.008319773 | 1.001657279 | 1.008766161 | +| 8 | 0.003279642 | 0.006555521 | 1.000001372 | 1.000031470 | +| 16 | 0.003275880 | 0.006551759 | 1.000000000 | 1.000000000 | +| 32 | 0.0032758796085 | 0.006551759 | 1.000000000 | 1.000000000 | + +**迹系数 → 1 正是 `l^T r = 1` 的精确恒等式**；而 `D(m) → Δ_w` 说明 +**幅度类修正在 `m→∞` 时消失**，不会改变 `Δ_w` 的幂次行为。【精确恒等式 + 数值】 +【解释】因此 `H1`/`H2` 需要重设：若 `−17/4` 与 5 个宽度不符，原因**不在** contact/seam/归一化， +而在**别的**（例如有限宽度修正的普适性或该宽度区间的拟合形式）。 + +--- + +## 2. 原始输出中的数字（逐项照抄，未重算） + +### 2.1 每个宽度的根与幅度 + +| `w` | `Δ_w(p_c)` | `Δ'_w(p_c)` | `A_open4` | `A_open8` | `p^ch_w` | `w^4·offset` | 耗时 | +|---|---|---|---|---|---|---|---| +| 4 | −3.2758796085e−03 | 2.4659477621 | 0.617686 | 2.506993 | 0.591417170853 | 0.340193 | 0.2 s | +| 5 | −1.1766036623e−03 | 2.3063319729 | 0.646017 | 3.764601 | 0.592235823205 | 0.318892 | 0.9 s | +| 6 | −5.2257299199e−04 | 2.1894276107 | 0.703049 | 5.885781 | 0.592507356206 | 0.309348 | 6.3 s | +| 7 | −2.6522246900e−04 | 2.0980576358 | 0.777206 | 9.372740 | 0.592619633400 | 0.303528 | 42.6 s | +| 8 | −1.4830712223e−04 | 2.0235972867 | 0.872497 | 15.174843 | 0.592672760575 | 0.300197 | 413.9 s | + +（`p^ch_w` = `I⁰_{4,w}(p) = I⁰_{8,w}(1−p)` 的交叉点；`p_c = 0.5927460507921`。 +`w = 4` 时 `p_c − p^ch_4 = 0.0013288799389615802`。【有限宽度表】） + +### 2.2 局部对数斜率（逐 `w`，未外推） + +``` +local slopes Δ_w : 4.5888 4.4516 4.3996 4.3532 （单调下降） +local slopes Δ'_w : 0.2999 0.2853 0.2765 0.2706 +local slopes offset: 4.2898 4.1667 4.1232 4.0826 +``` + +### 2.3 定标列（看是否常数 —— **都不是常数，都在漂移**） + +``` +w^{17/4}·Δ : −1.185995 −1.099646 −1.059961 −1.035803 −1.021632 ← 漂移 ~14% +w^4·Δ : −0.838625 −0.735377 −0.677255 −0.636799 −0.607466 ← 漂移 ~28% +w^4·offset : 0.340193 0.318892 0.309348 0.303528 0.300197 ← 漂移 ~12% +w^{1/4}·Δ' : 3.487377 3.448771 3.426639 3.412651 3.403271 ← 漂移 ~2.4% +``` + +### 2.4 `1/w` 外推（脚本给出的拟合） + +``` +slope_Delta_1overw : p_inf = 4.0347512749 c = 2.1777980826 rms = 0.0111613468 +slope_Delta_p_1overw : p_inf = 0.2310799605 c = 0.2738891726 rms = 0.0003914594 +slope_offset_1overw : p_inf = 3.8026421547 c = 1.9113610899 rms = 0.0108618002 +``` + +⚠️ **注意 `p_inf` 是斜率量级**（`Δ_w ~ w^{−p_inf}`）。 + +--- + +## 3. 裁定（对原 `W1–W4` 的回答） + +``` +W1 A_r 的定义与可分离性： + 良定义且已算出（归一化 / contact / seam 三类分量都在原始输出里）； + 但它【不能】进入 Δ_w —— Δ_w 只依赖 Perron 根。 +W2 修正后的 Δ_w / Δ'_w 序列与局部斜率： + "修正后" 与 "修正前" 逐位相同（identical_Delta_after_amplitude_removal = true）。 + 局部斜率见表 §2.2；外推见表 §2.4。 +W3 H1 / H2 的裁定： + ⚠️ 两个都不选 —— 【前提设错】。幅度不可能移动 Δ_w 的幂次。 + 可替代的表述：Δ_w 的外推指数 ≈ 4.035，更靠近整数 4，不靠近 17/4 = 4.25。 +W4 修正后局部斜率区间是否含 −17/4 = −4.25： + 含。实测局部斜率 4.5888 → 4.3532（量级）确实跨过 4.25； + 但那是因为它【整体在漂移】，不是因为它收敛到 4.25。 + 外推值 4.035 【不】在最近两个局部斜率（4.3996, 4.3532）之间，也不等于 4.25。 +``` + +--- + +## 4. 不能宣称什么（红线自查） + +1. **不是 exponent measurement。** 5 个宽度 + `1/w` 外推；`#775`/`#802` 的 stop rule 明确禁止这样表述。 +2. **`Δ_w`、`Δ'_w`、根偏移同源**，是**一项数据生产**，不是三份独立证据 + （`#801`：「同一原始系数表衍生的十个诊断是一次数据生产，不是十项独立证据」）。 +3. **`b/m → −Δ/2`、`d/m → +S/2` 引自 `#771`，未独立重算**（沿用前一位子代理的声明）。 +4. **控制表复现是复现别人的表**，不是新数据。 +5. **`A_open` 是开边界（contact）幅度**，其物理含义（是否等于 `#771` 约定下的 `A_r` 本体） + **未在此独立确认** —— 本文只报告原始输出把它算出来了、以及它进不去 `Δ_w`。 +6. 未识别 8-arm / spin±4；未改 `#275` 原始 U 合同；未做 L5–8 阶梯；GitHub 全程未写。 +7. `fits` 组里有三个字段为 `NaN`（`Delta_3param`、`Delta_pure`）， + **日志显示脚本在此处遇到 `log` 的 `invalid value` 警告** ⇒ 那两组拟合**不可用，已如实保留**。 + +--- + +## 5. 原始文件 + +- `results/research-dispatch/closure-amplitude-20260914.json` —— 原始输出（原样归档） +- `results/research-dispatch/closure-amplitude-telemetry-20260914.log` —— 脚本 stdout（含逐 `w` 行与拟合） +- `scripts/closure_amplitude.py` —— 脚本原文 diff --git a/docs/manuscripts/geometric-balance/common-label-marked-poisson-bridge-20260914.md b/docs/manuscripts/geometric-balance/common-label-marked-poisson-bridge-20260914.md new file mode 100644 index 000000000..8c175c7b6 --- /dev/null +++ b/docs/manuscripts/geometric-balance/common-label-marked-poisson-bridge-20260914.md @@ -0,0 +1,425 @@ +# Common-label marked Poisson bridge from the existing component-Poisson proof + +Date: 2026-09-14 + +Status: **candidate author-proof / proof-audit draft**. This note reorganizes estimates already written in `poisson-birth-windows.md` into the parameter-marked statement needed by #780. It does not claim independent verification. The point is to identify whether a genuinely new probability mechanism is missing, or whether the common-label bridge is already latent in the existing AGG/BK proof. + +## 1. Setup + +Fix a compact strictly subcritical interval + +```text +I compactly contained in (0,pc(G)). +``` + +Let `p0 in I`. On the infinite cylinder `C_w x Z`, use one common family of iid labels + +```text +U_v ~ Uniform(0,1), +``` + +and declare a site occupied at parameter `p` when `U_v<=p`. + +Let + +```text +nu_w(p) +``` + +be the once-per-complete-horizontal-winding-component anchor intensity per longitudinal row, with the same lowest-row/tie-break convention as `poisson-birth-windows.md`. + +The reused results are: + +1. for `p` in compact subcritical intervals, + +```text +-log nu_w(p)/w -> kappa(p) +``` + +uniformly along convergent parameter sequences; + +2. with localization height `H=w^2`, the intensity of components whose vertical span exceeds a generic `r>=w` obeys + +```text +rho_w^long(p;r) + := E[# {complete winding C: anchor_y(C)=0, L(C)>r}] + <= C_I w exp(-c_I r); +``` + +3. localized anchor indicators have dependency range `O(wH)` and, for two distinct full winding components in overlapping windows, + +```text +E[I_i I_j] <= U_w(p)^2, +U_w(p)=poly(w,H) exp[-(w-1) kappa(p)], +``` + +by two disjoint occupied winding witnesses and site BK; + +4. disjoint localization windows are functions of disjoint site-label sets, hence the AGG `b3` term is exactly zero. + +No Ornstein--Zernike prefactor, `p`-analyticity, or affine parameter clock is assumed below. + +## 2. Merger factorial intensity + +For `p_- 0 +``` + +whenever `p_-,p_+ -> p0` in a compact subcritical interval. For fixed `p_-H`, then trivially + +```text +n_C(p_-) <= |C| <= w L(C), +``` + +so + +```text +M_w^long + <= (w^2/2) + E sum_{C:anchor_y(C)=0} L(C)^2 1{L(C)>H}. +``` + +Use the discrete tail identity + +```text +L^2 1{L>H} + <= H^2 1{L>H} + + sum_{r>=H} (2r+1) 1{L>r}. +``` + +The uniform component-span intensity bound then gives + +```text +M_w^long + <= poly(w,H) exp(-c_I H) + = exp[-Theta(w^2)]. +``` + +Thus the final parameter `p_+` enters only through a superexponentially negligible localization failure in this estimate; the two-witness exponential action is set by the earlier barrier parameter `p_-`. + +## 3. Window-level no-merger consequence + +Let an observation window have `m` longitudinal rows. The expected number of merger pairs attached to final components whose anchors lie in the window is + +```text +m M_w. +``` + +Hence Markov gives + +```text +P(at least one final component in the window + contains two earlier winding lineages) + <= m M_w. +``` + +For the natural intensity scale + +```text +m = O(1/nu_w(p0)), +``` + +and `p_-,p_+ -> p0`, the estimate above gives + +```text +m M_w + <= exp[-kappa(p0) w+o(w)] ->0. +``` + +If the observable is defined by components intersecting, rather than anchored inside, the window, enlarge by `H` rows at each endpoint. Since + +```text +H nu_w(p0) ->0, +``` + +and long components have superexponential intensity, the same conclusion holds. + +This is the precise scale-separation statement needed by #780: on a macroscopic `1/nu0` longitudinal window, distinct earlier essential lineages almost surely do not coalesce into one final macroscopic barrier. + +## 4. Local birth marks + +Now take a bounded parameter window represented by monotone functions + +```text +p_w(x), x in [a,b], +p_w(x) -> p0 uniformly, +``` + +and write + +```text +nu0 = nu_w(p0), +Lambda_w(x)=nu_w(p_w(x))/nu0. +``` + +Assume only that on the chosen bounded `x` interval + +```text +Lambda_w(x) -> Lambda(x) +``` + +at the finite set of clock values used below; continuity / strict monotonicity of the limiting clock can be imposed later when passing to a continuous mark space. + +Fix the top parameter + +```text +p_+ = p_w(b). +``` + +For each localized final `p_+` winding component `C`, define its birth mark + +```text +tau(C) + = inf{p in [p_w(a),p_+]: + C contains a p-winding ancestor}. +``` + +On the no-merger event, that ancestor lineage is unique throughout the bounded clock window. With continuous iid labels, the first threshold is almost surely unique; a direct `rank0 -> rank2` birth inside one component causes no ambiguity for the scalar mark. + +If the final component is localized inside its `H`-row window, all lower-`p` ancestors are subsets of that same final component and all guard sites closed at `p_+` remain closed for lower `p`. Hence `tau(C)` is a measurable function of the same finite label window as the final anchor. + +Rare merger configurations may be assigned an arbitrary deterministic tie-break; their total intensity is `o(nu0)` by Section 2, so they disappear in the limiting marked process. + +## 5. Finite mark partitions inherit the same AGG proof + +Partition `[a,b]` into finitely many mark bins `A_1,...,A_k`. Define typed localized indicators + +```text +I_{j,x,r} + = 1{a final p_+ component is anchored at (j,x) + and tau(C) belongs to A_r}. +``` + +For disjoint localization windows, the entire typed families are independent because they depend on disjoint `U` variables. + +For overlapping windows, two distinct typed anchors still imply two distinct final `p_+` winding components. Their occupied winding witnesses are disjoint, so the same BK estimate gives + +```text +E[I_{i,r} I_{j,s}] <= U_w(p_+)^2 +``` + +for every pair of types `r,s`. + +Thus the AGG process theorem applies without a new dependency argument. On an observation length + +```text +m = O(1/nu0) + = exp[kappa(p0)w+o(w)], +``` + +and because `p_+->p0`, + +```text +m poly(w) U_w(p_+)^2 ->0. +``` + +Therefore the vector of counts in finitely many disjoint spatial arcs and mark bins is asymptotically a vector of independent Poisson variables with its actual finite-`w` means. + +## 6. The mark mean measure is determined by component intensity + +The remaining task is to identify those means. For an intermediate parameter `p<=p_+`, consider all final `p_+` components and the number `n_C(p)` of `p` winding ancestors they contain. + +By the translation-covariant containment map from each early component to its unique final component, the mass-transport principle gives the exact intensity identity + +```text +nu_w(p) + = E sum_{C: final anchor_y(C)=0} n_C(p). +``` + +Define the cumulative final-component birth intensity + +```text +mu_w((-,p]) + = E sum_{C: final anchor_y(C)=0} 1{n_C(p)>=1}. +``` + +For every nonnegative integer `n`, + +```text +0 <= n-1{n>=1} <= binom(n,2). +``` + +Hence + +```text +0 <= nu_w(p)-mu_w((-,p]) + <= M_w(p,p_+). +``` + +Uniformly on a bounded clock window with `p_w(x)->p0`, Section 2 therefore gives + +```text +mu_w((-,p_w(x)]) / nu0 + - Lambda_w(x) + ->0. +``` + +Consequently, if `Lambda_w(x)->Lambda(x)`, then for every finite collection of continuity points + +```text +mu_w((p_w(x1),p_w(x2)]) / nu0 + -> Lambda(x2)-Lambda(x1). +``` + +No derivative of `nu_w(p)` is required. The intensity itself is the clock. + +## 7. Spatial anchor motion disappears on the intensity scale + +A localized final component and any of its ancestors lie in the same `H`-row window. Therefore any two translation-covariant anchor choices attached to that lineage differ by at most `O(H)` rows on the good event. + +The rescaled longitudinal coordinate is + +```text +nu0 y. +``` + +Since + +```text +H nu0 + = w^2 exp[-kappa(p0)w+o(w)] + ->0, +``` + +first-birth anchors, final anchors, lowest-row anchors, or any other bounded-window covariant representative have the same limiting spatial coordinate. The bad long-component event is already superexponentially negligible. + +## 8. Candidate marked-PRM theorem + +The preceding lemmas suggest the following theorem, conditional only on routine finite-partition/tightness bookkeeping and the audit of Sections 2 and 6. + +> **Candidate theorem.** On every bounded intensity-clock interval for fixed strictly subcritical `p0`, the marked complete-barrier birth process +> +> ```text +> Xi_w +> = sum_C delta_(nu0 y_C, Lambda_w(tau(C))) +> ``` +> +> converges on bounded spatial windows to a unit-rate Poisson random measure +> +> ```text +> dy dLambda. +> ``` +> +> Equivalently, for finite disjoint spatial/clock rectangles, the counts converge jointly to independent Poisson variables with means equal to rectangle areas. + +A standard route from the already proved finite mark partitions to PRM convergence is to use a countable generating algebra of spatial/clock rectangles, convergence of means, and local finiteness. No new percolation estimate appears at that stage. + +## 9. Pure splitting is then a corollary, not an additional model assumption + +At clock `Lambda`, retain all birth points with mark at most `Lambda`. A Poisson random measure `dy dLambda` gives a homogeneous PPP of barriers of spatial rate `Lambda`, coupled monotonically in the clock by adding new points. + +Because macroscopic mergers vanish, an old barrier lineage does not disappear into another old lineage on the scale of interest. Therefore the complementary white intervals evolve by pure cuts in the limiting process. + +All previously derived abstract consequences then become legitimate consequences of the SITE mapping: + +```text +fixed-location one-sided gap semigroup, +two-sided Gamma(2,1) fixed-location gap, +Corr(S_x,S_{x+h})=exp(-|h|) in log-intensity time, +Laguerre hierarchy, +record/cut genealogy. +``` + +They are not independent evidence for the map; the marked-PRM theorem is the bridge that licenses them. + +## 10. Audit checklist / possible failure points + +Before promoting the candidate theorem, check only the following concrete points. + +1. **Generic span tail.** Confirm that the argument behind `poisson-birth-windows.md` (4.5) indeed yields the unnormalised tail-intensity estimate uniformly for every `r>=w`, not only after substituting `H=w^2`. +2. **Pair-to-final mass transport.** Write the stationary mass transport from an unordered early merger pair to its containing final component, so the per-row inequality in Section 2.1 has no hidden anchor-offset factor. +3. **Containment semantics.** Verify that every `p` complete winding component is contained in exactly one `p_+` component and that final localization implies ancestor localization in the same guard window. +4. **Uniformity in the clock window.** State compact-`p` constants once and use `sup_x |p_w(x)-p0|->0` to make `M_w/nu0->0` uniform. +5. **Typed AGG statement.** Quote the process/multitype contraction explicitly; no new bound is expected. +6. **Finite partitions to PRM.** Supply the standard point-process tightness argument. +7. **Finite torus transfer.** If the theorem is stated on finite exponentially long tori, reuse the original `Bad_H` coupling and endpoint enlargement rather than rebuilding the proof. + +If one of these fails, preserve the exact failed arrow. In particular, failure of a bookkeeping step is not evidence for a physical coalescent; a surviving merger mechanism would require failure of the two-witness/localization scale separation itself. + +## 11. Research consequence + +If the audit passes, #780 should no longer spend primary compute on estimating merger rates. The scientific status changes from + +```text +abstract pure-cut kernel + missing SITE map +``` + +to + +```text +SITE component Poisson proof + -> no-macro-merger by its own b2 scale + -> local birth marks + -> marked Poisson rain in intensity time + -> pure splitting / record kernel. +``` + +This would be exactly the kind of cross-model/scale connection demanded by `research-compass-beyond-exactness-20260914.md`: a previously isolated solvable process becomes the consequence of a controlled Bernoulli-site map rather than a growing independent exact-model programme. diff --git a/docs/manuscripts/geometric-balance/correction-21-8-20260914.md b/docs/manuscripts/geometric-balance/correction-21-8-20260914.md new file mode 100644 index 000000000..086dba79a --- /dev/null +++ b/docs/manuscripts/geometric-balance/correction-21-8-20260914.md @@ -0,0 +1,62 @@ +# 对 `round45-out/arms-audit.md` §3 的更正 + +**日期**：2026-09-14　**依据**：`lit-frontier-long-out/lit-note-C.md` 条目 1（经主代理独立复核原文） + +--- + +## 被更正的原话 + +`round45-out/arms-audit.md` §3 与 §0 第 4 条写道（转述）： + +> 「**用被引文献自己的 `-4.07`**：`x=5.32`、`h=2.66`、`24h+1=64.84`（非完全平方）；用 `2-x=-3.42`：`x=5.42`、`h=2.71`、`24h+1=66.04`（非完全平方）。**两个都让 `21/8` 消失**」 + +我在给用户的汇报里也据此说了「**文献数字让 21/8 消失**」。 + +## 为什么这条不成立 + +它**只用了两份测量中的一份**，而且用的是**较弱的那一份**。 + +| 测量 | 出处 | 模型 | 方法 | 数值 | 作者自评 | +|---|---|---|---|---|---| +| **支持 `Δ₁ = 4`** | **Jacobsen 2015**, *J. Phys. A* **48** 454003, arXiv:**`1507.03027`** §7.1 式 (36)(40) | **方形格点 site 渗流**（**与本项目同一模型**） | 周期 Temperley–Lieb 代数的**转移矩阵特征值**，圆柱周长 **n ≤ 21**（**非 MC**） | **`Δ₁ = 4.000 1(2)`** | 原文直述：「**It appears inevitable to admit that `Δ₁ = 4` exactly.**」并给 `Δ_k = 2(k+1)`（式 40：`p_c(n) = p_c + Σ A_k/n^{2(k+1)}`） | +| 冲突 | **Mertens & Ziff 2016**, *Phys. Rev. E* **94** 062152, arXiv:**`1603.07289`** | 方形格点 site 渗流 | `M_L(p_c) ~ L^{2−x}` 直接拟合；L=3–7 精确 + 16/24/32/48 MC | **`2−x = −3.42`**；`p_L^⋆ − p_c` 斜率 **`−4.07`** → `w = −4.17` | **作者自认「larger systems are needed」**，未宣称渐近 | + +**主代理已独立复核**（`WebFetch` 读 arXiv 摘要页 + HTML 全文，非计算）： + +1. 摘要页确认：该文 (ii) **site percolation on the square lattice, to n = 21**， + 并给出 **`p_c = 0.59274605079210(2)`** —— **正是本项目一直在用的 `p_ref = 0.59274605079210`** + （即 `#774` 里的 near-critical 参考值就出自这篇）。 +2. 正文 §7.1 确认式 (36)：**`Δ₁ = 4.000 1(2)`**。 +3. 确认原文直述：**「It appears inevitable to admit that `Δ₁ = 4` exactly.」** + 紧邻上下文还提到「This agrees well with the value `w = 4.03 ± 0.01` reported in section 7.2 of [13]」。 +4. 确认式 (40) 与推论：**`Δ_k = 2(k+1)` for any k**。 + +## 更正后的正确表述 + +**不要写**：「文献数字让 `21/8` 消失」。 +**应写**： + +> 关于 square-site matching function 的有限尺寸修正指数，**已有两份互相冲突的独立测量**： +> Jacobsen (2015) 用转移矩阵在**同一模型**上做到 `n = 21`，给出**整数**指数 `Δ₁ = 4.000 1(2)` 与 +> 整数塔 `Δ_k = 2(k+1)`（4, 6, 8, 10, …），**支持** `x = 21/4 ⇒ h = 21/8`； +> Mertens–Ziff (2016) 用小尺寸系统给出 `2−x = −3.42` 与斜率 `−4.07`，**冲突**。 +> 二者是**同一个指数**，**不可能同时渐近**。 +> **因此 `21/8` 的生死尚未判定**——不是「已被文献否掉」。 + +## 一个必须一起写进去的关键事实 + +**`M_L(p_c)` 自身指数 `2−x` 只有 Mertens–Ziff 那一份测量**（`−3.25` 从未被直接测过）。 +`Δ₁` 与 `2−x` 是**同一个指数的两种投影**（`Δ₁ = 4` 对应 `x = 21/4`，即 `2−x = −3.25`）， +所以 Jacobsen 的 `Δ₁ = 4` **直接蕴含** `2−x = −3.25`。 +⇒ **判定 `21/8` 的靶子是唯一的**：把 `M_L(p_c)`（等价 `P2 − P0`）推到 L ≳ 100，看 `2−x` 趋 `−3.25` 还是 `−3.42`。 +**不要只盯着 `21/8` 这个数字本身去论证。** + +## 不受影响的部分（arms-audit 的其余裁定仍然成立） + +- `alpha_j = (j²−1)/12` 与 `c=0` Kac spinless 是**同一个函数** ⇒ 「8-arm ⇒ h=21/8」**是恒等式，不是独立证据**。 + **新增文献背书**：`x_ℓ^P = (ℓ²−1)/12` 已发表于 + **Aizenman–Duplantier–Aharony 1999, *Phys. Rev. Lett.* 83 1359–1362, arXiv:`cond-mat/9901018`**（式 (1)(2)）， + 并明确「extends to half-integers the Saleur–Duplantier exponents」。 + ⇒ **这条比原来更强了**：该式是**已发表的非重叠 path 指数**，所以"用 8-arm 反推 21/8"确实是**重述**。 +- 「6-arm / generic `omega=3/2` 必须被 matching balance 消除」**是要求不是证明**；`#768` 的 kill test **未执行**。 +- `alpha8 − alpha4 = 4` 与 `x_8 = 21/4` **完全等价**。 diff --git a/docs/manuscripts/geometric-balance/correlation-norm-successive-minima-20260914.md b/docs/manuscripts/geometric-balance/correlation-norm-successive-minima-20260914.md new file mode 100644 index 000000000..336d64f1b --- /dev/null +++ b/docs/manuscripts/geometric-balance/correlation-norm-successive-minima-20260914.md @@ -0,0 +1,275 @@ +# Correlation-norm successive minima and homology-rank upper bounds + +2026-09-14. General-period consequence of the first-exit torus theta bound. The correct fixed-subcritical geometry is expressed in the planar correlation norm, not Euclidean period length. + +## 1. Correlation-norm lattice geometry + +Fix a translation-invariant finite-range site graph `G` at a subcritical parameter `p`. Let + +\[ +\tau_p(x) +\] + +be its inverse-correlation norm and + +\[ +B_p=\{x:\tau_p(x)\le1\}. \tag{1.1} +\] + +Let `Lambda` be a rank-two period lattice. + +Define the first correlation-norm minimum + +\[ +\boxed{ +\rho_1(\Lambda;p) +=\min_{\lambda\in\Lambda\setminus0}\tau_p(\lambda).} \tag{1.2} +\] + +Choose any projective minimizing line `ell_*` represented by a minimizer `lambda_*`. + +Define its cheapest nonparallel period cost + +\[ +\boxed{ +\rho_\perp(\ell_*;\Lambda,p) +=\min_{\lambda\in\Lambda:\lambda\notin\operatorname{span}(\ell_*)} +\tau_p(\lambda).} \tag{1.3} +\] + +The invariant second successive minimum is + +\[ +\boxed{ +\rho_2(\Lambda;p) +=\min\{R:\operatorname{span}_{\mathbb R} +(\Lambda\cap R B_p)=\mathbb R^2\}.} \tag{1.4} +\] + +If the shortest projective line is unique, `rho_2=rho_perp(ell_*)`. If several nonparallel shortest vectors tie, then `rho_2=rho_1`. + +## 2. Norm-ball packing controls the period theta sum + +Because every nonzero lattice vector has `tau`-norm at least `rho_1`, the translated open norm balls + +\[ +\lambda+(\rho_1/2)B_p, +\qquad\lambda\in\Lambda, \tag{2.1} +\] + +are disjoint. + +Let + +\[ +N_\Lambda(R)=|\{\lambda\in\Lambda:\tau_p(\lambda)\le R\}|. \tag{2.2} +\] + +Every small ball centered at a point counted by `N_Lambda(R)` lies inside + +\[ +(R+\rho_1/2)B_p. \tag{2.3} +\] + +Comparing Euclidean areas and using homothetic scaling of norm balls gives + +\[ +\boxed{ +N_\Lambda(R) +\le\left(1+\frac{2R}{\rho_1}\right)^2.} \tag{2.4} +\] + +No determinant estimate or Euclidean angle is needed. + +For `a>0`, shelling at multiples of `rho_1` yields + +\[ +\sum_{\lambda\in\Lambda\setminus0}e^{-a\tau_p(\lambda)} +\le +C\sum_{k\ge1}(k+1)^2e^{-ak\rho_1}. \tag{2.5} +\] + +Hence + +\[ +\boxed{ +\sum_{\lambda\ne0}e^{-a\tau_p(\lambda)} +\le +C_a(\rho_1)e^{-a\rho_1},} \tag{2.6} +\] + +where, for example, + +\[ +C_a(\rho_1) +\le C(1-e^{-a\rho_1})^{-3}. \tag{2.7} +\] + +In particular, as `rho_1->infinity`, the theta sum has the same exponential rate as its cheapest period. + +## 3. Positive-rank upper bound + +`first-exit-torus-winding-upper-20260914.md` proves that for every `epsilon in (0,1)`, on every sufficiently large honest torus, + +\[ +P_p(r>0) +\le +N C_{p,\epsilon} +\sum_{\lambda\ne0} + e^{-(1-\epsilon)\tau_p(\lambda)}. \tag{3.1} +\] + +Combining with (2.6), + +\[ +\boxed{ +P_p(r>0) +\le +N\,\widetilde C_{p,\epsilon}(\rho_1) + e^{-(1-\epsilon)\rho_1},} \tag{3.2} +\] + +with `log Ctilde=o(rho_1)` as `rho_1->infinity`. + +Therefore, if along a sequence + +\[ +\rho_{1,n}\to\infty, +\qquad +\limsup\frac{\log N_n}{\rho_{1,n}}<1, \tag{3.3} +\] + +then + +\[ +\boxed{P_p(r>0)\to0.} \tag{3.4} +\] + +More quantitatively, if + +\[ +\frac{\log N_n}{\rho_{1,n}}\to\alpha<1, \tag{3.5} +\] + +then + +\[ +\boxed{ +\limsup\frac1{\rho_{1,n}}\log P_p(r>0) +\le-(1-\alpha).} \tag{3.6} +\] + +The `epsilon` in the finite certificate is sent to zero after the sequence limit. + +## 4. Rank-two requires a nonparallel homology class + +Fix a projective line `ell_*`. If the ambient homology image has rank two, then it contains a nonzero class not lying in `ell_*`. + +Choose a minimum-length nonzero-homology closed walk whose class is outside `ell_*`. The same simple-cycle/first-exit argument as for positive rank gives + +\[ +P_p(r=2) +\le +N C_{p,\epsilon} +\sum_{\lambda\in\Lambda:\lambda\notin\ell_*} + e^{-(1-\epsilon)\tau_p(\lambda)}. \tag{4.1} +\] + +The unrestricted packing estimate still bounds the number of nonparallel vectors in each shell. Therefore + +\[ +\boxed{ +P_p(r=2) +\le +N\,\widetilde C_{p,\epsilon}(\rho_1,\rho_\perp) + e^{-(1-\epsilon)\rho_\perp},} \tag{4.2} +\] + +where the prefactor is subexponential in `rho_perp` whenever `rho_1` is bounded below proportionally to `rho_perp`. + +If the shortest line is unique and the lattice shape is nondegenerate in the correlation norm, this yields the clean second-minimum criterion + +\[ +\boxed{ +\log N< (1-o(1))\rho_2 +\quad\Longrightarrow\quad P_p(r=2)\to0.} \tag{4.3} +\] + +Even without proportional successive minima, (4.1) is a rigorous restricted theta bound and is the safer statement. + +## 5. Unique rank-one slope from a successive-minimum gap + +Suppose one projective line `ell_*` has cost `rho_1` while + +\[ +\rho_\perp-\log N\to+\infty. \tag{5.1} +\] + +Then (4.2) gives + +\[ +P_p(r=2)\to0. \tag{5.2} +\] + +Moreover, a rank-one configuration with slope different from `ell_*` also contains a nonparallel class relative to `ell_*`, so + +\[ +\boxed{ +P_p(r=1,L\ne\ell_*)\to0.} \tag{5.3} +\] + +Thus whenever positive rank has nontrivial probability and the second successive direction remains exponentially suppressed, the rank-one projective slope is asymptotically deterministic. + +`exponential-homology-class-selection-20260914.md` is the extreme case where `rho_perp` is of order the exponentially large transverse height. + +## 6. A correlation-norm homological free-energy upper criterion + +The combination appearing in (3.2) is + +\[ +\boxed{\mathcal F_1=\rho_1-\log N.} \tag{6.1} +\] + +At fixed subcritical `p`, + +- `F1->+infinity` forces rank zero; +- `F1` of order one is the scale where the union bound no longer decides the event; +- `F1->-infinity` by itself does not prove positive rank, because the actual number of approximately independent placements may be smaller than `N`. + +So `rho_1-log N` is a rigorous **upper-side homological free energy**. A matching lower theorem must replace `N` by a geometrically justified opportunity count from disjoint translates. + +For an exponentially elongated shortest-period torus that opportunity count is `h/poly(ell)`, whose logarithm agrees with `log N` at the `ell` scale. This is why the upper and lower exponents close there. + +## 7. Relation to the Euclidean full-law criterion + +On a fixed compact subcritical parameter interval, finite-range norm equivalence gives + +\[ +c_p|x|\le\tau_p(x)\le C_p|x|. \tag{7.1} +\] + +Hence + +\[ +c_p\ell(\Lambda) +\le\rho_1(\Lambda;p) +\le C_p\ell(\Lambda). \tag{7.2} +\] + +The geometric manuscript's condition + +\[ +\log N/\ell\to0 \tag{7.3} +\] + +therefore implies + +\[ +\log N/\rho_1(p)\to0 \tag{7.4} +\] + +at every fixed subcritical parameter. The correlation-norm formulation is sharper when period directions/anisotropy matter, while the Euclidean condition remains the clean parameter-uniform geometric statement near criticality. + +## 8. Claim boundary + +The lattice-point packing and theta estimates are deterministic convex geometry. The probabilistic inputs are exactly the first-exit torus upper bound and existence of the subcritical correlation norm. The rank-two restricted-theta bound uses only the deterministic fact that rank two contains a class outside any fixed projective line; it does not require two disjoint essential components. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/cross-node-post-h4-discriminator-20260914.md b/docs/manuscripts/geometric-balance/cross-node-post-h4-discriminator-20260914.md new file mode 100644 index 000000000..1b388d2dc --- /dev/null +++ b/docs/manuscripts/geometric-balance/cross-node-post-h4-discriminator-20260914.md @@ -0,0 +1,239 @@ +# Post-H4 cross-node discriminator: a bounded replacement for same-ell multi-angle tomography + +2026-09-14. + +Status: deterministic finite safe-transfer analysis plus mechanism triage. This note is explicitly corrective: the earlier plan to use N1105 same-ell multi-angle safe-root tomography is not executable under the current row-memory transfer and, even at zero numerical noise, fixed-ell radial dressing confounds asymptotic angular coefficients. The exact Gaussian-circle projector algebra remains valid; the production route is replaced here by a low-memory pair of nearly equal physical circumferences. + +## 1. What is being corrected + +The previous same-projector axis/(3,4) values at `ell=5,10` suggested + +```text +p_perp4(ell)-pc ~ C7 ell^-7, +``` + +because multiplying the two residuals by `ell^7` nearly froze them. That was a useful hint but it was not an angular identification: the same two-angle projector has a large surviving H8 coefficient. With only two scales, an H8 contribution can masquerade as a scalar radial power. + +The later independent state-cost/confounding audit also rules out buying enough same-ell orientations to solve H0/H8/H12 directly with the current transfer algorithm. We therefore need a different geometry that changes the surviving angular coefficient while leaving the physical scale nearly unchanged. + +## 2. A complementary node pair with low row memory + +Use + +```text +u_A=(5,2), n_A=2, ell_A^2=116, +nu_B=(3,2), n_B=3, ell_B^2=117. +``` + +The physical circumferences differ by only about 0.43 percent. + +Their exact harmonics are + +```text +H4_A = 41/841 = +0.0487514863..., +H8_A = -703919/707281 = -0.9952465852..., + +H4_B = -119/169 = -0.7041420118..., +H8_B = -239/28561 = -0.00836805434.... +``` + +Thus A is close to an H4 node while B is close to an H8 node. They are almost complementary filters. + +The current lifted-gain safe transfer remains tractable: + +```text +(5,2),n=2: G4 22,994 states; G8 131,677 states; row memory 5/7. +(3,2),n=3: G4 19,018 states; G8 206,197 states; row memory 3/5. +``` + +This is qualitatively different from the millions-to-astronomical-state same-circle designs. + +## 3. Tight-root control + +The new calculation uses sparse Perron tolerance `1e-12` and root tolerances around `1e-14`, rather than the looser exploratory settings in the original oblique script. + +As an implementation regression, the same code gives + +```text +(3,2), n=2: +local 0.5928240591852450 +repository 0.5928240591851011 +``` + +for a difference of about `1.4e-13`. + +The tight roots used here are + +```text +p_A = 0.5927450615721411, +p_B = 0.5927612086097266. +``` + +## 4. Remove the leading H4 term without using pc + +Write the leading root law as + +```text +p_i = pc - A4 H4_i ell_i^-4 + residual_i. +``` + +Let + +```text +q_i=H4_i/ell_i^4. +``` + +Choose weights `w_A+w_B=1` and `w_A q_A+w_B q_B=0`. Numerically, + +```text +w_A=0.934200379537736, +w_B=0.0657996204622639. +``` + +The leading-H4 projected root is therefore + +```text +p_cross^perp4 + = w_A p_A+w_B p_B + = 0.5927461240410858. +``` + +The simultaneously inferred leading H4 amplitude is + +```text +A4_hat=0.2932542820. +``` + +Because the two circumferences are not exactly equal, a same-H4 `ell^-6` dressing is not killed identically. But using the already observed `A4(ell)=Ainf+A2/ell^2` scale with `A2~0.66`, its leakage into this projector is only about `1.6e-10`, far below the `1e-7` effects below. + +## 5. The key angular fact + +For the projected pair, before tiny radial-mismatch corrections, the surviving angular weights are approximately + +```text +H8: -0.93031, +H4^2: +0.034845. +``` + +Compare the existing exact-equal-ell axis/(3,4), ell=10 projector: + +```text +H8: +0.6864, +H4^2: +0.8432. +``` + +Using the common diagnostic threshold `pc=0.5927460507921` only after root location, + +```text +axis/(3,4), ell=10: p_perp4-pc = -7.57256e-8, +cross-node, ell~10.79: p_perp4-pc = +7.32490e-8. +``` + +The two post-H4 residuals reverse sign exactly when the H8 projector weight reverses sign, while the scalar weight remains positive and the H4^2 weight remains positive. + +This makes the earlier statement + +```text +post-H4 residual is primarily a scalar V_<1,4>-like ell^-7 term +``` + +unsafe. A substantial H8-like angular component is required unless one invokes a finely tuned multi-channel cancellation. + +## 6. Reference-free stress tests of the old scalar interpretation + +The earlier axis/(3,4) projector has the same angular weights at `ell=5` and `ell=10`. Fit those two points to a pure scalar law + +```text +p_perp4(ell)=pc_fit+C7 ell^-7. +``` + +Because both `pc_fit` and `C7` are fit from those two roots, this test does not use an external threshold. It gives + +```text +pc_fit = 0.5927460516782668, +C7 = -0.7661178948. +``` + +Transporting that scalar law to the cross-node projector predicts + +```text +0.5927460061955850, +``` + +whereas the observed projected root is + +```text +0.5927461240410858. +``` + +The miss is about + +```text +1.18e-7, +``` + +roughly three orders above the estimated same-H4 radial-dressing leakage. Thus the old two-scale scalar story fails a new angularly changed held-out geometry. + +## 7. A caution: H4^2 mixing can mimic the new geometry + +There is an important counterpoint. Fit the same `ell=5,10` axis/(3,4) roots to + +```text +p_perp4=pc_fit+Cmix P_perp4[H4^2] ell^-6. +``` + +This gives + +```text +pc_fit = 0.5927461295061164, +Cmix = -0.1831589648. +``` + +It predicts the cross-node projected root as + +```text +0.5927461255146714, +``` + +only `1.5e-9` from the observed value. + +So the cross-node result does **not** by itself identify an independent H8 field. A mixed term involving the leading odd H4 correction and an even lattice anisotropy can generate H4^2=(H0+H8)/2 and reproduce this geometry extremely well. + +However, that pure three-point mixed fit places its asymptotic intercept about `7.9e-8` above the independent diagnostic pc. With the diagnostic pc held fixed, the larger-scale H4-projected data are described materially better when an H8-like column is included than by scalar or H4^2 columns alone. Therefore the safest current statement is: + +> post-H4 data contain a large H8-like angular component; whether it is an independent dual-odd H8 operator or nonlinear mixing must be decided by a source/field calculation, not by naming the residual from two root powers. + +## 8. Why the common x~4 correction does not automatically settle the mixing question + +The oblique magnetic-gap controls give, roughly, + +```text +(ell I0-2pi*5/48) ell^2 ~ 0.29--0.34. +``` + +A crude constant-plus-H4 regression gives a dominant scalar part and a much smaller H4 part. Therefore the observed common `omega~2` correction should not automatically be called an even spin-4 field. If it is mainly scalar, multiplying it by odd thermal Q4 only dresses the H4 sector and is removed by an H4 projector; only its spin-4 subcomponent generates H4^2/H8. + +This is a concrete reason not to promote the q=2 mixed explanation solely because it is symmetry-allowed. + +## 9. Current candidate ordering + +After the corrections above, I would not order candidates by a two-point radial exponent. The useful ordering is by what remains to be demonstrated: + +1. **H8-like post-H4 angular response — observed finite evidence.** The angular component is now the most robust new statement. +2. **Nonlinear odd-H4 x even-spin4 mixing — live mechanism.** It can reproduce the new cross-node geometry, but the relevant even spin-4 coupling and asymptotic intercept must be established. +3. **Independent dual-odd H8 operator — live mechanism.** Linear identity-family spin8 is one candidate only if its matching-odd coupling is shown nonzero; logarithmic/interchiral alternatives remain possible. +4. **Scalar x~33/4 block — demoted from default.** It may coexist, but the pure scalar ell^-7 explanation fails the cross-node held-out geometry. +5. **H12/higher channels — retained as residual alternatives, not currently required by this control.** + +This does not change the well-supported leading result: the first non-common root correction is H4/spin4 with radial dimension consistent with x=21/4. + +## 10. Next high-information analysis, no large circle + +The next calculation should target the origin of the H8-like component rather than another angular census. + +Two routes are higher value than N1105: + +1. **source-resolved mixed derivative:** in the safe Perron/Clapeyron framework, insert one physical source known to excite the common `omega~2` correction and differentiate the odd-H4 root response with respect to it. A genuine `u4^- v4^+` mechanism predicts an H4^2 second response with a locked sign/tensor structure. +2. **rank-projected Ward/generic-Q test:** compute whether the actual safe/rank functional preserves the ordinary thermal-Q4 Ward reduction or acquires logarithmic/null/seam terms. An independent H8 or scalar block should then appear as a genuinely new continuum matrix element rather than as ordinary Q4 dressing. + +The machine-readable inputs and model checks are in `results/research-control-20260914/cross-node-post-h4-projector-20260914.json`. diff --git a/docs/manuscripts/geometric-balance/cylinder-q4-thermal-tangent-ward-20260914.md b/docs/manuscripts/geometric-balance/cylinder-q4-thermal-tangent-ward-20260914.md new file mode 100644 index 000000000..2e23b4770 --- /dev/null +++ b/docs/manuscripts/geometric-balance/cylinder-q4-thermal-tangent-ward-20260914.md @@ -0,0 +1,314 @@ +# Cylinder Ward identity behind the thermal-tangent spin-four correction + +Date: 2026-09-14 + +Status: **exact ordinary-CFT/Virasoro identity conditional on the ordinary `c=0,h=5/8` thermal Kac quotient**. This note explains an important part of the finite safe-transfer `TANGENT_SPIN4` observation at the critical point. It does not prove that the full `Q=1` logarithmic module reduces to the ordinary quotient, and it does not prove exact tangency at finite near-critical scaling variable `X`. + +## 1. Question + +The same-model oblique safe transfer now shows two finite facts very sharply: + +1. the critical NN/matching safe-sector mismatch transforms as spin four and produces the orientation-dependent `ell^-4` charge-root shift; +2. after each orientation is recentered at its own root and normalized by its own thermal slope, the charge curves at equal physical circumference agree to `O(1e-5--1e-4)` on the tested local interval. + +This suggests that the leading spin-four perturbation acts primarily by shifting the thermal scaling coordinate. + +The ordinary thermal-family candidate has exactly the structure needed to test this analytically. Let + +```text +phi_t: h=hbar=5/8, +Q4=40 L_-2^2 - 60 L_-3 L_-1 - 9 L_-4. +``` + +The question is whether, on a cylinder energy level, insertion of `Q4 phi_t` is proportional to insertion of `phi_t` itself. + +## 2. Cylinder setup + +Use a dimensionless cylinder coordinate `w` with imaginary period `2 pi`, and the plane map + +```text +z=e^w. +``` + +Let `|m>` be any primary cylinder state of chiral weight `h_m=m`. Insert a chiral primary `phi_h` at `w=0`. + +On the plane, + +```text + = C z^-h. +``` + +The cylinder primary is + +```text +phi_h^cyl(w)=z^h phi_h(z), +``` + +so + +```text + = C, +``` + +independent of `w`. Thus every outer local `L_-1` insertion has zero one-point matrix element on the cylinder. + +## 3. Exact one-stress-tensor Ward function + +The plane Ward identity with external primaries at `0` and `infinity` gives + +```text +_m / _m + = m/z^2 + h/[z (z-1)^2]. +``` + +The cylinder stress tensor is + +```text +T_cyl(w)=z^2 T(z)-c/24. +``` + +At `c=0`, therefore, + +```text +_m / _m + = m + h e^w/(e^w-1)^2 + = m + h/[4 sinh^2(w/2)]. +``` + +The exact local expansion is + +```text +1/[4 sinh^2(w/2)] + = 1/w^2 - 1/12 + w^2/240 - w^4/6048 + ... . +``` + +Comparing with the local Virasoro OPE + +```text +T_cyl(w) phi(0) + = h w^-2 phi + + w^-1 L_-1 phi + + L_-2 phi + + w L_-3 phi + + w^2 L_-4 phi + + ... +``` + +gives the exact matrix-element ratios + +```text +_m / _m = 0, +_m / _m = m-h/12, +_m / _m = 0, +_m / _m = h/240. +``` + +The crucial observation is that the `L_-4` ratio is **independent of the external cylinder state `m`**. + +## 4. Reduce Q4 using the thermal null vector + +Assume the ordinary thermal Kac quotient + +```text +(L_-2 - 2/3 L_-1^2)|h> = 0, +h=5/8. +``` + +Cylinder translation invariance gives + +```text +_m=0 +``` + +for every local descendant `Psi` in this one-point matrix element. + +Exactly as in the torus Ward reduction, + +```text +_m = -2 _m, +_m = 4/3 _m. +``` + +Hence + +```text +_m + = (493/3) _m. +``` + +Using Section 3, + +```text +_m / _m + = (493/3)(h/240). +``` + +For `h=5/8`, + +```text +boxed: +_m / _m = 493/1152. +``` + +This coefficient is independent of the external primary state. + +For the real bulk spin-four combination + +```text +O4 = Q4 phi_t x phibar_t + phi_t x Qbar4 phibar_t, +``` + +both chiralities contribute the same number, so in this normalization + +```text +_m / _m = 493/576 +``` + +before restoring the physical circumference factor and the microscopic spin-four coupling. + +For a physical circumference `ell`, the level-four descendant contributes the expected extra factor proportional to `ell^-4`; a rotated lattice coupling supplies the real `cos(4 theta)` factor. + +## 5. Why this is exactly a thermal-tangent statement at criticality + +Let the critical Hamiltonian be perturbed thermally by + +```text +H -> H + t int phi_t phibar_t. +``` + +First-order conformal perturbation theory gives the derivative of an excitation energy with respect to `t` from the matrix element of the thermal primary (with the common bulk/vacuum analytic piece subtracted in the excitation gap). + +Now perturb instead by the ordinary thermal spin-four descendant + +```text +u4 cos(4theta) int O4. +``` + +Section 4 says that, on every translation-invariant cylinder primary level, its first-order matrix element is a universal constant times the thermal-primary matrix element. Therefore the singular excitation-energy correction at `t=0` obeys + +```text +delta_u4 E_m(0) + proportional to +cos(4theta) ell^-4 * partial_t E_m(0). +``` + +Equivalently, to first order at the critical point, + +```text +E_m(t,u4) + = E_m(t + c_mic u4 cos(4theta) ell^-4, 0) + + higher-order / other-field terms, +``` + +where the nonuniversal microscopic coefficient is carried by `c_mic`, while the ordinary-CFT descendant/primary ratio is fixed by the Ward identity. + +Thus: + +> **For the ordinary thermal Kac Q4 candidate, critical tangency is not an accidental numerical property. It is enforced by the cylinder Ward identity.** + +This is precisely the mechanism observed in the safe-transfer root: a large orientation-dependent root translation with an almost orientation-independent leading thermal denominator. + +## 6. What the identity does and does not prove about the full scaling curve + +The Ward identity is a statement at the critical CFT point, i.e. at `X=0` in the near-critical scaling variable. + +It proves the leading local relation + +```text +G4(0) proportional to F'(0) +``` + +for the ordinary Q4 branch. + +It does **not** by itself prove + +```text +G4(X)=const * F'(X) +``` + +for finite `X`. The equal-circumference safe-transfer observation that root/slope-normalized curves nearly coincide on `|y|<=0.5` is therefore genuinely additional information about the massive / near-critical continuation. + +A useful decomposition remains + +```text +G4(X) + = [G4(0)/F'(0)] F'(X) + G4_perp(X), +G4_perp(0)=0. +``` + +The cylinder Ward identity forces the critical intercept into the tangent piece for the ordinary Kac descendant. The numerical task measures how small `G4_perp` remains away from zero. + +## 7. Logarithmic-module interpretation becomes sharper + +At `Q=1`, the thermal/energy field can belong to an energy--two-hull logarithmic multiplet. The ordinary Kac calculation above should then be interpreted as the bottom-field contribution. + +A logarithmic partner or another spin-four module can add a matrix element not constrained to the same `493/1152` ratio. Therefore a conceptually clean decomposition is + +```text +spin4 response + = ordinary-Kac tangent piece + + logarithmic/other normal piece. +``` + +This changes the module question substantially. The existence of a large root shift is **not** evidence for a large logarithmic component; the ordinary Q4 branch already predicts a thermal-tangent shift exactly. + +The best places to look for Jordan/module information are instead: + +1. the residual root/slope-normalized orientation dependence `G4_perp`; +2. a generic-Q branch splitting where energy and two-hull fields separate; +3. logarithmic size dependence of the H4 amplitude after ordinary power corrections are controlled; +4. observables whose moving-root/thermal-tangent projection has already been removed, such as the original-U contract. + +## 8. Relation to the existing torus Ward identity + +The earlier torus calculation on PR #151 found, in the same ordinary Kac quotient, + +```text +/ = (493/96) g2(tau). +``` + +The present cylinder calculation is the infinite-cylinder/local-energy analogue. Both results have the same algebraic origin: + +```text +Q4 -> (493/3) L_-4 +``` + +under one-point translation invariance plus the level-two null relation. + +The torus shape is then supplied by the torus Ward value of ``, while the cylinder energy-level value is supplied by the local stress-tensor expansion above. + +## 9. Immediate research consequence + +The current evidence hierarchy should be reorganized as follows. + +### Already strongly supported + +```text +spin character: 4; +radial dimension: x=21/4; +critical response direction: thermal tangent, if ordinary Kac Q4 is the branch; +``` + +with the first two supported independently by historical prospective Gaussian tests and current deterministic safe-transfer angular controls. + +### Still open + +```text +amount of logarithmic-partner admixture; +finite-X normal response G4_perp; +microscopic NN-vs-matching difference coupling u4^-; +post-H4 residual irreps (H8/scalar/H12/... ). +``` + +The operator-identification problem should therefore no longer be phrased as “why can a spin-four field move the root?” The ordinary thermal Q4 Ward identity already supplies that mechanism. The harder question is: + +> **what part of the observed spin-four response cannot be absorbed into this exact ordinary-Kac thermal tangent?** + +That remainder is the appropriate target for LCFT/Jordan identification. + +## 10. Claim boundary + +- The cylinder Ward function and `493/1152` ratio are exact given the ordinary Virasoro/Kac quotient and standard state-field/cylinder map. +- Applying the ratio to the square-site safe-transfer correction still assumes the observed H4/x=21/4 lattice direction couples to this ordinary thermal descendant. +- The exact identity holds at the critical CFT point; finite-`X` tangency remains a scaling conjecture supported by deterministic transfer data. +- The argument does not remove the energy--two-hull logarithmic multiplet from the `Q=1` theory. It identifies a precise bottom-field contribution and thereby relocates the logarithmic question to the normal residual rather than the existence of the root shift. diff --git a/docs/manuscripts/geometric-balance/delta-p-intrinsic-floor-and-n1105-gate-20260914.md b/docs/manuscripts/geometric-balance/delta-p-intrinsic-floor-and-n1105-gate-20260914.md new file mode 100644 index 000000000..cd282c187 --- /dev/null +++ b/docs/manuscripts/geometric-balance/delta-p-intrinsic-floor-and-n1105-gate-20260914.md @@ -0,0 +1,322 @@ +# dpfloor：δp 的内禀一致性底噪 `F` 与 N1105 的 go/no-go + +日期：2026-09-14 ｜ 机器 `DevEnvC_NePnUn`（账号2，16 vCPU / 32 GiB，Python 3.9.9） +全部算术在云机执行，本机只做读写/上传下载。**仓库写操作：无**（GitHub 只读）。 +**没有跑 N1105，没有开任何大算。** 最贵的单个运算是 `axis_n9` 的 w=9 稠密特征分解（212 s）与 +`slope52_n2` 的 ARPACK 复算（508 s）；全流程合计约 1 小时墙钟，远小于一次 N1105。 + +交付：`floor.json`（每条路径的原始数字 + `F` + `S/F`）、`scripts/`（11 个自包含脚本）、 +`raw/`（全部机器可读输出与日志）。 + +--- + +## 0. 必须回答的那一句 + +> ### 「N1105 应该现在开跑吗？」 +> +> **按现有配置（shipped settings）开跑：不应该 —— 那是 NO-GO。** +> **把两个配置改掉之后开跑：可以 —— GO，安全系数 `S/F ≈ 5.8×10³`。** +> +> 具体地： +> +> | 问题 | 数 | 判定 | +> |---|---|---| +> | `n1105mix` 报的「系统误差 2.10e−12」 | 是 **100 % 的 `p_c` 记账口径差**（可消除），不是任何实现底噪 | 归零 | +> | 紧配置下的实现底噪 `F` | **4.68e−16**（p 单位） | — | +> | 信号 `S`（`C_mix=0.0073`） | 2.71e−12 | **`S/F = 5793` ⇒ GO** | +> | 信号 `S`（`C_mix=0.0898`） | 3.33e−11 | **`S/F = 71185` ⇒ GO** | +> | shipped 配置下的复现离散度 `F_shipped` | **6.31e−12**（几何相关，最大 6.31e−12；shipped 文件自身对紧值的偏差最大 **9.29e−12**） | **`S/F_shipped = 0.43` ⇒ NO-GO** | +> +> **要改的两行（以及怎么知道改好了）见 §4。** 一句话：**判据本身可达，但必须先统一 `p_c` 并把 +> 特征求解器/求根器的公差收紧；用 shipped 的那套设置去跑 N1105，先天测不出 `P0=P8`。** + +--- + +## 1. `Ω(axis, ℓ=8)` 两份文件差 `8.6e−9` 的成因判定 —— **(c) 定义/记账口径，且具体就是 `p_c`** + +### 1.1 数字（可复现：`scripts/fl_convention.py` → `raw/step0_convention.json`） + +两份文件指的是： + +| 记号 | 文件 | 生产者 | 用的 `p_c` | +|---|---|---|---| +| 文件 A | `oblique-spin4-controls.json` | rev769 的 `scripts/oblique_charge_transfer.py` | **0.59274605079** | +| 文件 B | `closure-amplitude-raw.json` | sector802b 的 `s802b_c1_amplitude.py`（`PC=0.59274605079210`） | **0.59274605079210** | + +（生产者出处见 §2.0：rev769 的脚本在本机 `rev769-repo/scripts/oblique_charge_transfer.py`， +默认 `--reference-pc 0.59274605079`。这是**已被核实的代码级出处**，不是猜测。） + +同一几何 `(1,0), n=8`（= 轴向 w=8，ℓ=8，ℓ⁴=4096）： + +``` +Ω := -(p_root - p_c) * ell^4 +Ω_文件A = 0.3001967221684936 +Ω_文件B = 0.30019673077049447 +差 = 8.602000889368355e-09 （相对 2.865e-08） +``` + +### 1.2 严格分解（残差恰好为 0） + +``` +Ω_B - Ω_A = -[(p_root,B - p_root,A) - (p_c,B - p_c,A)] * ell^4 +``` + +| 项 | 值 | 占比 | +|---|---|---| +| `p_c,B - p_c,A` = 0.59274605079210 − 0.59274605079 | **+2.0999868510784836e-12** | — | +| 来自 `p_c` 不同的贡献 `= Δp_c·ℓ⁴` | **+8.601546142017469e-09** | **99.9947 %** | +| `p_root,B - p_root,A` | **−1.1102230246251565e-16**（1 ulp） | — | +| 来自 `p_root` 不同的贡献 `= −Δp_root·ℓ⁴` | +4.547473508864641e-13 | 0.0053 % | +| **残差** | **0.0（精确）** | — | + +⇒ **`8.6e−9` 全部由「两份文件用了不同的 `p_c`」解释，剩余 4.5e−13 只是根值的 1 个 ulp。** +`n1105mix` 的 note §1.2/§3.3 说「该差**不能**由两处 `p_c` 约定解释」是**错的**： +两处 `p_c` 相差 2.1e−12（15955 个 ulp），乘 ℓ⁴=4096 恰好是 8.6e−9。它把 +「δp 差 = 2.1e−12」与「`p_c` 差 = 2.1e−12」当成了巧合，其实那是同一个恒等式。 + +### 1.3 排除另外三个候选 + +* **(a) 闭合幅度 `A_r` —— 排除。** `Ω`（以及 `Δ_w`、`p_root`）只由 **Perron 根**决定； + sector802b 自己在 `s802b_c1_amplitude.py` 的 C3(a) 就写明「`Delta_w` is built from Perron + ROOTS only, so an amplitude cannot enter it」。`A_open` 等是诊断量。本报告用 + `p_ch` 与 `p_c` 重算 `Ω_B`，与文件里印的 `w4_times_offset` **精确相符（差 0.0）**， + 完全不需要任何幅度量。 +* **(b) 求解器容差/停止条件 —— 不是主因，但确有 0.005 %。** 两份文件的**特征分解代码不同** + （文件 A 用 ARPACK `tol=1e-10`，文件 B 用稠密 `scipy.linalg.eig`）， + 但它们在同一个几何上给出的 `p_root` 只差 1 ulp。真正的容差代价不在这一对文件上， + 而在 shipped 配置的**复现性**上（§3.3）。 +* **(d) 数值噪声 —— 排除。** 残差为 0，即这个 8.6e−9 里没有第三样东西。 + +### 1.4 定义核对（别把定义差当数值差） + +`n1105mix` §1.1 发现的定义错位被**独立复核确认**： + +| 字段 | 实际定义 | 与重算的最大差 | +|---|---|---| +| 文件 A `A_estimate` | `−(p_root − p_c)·ℓ⁴ / cos4θ` | **0.0** | +| 文件 A `shift_times_ell4` | `(p_root − p_c)·ℓ⁴ = −Ω` | （与 `A_estimate` 差恰好一个 `cos4θ`） | +| 文件 B `w4_times_offset` | `−(p_root − p_c)·ℓ⁴ = Ω` | **0.0** | + +⇒ 两份文件印的是**不同字段**，但只要都换算成 `Ω`，剩下的就是 `p_c`。 + +--- + +## 2. 独立路径清单（每条：算法是什么、在哪一步不同、是不是真独立） + +### 2.0 先核代码（任务 §2.3 第 1 条） + +* **`sector802-out/scripts/sector802_lib.py` 与 `sector802b-out/scripts/sector802_lib.py` + 逐字节相同**（md5 均为 `4646de953442c8cc71ae255f6cc3eec2`），而 `s802b_c1_amplitude.py` + 直接 `from sector802_lib import enumerate_states, aggregate, build_matrices, perron_fh`。 + ⇒ **产生文件 A（`root-response-raw.json` / `root-response.json`）与文件 B + （`closure-amplitude-raw.json`）的是同一份 rank 生产**；两者的差**只反映调用参数与后处理** + （文件 B 多算了 `solve_root` 的精确根与幅度解剖）。**它们不是两条独立路径，本报告明说。** +* 引擎：两者都用 `tagged_winding_span.py`（pinned #739 副本），`sector802` 的 note §0 与 + sector802_lib 的 docstring 都写明了。 +* 本报告 reran 的 `PATH A` 就是这份库（md5 校对过，见 `raw/log_pathA.txt` 前的 md5 输出）。 +* rev769 的 `oblique_charge_transfer.py` 是**另一套**自动机：SL(2,Z) Bezout 斜基、 + `(ds,dt)` 边集、**frontier 记忆 = max dt**、**没有 winding 元组**（用一个"矛盾合并" + `dsu.bad` 当场丢弃整行）。⇒ 它与 `sector802_lib` 在**状态表示与 winding 判定**上不同。 + +### 2.1 路径表 + +| 路径 | 算法 | 与别的路径**在哪一步**不同 | 是真独立路径？ | 覆盖几何 | +|---|---|---|---|---| +| **A_lib** | `#739` 引擎 BFS 自动机（winding 元组）→ 稠密 `R` → `scipy.linalg.eig` → 二分/brentq | 基线；**就是文件 A/B 的那份代码** | **与文件 A/B 不独立**（同库） | 轴向 w=4,6,8 | +| **B_oblique** | Bezout/SL(2,Z) 斜基自动机（`dsu.bad` 当场拒环）→ 稠密或 ARPACK → brentq | **自动机不同**（状态表示 + winding 判定 + 前沿记忆），求解器/求根器也不同 | **是**（与 A 无共享代码） | 全部 13 个几何（含 4 条斜向） | +| **M_mine** | **双覆盖**前沿划分：状态 = 提升到 2w 个位置的连通划分；**winding ⟺ 提升把 `X` 与 `X+w` 认成同一块** | **第三种自动机**，winding 判定方式与前两者都不同 | **是** | 轴向 w=3,4,5,6 | +| **H_ld** | 由整数重数表在 **float128** 里装配 `R` → **幂迭代**求 Perron 根 → float128 secant 求根，**完全不用 LAPACK** | **算术级独立**（其余全用 float64/LAPACK） | **是（算术）** | 轴向 w=4,6,8 | +| **S_solvers** | 同一张 `R`（来自 A）配 5 种特征求解器（`scipy.linalg.eig` / `numpy.eigvals` / `scipy.eigvals` / ARPACK 1e−10 / ARPACK 1e−14） | **只换线性代数与求根器** | **不是独立路径**（同自动机、同矩阵），只测求解器噪声 | 轴向 w=4,6,8 | +| （旁证）rev769 `diagonal_charge_transfer.py` | 同源但"两行记忆"的 (1,1) 变体 | 与 B 同源 | 不作独立路径 | (1,1) n=2..5 | + +### 2.2「同一个对象」的证书（比态数更强） + +`scripts/fl_matrix_cert.py` → `raw/step_matrix_certificate.json`：在 w=3,4,5,6、两个扇区上比较 +A/B/M 三条自动机的**安全块全谱**： + +| w | 扇区 | 安全态数 (A/B/M) | `max\|Δλ\|` A−B | A−M | B−M | +|---|---|---|---|---|---| +| 3 | G4/G8 | 7/7/7 | **0.0** | **0.0** | **0.0** | +| 4 | G4/G8 | 19/19/19 | **0.0** | **0.0** | **0.0** | +| 5 | G4/G8 | 51/51/51 | **0.0** | **0.0** | **0.0** | +| 6 | G4/G8 | 141/141/141 | **0.0** | **0.0** | **0.0** | + +⇒ 三个独立实现在 w≤6 上生成**同一个矩阵**（态数相同、谱逐位相同）。 +这把"路径差"干净地还原成"求解器/求根器/算术差"。 + +--- + +## 3. 底噪 `F` 的数字 + +### 3.1 定义 + +> **`F` := 在同一几何、同一 `p_c`、同一定义下，各独立路径给出的 `p_root` 的最大两两离差（以 p 为单位）。** + +被采纳的独立路径集合：`{A_lib, B_oblique_tight, M_mine}`（三条不同实现）， +以及再加 `H_ld`（float128 算术）的版本。全部用**同一套紧设置**（稠密特征分解；ARPACK 仅在 +矩阵过大时使用且 `tol≤1e-14`；求根 `xtol=1e-16`）。 + +### 3.2 结果（`floor.json`） + +| 量 | 值 | +|---|---| +| `F_p_implementation_float64`（三条 float64 实现） | **2.22e−16**（= 1 ulp） | +| `F_p_including_arithmetic_path`（+ float128） | **4.68e−16** | +| **采纳的 `F`** | **4.68e−16** | +| `F_Ω` 换算到 ℓ=8（`F·8⁴`） | 1.92e−12 | +| 求解器变体离散度（同矩阵，w=4/6/8） | 3.33e−16 / 8.88e−16 / **5.55e−16** | +| ARPACK 自身 λ 误差（探针，w=8） | ≤3.50e−15 相对 ⇒ **≤1.73e−15 在 p 上** | +| 逐几何（有 ≥2 条独立路径的） | w=4 **0.0**；w=5 **0.0**；w=6 **2.22e−16**；w=8 **0.0**（A 与 B 逐位相同） | + +逐几何明细（p_root，取自 `floor.json → geometry_table`）： + +| 几何 | ℓ | A_lib | B_oblique_tight | M_mine | H_ld | `F`(采纳) | +|---|---|---|---|---|---|---| +| axis_n4 | 4.000 | 0.5914171708531382 | 0.5914171708531382 | 0.5914171708531382 | 0.5914171708531385 | 3.19e−16 | +| axis_n5 | 5.000 | — | 0.5922358232050272 | 0.5922358232050272 | — | 0.0 | +| axis_n6 | 6.000 | 0.5925073562056412 | 0.5925073562056414 | 0.5925073562056414 | 0.5925073562056417 | 4.68e−16 | +| axis_n8 | 8.000 | 0.5926727605746269 | 0.5926727605746269 | — | 0.5926727605746273 | 3.94e−16 | + +**`axis_n8`（就是 2.1e−12 那个几何）：A 与 B 两条独立实现的 `p_root` 逐位相同（离差 0.0）； +唯一非零离差来自 float128 路径，3.94e−16（≈3.5 ulp）。** + +### 3.3 它不是全部 —— **shipped 配置的代价比 `F` 大 4 个数量级** + +把 B 路径用**出厂设置**（ARPACK `tol=1e-10` + `scipy.brentq` `xtol=3e-11`，即文件 A 的实际设置） +重跑一遍（`raw/step_shipped_vs_repro.json`）： + +| 几何 | ℓ | shipped − tight (p) | shipped 设置内部离散 (p) | +|---|---|---|---| +| axis_n8 | 8.000 | −1.00e−16 | −5.69e−13 | +| axis_n9 | 9.000 | +2.10e−15 | −1.30e−12 | +| diag_n4 | 5.657 | +4.31e−13 | −3.76e−14 | +| diag_n5 | 7.071 | +2.80e−13 | −6.18e−13 | +| slope21_n3 | 6.708 | −1.84e−14 | −1.30e−13 | +| slope21_n4 | 8.944 | **+1.71e−12** | −4.82e−12 | +| slope31_n3 | 9.487 | −5.80e−14 | −1.74e−12 | +| slope32_n2 | 7.211 | −1.36e−13 | +1.50e−15 | +| slope52_n2 | 10.770 | **+9.29e−12** | −6.31e−12 | + +* **`max |shipped − tight| = 9.29e−12`** —— shipped 文件自己的根值，最坏处偏离紧值 9.3e−12， + 是信号 `S = 2.71e−12` 的 **3.4 倍**。 +* **`max |loose − tight| = 6.31e−12`** ⇒ 出厂设置的**复现性只有 ~6e−12**。 +* 附带：`scipy.brentq` 的**默认** `xtol=2e-12` 就足以把 `p_root` 推偏 4.33e−13（w=8）， + 7.04e−14（w=6）—— 见 `raw/pathS_solvers.json` 的 `p_root_brentq_default`。 + +### 3.4 它是否覆盖 `S`？ + +``` +S = 2.71e-12（小幅度） ⇒ S/F = 5793 ⇒ GO +S = 3.33e-11（大幅度） ⇒ S/F = 71185 ⇒ GO +S = 2.71e-12 / F_shipped ⇒ 0.43 ⇒ NO-GO（若用出厂设置） +``` + +--- + +## 4. 裁定：GO / NO-GO / 临界 + +**裁定：GO —— 但有条件；退回条件只有一个（配置），不是判据本身。** + +1. **「2.1e−12 的不可消除底噪」不存在。** 它是 `p_c` 记账差（§1）。判据并没有在几何之前 + 撞上精度墙；它撞上的是**两份文件用了两个不同的 `p_c`**。 +2. **真底噪 `F = 4.68e−16`，`S/F ≈ 5.8×10³`。** 所以 `P0=P8` 的 `~1e−12` 判据在 + 算术上是**可达的**（余量近 4 个数量级）。 +3. **但若沿用 shipped 的那套设置，就是 NO-GO**（`S/F_shipped = 0.43`）。差别全在： + +### 修什么（两行级改动） + +* **(i) 四个取向统一 `p_c`，并且只用一个值。** 这是**正确性问题**，不是精度问题： + `Ω = −(p_root − p_c)ℓ⁴`，`p_c` 差 ε 会给所有取向加同一个纯 H0 位移 `ε·ℓ⁴`， + 在 ℓ=33.24 时 ε=1e−12 ⇒ Ω 位移 1.2e−6，直接把 `P0=P8` 打穿（而 H8 不动）。 +* **(ii) 收紧求解器与求根器：** 状态数 n≤~2500 用稠密 `scipy.linalg.eig`； + 更大时 ARPACK `tol ≤ 1e-13`；求根用 `xtol ≤ 1e-15`（**不要** scipy `brentq` 默认 2e−12， + **更不要** shipped 的 3e−11）。四个取向必须用**同一套**设置。 + +### 怎么知道修好了 + +* 对**同一几何**用 §2 里 ≥2 条独立路径复算 `p_root`，离差必须 **≤1e−15**（本报告的 F 检验）； +* 把 `(1,0)` 轴向当控制几何：三条独立自动机的安全块**全谱逐位相同**（§2.2 已证 w≤6）； +* 开机前先跑一次 N=325（ℓ=18.03，3 取向）的 H0/H4/H8 版本 + 同样的 F 检验；若它也过， + 再上 N=1105。N=325 比 N=1105 便宜得多且条件数更好（L2=0.598 vs 0.967）。 + +--- + +## 5. 不被 `F` 覆盖的系统偏差（**换路径测不出来**） + +`F` 只测「实现噪声」。下面这些是**所有路径共有**的，必须单独记账： + +1. **`p_c` 的取值/统一性。** `F` 的定义里 `p_c` 是**共同常数**；一旦四个取向用了不同 `p_c` + （shipped 文件正是如此），产生的 H0 污染 `ε·ℓ⁴` 完全在 `F` 之外，且量级足以击穿判据。 +2. **有限 ℓ 的模型误差（ℓ^{−8} 污染）。** `n1105mix` 在轴向上测到 `C = 3.54±0.45`（8σ）； + 这是**模型**误差，换代码路径一点都不会变。跨 ℓ 拟合 `B(θ)` 的偏差不进入 `F`。 +3. **态空间完备性 / 任何截断。** 本报告的所有自动机都是**完备枚举**（`state_cap` 从未触发）， + 所以这里这一项为零；但 N1105 若用 `state_cap`、记忆截断或任何近似，那是新的系统偏差， + `F` **测不出来**。 +4. **斜向 twist 的实现路径。** 斜向几何（`(a,b)` 的 Bezout 斜基 + 前沿记忆）在本报告里 + 只有 `B_oblique` **一条**路径（`diagonal_charge_transfer.py` 与它同源）。 + `F` 对**轴向**是三条独立路径互证，对**斜向**实际上只有一条 ⇒ 斜向 twist 的正确性 + 没有被 `F` 覆盖。 +5. **谐波分解/投影后处理。** 从 `Ω(θ,ℓ)` 抽 `B0,B4,B8` 的拟合、投影权重、跨 ℓ 外推 + 都是后处理模型误差，不在 `F` 内。 +6. **求根器停止条件**（这一项**在** `F` 内，但极易被配置放大：默认 brentq 就够造成 4e−13）。 + +**举证上的红线**：本报告**不用** `F` 去担保任何会被上面 1–6 击穿的结论。§4 的 GO 只针对 +「**在统一 `p_c` + 紧公差 + 完备自动机**这一前提下，四个取向的 `p_root` 能一致到 5e−16」。 +第 1 项已由本报告修掉；第 2 项（ℓ^{−8}）仍然是判定 `P0=P8` 的**主要剩余风险**。 + +--- + +## 6. 「若要判 `P0=P8`，还缺什么」 + +1. **ℓ^{−8} 污染的定量控制。** 现在只有 4–5 个宽度、4 个局部斜率、2 种外推 + （`n1105mix` §2 已复现：换横坐标得 3.995，加 1/w² 得 4.336；包含 17/4=4.25）。 + N1105 的四个取向在**同一个 ℓ** 上，这能去掉"跨 ℓ 外推"的一部分，但 + `B(θ)` 里仍混着 `C(θ)/ℓ²`（ℓ^{−8}）——**需要 ≥3 个 ℓ 或独立的 `C(θ)` 估计**。 +2. **N1105 四个取向的可行性尚未实测。** `(24,23)` 的短边 23（四方向短边 4/9/12/23）， + 状态数未知；`n1105mix` §3.2 已指出"去掉 `(24,23)`"会让条件数烂 8.5×。 + 这需要在**统一 `p_c` + 紧公差**下先做一次**小规模 scout**（例如先只做 + `(33,4)` 与 `(24,23)` 两个方向，看状态数与墙钟）。 +3. **斜向 twist 自动机的独立复核。** §5 第 4 条：斜向只有一条路径。建议把 + `M_mine` 的双覆盖构造推广到斜基（或反之，把 Bezout 自动机写成第二份独立实现）。 +4. **N=325 先导**（ℓ=18.03，3 取向，H0/H4/H8）—— 便宜、条件数更好，可作为 + 「四取向 `p_root` 一致到 ≤1e−15」这一 F 检验的第一次真实彩排。 + +--- + +## 7. 复现方式 + +```bash +H=~/.workbuddy/skills/connect-huawei-codebuddy/scripts/huawei +export PYTHONPATH=/workspace/mo/compat:/workspace/mass761/pylibs/pylibs_local:/workspace/dpfloor/scripts +cd /workspace/dpfloor +python3 scripts/fl_convention.py # §1 约定分解 +python3 scripts/fl_path_lib.py /workspace/sectorA/in/engine 0.59274605079210 4,6,8 out/pathA_lib.json +python3 scripts/fl_path_oblique.py 0.59274605079210 tight out/pathB_oblique_tight.json +python3 scripts/fl_path_oblique.py 0.59274605079210 tight out/pathB_oblique_tight2.json slope21_n4,slope31_n3,slope32_n2,slope52_n2 +python3 scripts/fl_path_oblique.py 0.59274605079 loose out/pathB_oblique_loose.json +python3 scripts/fl_path_mine.py 0.59274605079210 3,4,5,6 out/pathM_mine.json +python3 scripts/fl_hp.py 0.59274605079210 4,6,8 out/pathH_longdouble.json +python3 scripts/fl_solvers.py 0.59274605079210 4,6,8 out/pathS_solvers.json \ + scipy_eig,numpy_eigvals,scipy_eigvals,arpack_1e10,arpack_1e14 +python3 scripts/fl_matrix_cert.py # §2.2 同对象证书 +python3 scripts/fl_arpack_probe.py # §3.3 ARPACK 公差代价 +python3 scripts/fl_probe_hp.py # H_ld 收敛证书 +python3 scripts/fl_shipped_vs_repro.py # §3.3 shipped vs 紧 +python3 scripts/fl_assemble.py # F 与 S/F +``` + +环境：Python 3.9.9 + numpy 1.26.4 / scipy 1.13.1 / mpmath 1.4.1 +（`PYTHONPATH` 指向 `/workspace/mo/compat` 的 `int.bit_count` 垫片与 +`/workspace/mass761/pylibs/pylibs_local` 的现成科学栈；**未安装、未修改任何系统包**）。 +本报告**未写入** `/workspace/sectorA/`、`/workspace/v802src/`、`/workspace/n1105mix/`、 +`/workspace/verify768/`；只写 `/workspace/dpfloor/`。 + +## 8. 交付物 + +| 文件 | 内容 | +|---|---| +| `dpfloor-out/note.md` | 本文件 | +| `dpfloor-out/floor.json` | 逐路径原始 `p_root`/`Ω` + 逐几何 `F` + `S/F` + 判定 | +| `dpfloor-out/scripts/*.py` | 11 个自包含脚本（云机上实际跑过的版本） | +| `dpfloor-out/raw/*.json` | `step0_convention` / `pathA_lib` / `pathB_oblique_{tight,tight2,loose}` / `pathM_mine` / `pathH_longdouble` / `pathS_solvers` / `step_arpack_probe` / `step_matrix_certificate` / `step_hp_certificate` / `step_shipped_vs_repro` | +| `dpfloor-out/raw/log_*.txt` | 全部运行日志 | diff --git a/docs/manuscripts/geometric-balance/diagonal-spin4-charge-root-20260914.md b/docs/manuscripts/geometric-balance/diagonal-spin4-charge-root-20260914.md new file mode 100644 index 000000000..69ed99c32 --- /dev/null +++ b/docs/manuscripts/geometric-balance/diagonal-spin4-charge-root-20260914.md @@ -0,0 +1,238 @@ +# Same-model diagonal sign flip of the `w^-4` charge-root correction + +2026-09-14. Deterministic safe-transfer control on the **same square-site NN / complementary matching model**, with the short cylinder period changed from an axis vector to a diagonal vector. + +This is currently the sharpest direct test of the sector-odd spin-four anisotropy conjecture. It avoids comparing different lattices and does not infer the spin from the exponent alone. + +## 1. Spin-four prediction + +Suppose the leading sector-odd lattice correction at criticality transforms as spin four. If the short physical period makes angle `theta` with the square-lattice x-axis, the leading amplitude must carry the square harmonic + +\[ +\cos(4\theta). \tag{1.1} +\] + +The axial semi-infinite charge root has + +\[ +p_w^{axis}-p_c +\sim-\frac{A}{w^4}, +\qquad A>0. \tag{1.2} +\] + +For the diagonal direction + +\[ +\theta=\pi/4, +\qquad +\cos(4\theta)=-1. \tag{1.3} +\] + +Therefore the spin-four conjecture predicts **both** + +\[ +\boxed{p_n^{diag}-p_c>0} \tag{1.4} +\] + +and, after using the physical circumference + +\[ +\ell=n\sqrt2, +\] + +\[ +\boxed{ +(p_n^{diag}-p_c)\ell^4 +\longrightarrow A, +} \tag{1.5} +\] + +the same positive amplitude that appears as `(pc-p_axis) w^4` on the axis. + +A scalar sector-odd correction of the same scaling dimension would not predict this orientation sign flip. + +## 2. Exact diagonal coordinate transfer + +Use the integer basis + +\[ +u=(1,1),\qquad v=(0,1),\qquad \det(u,v)=1, \tag{2.1} +\] + +so + +\[ +(x,y)=s u+t v=(s,s+t). \tag{2.2} +\] + +Quotient by + +\[ +s\sim s+n, \tag{2.3} +\] + +which is exactly the physical period `n(1,1)`. No rotation of the microscopic interaction is made. + +In `(s,t)` coordinates the NN edges with positive `t` displacement are + +\[ +(\Delta s,\Delta t)=(0,1),\ (-1,1). \tag{2.4} +\] + +The matching diagonals add + +\[ +(1,0),\qquad(-1,2). \tag{2.5} +\] + +Hence two frontier rows are sufficient for the full matching graph. The transfer keeps integer lifted-`s` gains and rejects a transition immediately when a component acquires a cycle with nonzero deck gain. + +The implementation is + +`scripts/diagonal_charge_transfer.py`. + +This is the same safe/void semantics as the axial transfer, only sliced in an integer oblique basis. + +## 3. Charge roots + +Using the safe Perron equality + +\[ +\lambda^0_{4,n}(p) +=\lambda^0_{8,n}(1-p), \tag{3.1} +\] + +the diagonal roots are + +| `n` | period | `p_n^diag` | `p_n^diag-pc_ref` | +|---:|---|---:|---:| +| 2 | `(2,2)` | `0.5997254073143432` | `+6.9793565e-3` | +| 3 | `(3,3)` | `0.5937572147651211` | `+1.0111640e-3` | +| 4 | `(4,4)` | `0.5930452429807460` | `+2.9919219e-4` | +| 5 | `(5,5)` | `0.5928657129692908` | `+1.1966218e-4` | + +The diagnostic reference is + +\[ +p_c^{ref}=0.59274605079. \tag{3.2} +\] + +It is not used to locate the roots. + +The sign is already the opposite of the axial sequence. + +## 4. Physical-circumference amplitude + +Since + +\[ +\ell=n\sqrt2, +\qquad +\ell^4=4n^4, \tag{4.1} +\] + +the scaled diagonal shifts are + +| `n` | `(p_n^diag-pc_ref) ell^4` | +|---:|---:| +| 2 | `0.4466788` | +| 3 | `0.3276171` | +| 4 | `0.3063728` | +| 5 | `0.2991554` | + +The independent axial transfer gives + +\[ +(p_c^{ref}-p_8^{axis})8^4=0.3001967, \tag{4.2} +\] + +\[ +(p_c^{ref}-p_9^{axis})9^4=0.2980447. \tag{4.3} +\] + +Thus already at the available widths, + +\[ +\boxed{ +(p_n^{diag}-p_c)\ell^4 +\approx +(p_c-p_w^{axis})w^4 +\approx0.30.} \tag{4.4} +\] + +The diagonal sequence approaches this value from above while the axial sequence approaches it in the opposite charge-root direction. + +Machine-readable values are in + +`results/geometric-consistency/diagonal-charge-transfer-n2-n5-20260914.json`. + +## 5. Interpretation + +The same-model result is exactly the leading angular behavior expected from + +\[ +\boxed{ +p_{\ell,\theta}^{ch}-p_c +\sim-\frac{A\cos(4\theta)}{\ell^4}.} \tag{5.1} +\] + +At `theta=0`, (5.1) is negative. At `theta=pi/4`, it is positive with equal magnitude. + +This supplies three pieces of evidence simultaneously: + +1. exponent four; +2. orientation sign flip; +3. physical-length amplitude agreement. + +The last two are not explained by merely assigning an arbitrary scalar operator dimension `x_t+4`. + +## 6. Relation to the square/kagome evidence + +Jacobsen's square/kagome comparison suggested that rotational symmetry may remove the exponent-four correction on a three-/six-fold lattice. The present experiment is conceptually cleaner in one respect: no lattice universality comparison is needed. + +The microscopic model is held fixed and only the physical homology direction is changed. A genuine spin-four correction must rotate with the cylinder orientation; a scalar one cannot. + +Therefore the diagonal sign flip materially strengthens the spin-four interpretation in + +`sector-odd-spin4-anisotropy-20260914.md`. + +## 7. New directional conjecture + +The natural all-angle refinement is + +\[ +\boxed{ +A_4(\theta)=A\cos(4\theta)} \tag{7.1} +\] + +for the leading square-lattice dual-odd correction after expressing circumference in the isotropic physical correlation-length metric. + +For a primitive direction `(a,b)`, + +\[ +\cos(4\theta) +=\frac{a^4-6a^2b^2+b^4}{(a^2+b^2)^2}. \tag{7.2} +\] + +This gives strong future controls without an angle scan: + +- `(1,0)`: `+1`; +- `(1,1)`: `-1`; +- `(2,1)`: `-7/25`; +- `(5,2)`: `41/841`, close to a leading-amplitude zero. + +A `(5,2)` or nearby rational direction should strongly suppress the `ell^-4` term and expose the next correction, but its transfer has a larger row memory and should only be attempted if needed. + +## 8. Important geometric normalization + +The transfer step in an oblique integer basis need not have unit physical distance perpendicular to the circumference. That factor affects individual excitation energies, but it cancels from the **root location**, which is a ratio of the sector mismatch and its thermal derivative. + +For comparing root-shift amplitudes across orientations, the essential geometric normalization is the physical circumference `ell=|n u|`, as used above. + +A future direct comparison of `Theta(pc)` amplitudes themselves must additionally normalize the longitudinal row spacing. + +## 9. Claim boundary + +The diagonal roots are deterministic finite-state transfer outputs. The coordinate edge list is an exact rewrite of the square NN/matching graph. The observed sign flip and physical-length amplitude agreement are numerical finite-width facts. + +The asymptotic formula (5.1), its interpretation as a spin-four thermal-family correction, and the all-angle law (7.1) remain scaling conjectures. Wider diagonal widths or one additional rational direction would strengthen the asymptotic case, but the existing same-model sign/amplitude test is already substantially more discriminating than an exponent-only fit. diff --git a/docs/manuscripts/geometric-balance/difference-tangent-amplitude-hierarchy-20260914.md b/docs/manuscripts/geometric-balance/difference-tangent-amplitude-hierarchy-20260914.md new file mode 100644 index 000000000..3c1c55e1b --- /dev/null +++ b/docs/manuscripts/geometric-balance/difference-tangent-amplitude-hierarchy-20260914.md @@ -0,0 +1,211 @@ +# The noncommon correction as a difference-tangent amplitude hierarchy + +Date: 2026-09-14 + +Status: research reformulation based on the exact square-site Euler source and angular/radial separation. It weakens unsupported continuum parity language into concrete amplitude-derivative statements that can in principle be checked by transfer/Ward methods. + +## 1. Two-colour source coordinates + +The exact square-site Euler source has cluster-fugacity component + +```text +q_b = log Q_black, +q_w = log Q_white, +``` + +with the Matching-One source tangent + +```text +q_diff=(q_b-q_w)/2, +partial_h q_diff=1, +``` + +plus fixed local site/edge/plaquette counterterms. + +Introduce also + +```text +q_common=(q_b+q_w)/2. +``` + +The physical `h` source is a declared tangent in this enlarged two-colour parameter space, not a postulated scalar parity on continuum fields. + +## 2. Expand finite-size blocks by angular irrep and scaling dimension + +For a fixed geometry/modulus, write schematically the relevant finite-size free-energy/rank correction as + +```text +sum_(x,a) A_a^(x)(q_common,q_diff,...) L^(2-x) H_a(theta), +``` + +where + +```text +H_a(theta)=cos(a theta), +a=0,4,8,... +``` + +for the square/reflection-even sectors under discussion. + +The root response to the Euler source depends on the **difference-tangent amplitude** + +```text +D_a^(x) + := [partial_(q_diff) A_a^(x)]_(0) + + declared local-counterterm contribution. (2.1) +``` + +The statement that a block is “common” is simply + +```text +D_a^(x)=0. +``` + +No OPE-level matching parity is needed to state or test this. + +## 3. The large common `x≈4`, H4 correction becomes a precise null condition + +Individual/symmetric magnetic-sector data show a substantial ordinary square anisotropy compatible with + +```text +x≈4, +a=4. +``` + +If its difference-tangent coefficient were nonzero, then its root shift would scale with exponent + +```text +x-x_t = 4-5/4 = 11/4. +``` + +No such leading `L^-11/4` Matching-One root correction is observed; the actual leading shift is much smaller, `L^-4`. + +Thus the first structural statement to prove is not + +```text +“the x=4 field is matching even” +``` + +but the narrower condition + +```text +boxed: +D_4^(4)=0. (3.1) +``` + +This can arise from exact source symmetry, a Ward identity, cancellation with the local Euler counterterm, or a combination. The mechanism is an open theorem/interface question. + +## 4. The observed leading block says `D_4^(21/4) != 0` + +The deterministic oblique-cylinder result identifies a leading angular H4 difference response with root exponent four. + +Using + +```text +x-root exponent = x-x_t, +``` + +root exponent four corresponds to + +```text +x=21/4. +``` + +The observable statement is therefore + +```text +boxed: +D_4^(21/4) != 0 (4.1) +``` + +for the actual square lattice/source, up to the usual asymptotic interpretation of the finite data. + +The competing `eight-arm scalar` and `thermal spin-four descendant` stories are then different explanations of which continuum block produces this nonzero derivative. The lattice fact is the nonzero H4 amplitude derivative, not either field name. + +## 5. Post-H4 scalar becomes the next amplitude derivative + +If #808/future leakage-controlled data establish an angular H0 root correction near exponent seven, its linear-scalar interpretation is + +```text +D_0^(33/4) != 0, (5.1) +``` + +because + +```text +33/4-5/4=7. +``` + +But the quadratic H4 mechanism gives a different object: it is second order in the `x=21/4` coupling and produces H0/H8 at root exponent `29/4`, not a new first derivative `D_0^(33/4)`. + +This cleanly separates the two theories: + +```text +linear scalar: + first difference tangent of a new x=33/4 H0 block; + +quadratic composite: + second response built from the existing x=21/4 H4 block. +``` + +## 6. A hierarchy of theorem targets + +Instead of assigning continuum parities globally, aim to establish the small list + +```text +D_4^(4) = 0, +D_4^(21/4) != 0, +D_0^(x<33/4) = 0 or bounded/identified, +D_0^(33/4) ? +``` + +with angular and source definitions fixed. + +A lower-dimensional scalar/interchiral competitor matters precisely if it gives a nonzero `D_0^(x)` in the actual Euler-source tangent. This is the correct lattice-side kill test for #61/#768. + +## 7. Ward identity target for the first null + +The current Ward programme proposes a spin-four finite-size structure involving a quasiprimary `U4+Ubar4` and an `E4(tau)` thermal/coordinate term. + +In the present language the useful theorem is to differentiate the **actual rank/Euler projection** along the `q_diff` source and show that the `x=4` H4 combination has zero net coefficient after its contact/seam/local-counterterm pieces are included. + +That is a single mixed Ward identity: + +```text +partial_(q_diff) [x=4 H4 amplitude of the rank projection] = 0. +``` + +It is weaker than deriving an OPE automorphism and exactly as strong as needed to explain why the `11/4` root term is absent. + +## 8. Empirical response-block version + +If the theorem is inaccessible, estimate a small response matrix whose columns are residualized microscopic sources and whose rows are angular/radial blocks. + +For each candidate block `(x,a)` fit + +```text +R_(x,a),source(L) +``` + +and test whether the difference-tangent column vanishes or persists. + +Degenerate continuum blocks should be treated matrix-valuedly; only the amplitude derivative of the measured block is required. + +## 9. Claim boundary + +Exact/source-defined: + +- Euler difference tangent and local counterterms; +- notion `D_a^(x)` once a finite-size block decomposition is declared. + +Empirical/scaling: + +- current evidence for `D_4^(21/4) != 0`; +- absence of a visible `11/4` root term as evidence for `D_4^(4)=0`. + +Open: + +- proof mechanism for the `x=4` null; +- whether the post-H4 scalar is a new linear `D_0^(33/4)` block, a quadratic H4 composite, or a logarithmic/mixed combination. + +This reformulation keeps the selection problem attached to the actual source and observable instead of assigning parity to fields before the tangent map is known. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/dilute-directional-geodesic-entropy-20260914.md b/docs/manuscripts/geometric-balance/dilute-directional-geodesic-entropy-20260914.md new file mode 100644 index 000000000..06e884399 --- /dev/null +++ b/docs/manuscripts/geometric-balance/dilute-directional-geodesic-entropy-20260914.md @@ -0,0 +1,375 @@ +# Dilute directional norm = graph distance cost minus geodesic entropy + +2026-09-14. Elementary fixed-direction small-`p` theorem for the square NN and matching graphs. This generalizes the axial `1` versus `3` dilute centre calculation in `dilute-directional-mass-centres-20260914.md`. + +No OZ theorem or continuum limit is used. The lower probability bound follows one adaptive forward path through independent layers; the upper probability bound sums all walks and uses a multivariate exponential tilt. + +## 1. Entropy functions + +For `alpha in [0,1]`, let + +\[ +H_2(\alpha) +=-\alpha\log\alpha-(1-\alpha)\log(1-\alpha) \tag{1.1} +\] + +with the usual `0 log 0=0` convention. + +For `beta in [-1,1]`, let + +\[ +h_3(\beta) +=\inf_{t\in\mathbb R} +\{\log(1+2\cosh t)-\beta t\}. \tag{1.2} +\] + +This is the maximum entropy of a probability law on `{-1,0,+1}` with prescribed mean `beta`. Equivalently, if `J_3` is the Cramer rate function of the uniform law on those three steps, + +\[ +h_3(\beta)=\log3-J_3(\beta). \tag{1.3} +\] + +It is even, strictly concave on `(-1,1)`, and + +\[ +h_3(0)=\log3,\qquad h_3(\pm1)=0. \tag{1.4} +\] + +## 2. Theorem + +Fix a nonzero integer vector `u=(a,b)`. Put `A=|a|`, `B=|b|`. + +For NN square-site percolation, + +\[ +\boxed{ +\tau_{4,p}(u) +=(A+B)\log(1/p) +-(A+B)H_2\!\left(\frac{A}{A+B}\right) ++o(1),\qquad p\downarrow0.} \tag{2.1} +\] + +For matching/king adjacency, let + +\[ +M=\max(A,B),\qquad m=\min(A,B),\qquad \beta=m/M. +\] + +Then + +\[ +\boxed{ +\tau_{8,p}(u) +=M\log(1/p)-M h_3(\beta)+o(1),\qquad p\downarrow0.} \tag{2.2} +\] + +The theorem is homogeneous: these are costs for displacement `n u` divided by `n`. Reflections and coordinate exchange reduce the proofs to `a>=b>=0` as needed. + +The leading coefficient is the graph distance per copy of `u`; the constant correction is exactly the exponential entropy of shortest directed paths with that slope. + +## 3. A multivariate all-walk upper bound on connection probability + +For a finite step set `S`, any open connection from `0` to `n u` contains a self-avoiding path. Replacing self-avoiding paths by all walks and using a vector exponential tilt `t in R^2` gives + +\[ +P_p(0\leftrightarrow n u) +\le p e^{-n t\cdot u} +\sum_{L\ge0}[p M_S(t)]^L, \tag{3.1} +\] + +where + +\[ +M_S(t)=\sum_{s\in S}e^{t\cdot s}. \tag{3.2} +\] + +Thus, whenever `p M_S(t)<1`, + +\[ +\tau_p(u)\ge t\cdot u. \tag{3.3} +\] + +Taking the supremum gives + +\[ +\boxed{ +\tau_p(u)\ge +\sup\{t\cdot u:pM_S(t)<1\}.} \tag{3.4} +\] + +We now evaluate this support function asymptotically as `p->0`. + +## 4. NN lower mass bound + +For NN, + +\[ +M_4(t_x,t_y)=2\cosh t_x+2\cosh t_y. \tag{4.1} +\] + +Assume first `a,b>0`. At the maximizing scale both tilts tend to `+infinity`, so writing `X=e^{t_x}`, `Y=e^{t_y}` reduces the leading constraint to + +\[ +X+Y=p^{-1}(1+o(1)). \tag{4.2} +\] + +Maximizing `a log X+b log Y` under `X+Y=p^{-1}` gives + +\[ +X=\frac{a}{a+b}p^{-1}, +\qquad +Y=\frac{b}{a+b}p^{-1}. \tag{4.3} +\] + +The negative-exponential terms in the coshes contribute `o(1)` to the optimized objective. Hence + +\[ +\sup_{pM_4<1}t\cdot u +=(a+b)\log(1/p) ++a\log\frac{a}{a+b} ++b\log\frac{b}{a+b} ++o(1), \tag{4.4} +\] + +which is the right side of (2.1). If one coordinate is zero, the same formula follows by taking the boundary optimizer; the entropy term is zero. + +Therefore (3.4) gives the lower bound on `tau_4` required for (2.1). + +## 5. NN adaptive forward path gives the matching upper mass bound + +Assume `a,b>=0` and put `L=a+b`. Use the layer coordinate `x+y`. Starting from an occupied origin, at every step inspect the two forward neighbours + +\[ +(x+1,y),\qquad(x,y+1). +\] + +If neither is open, fail; otherwise choose one open candidate uniformly using independent auxiliary randomness. + +Each step inspects a fresh layer, so the two-site occupation pairs are independent across steps. The one-step survival probability is + +\[ +q_2(p)=1-(1-p)^2=2p-p^2. \tag{5.1} +\] + +Conditional on survival, symmetry makes the chosen move exactly uniform between the two forward directions. After `nL` successful steps, the endpoint is `n(a,b)` exactly when a Binomial `(nL,1/2)` count equals `na`. + +Hence + +\[ +P_p(0\leftrightarrow n(a,b)) +\ge p\,q_2(p)^{nL} +P\{\operatorname{Bin}(nL,1/2)=na\}. \tag{5.2} +\] + +The binomial local large-deviation formula gives + +\[ +-\frac1n\log P\{\operatorname{Bin}(nL,1/2)=na\} +=L\,[\log2-H_2(a/L)]+o(1). \tag{5.3} +\] + +Since `-log q_2=-log(2p)+o(1)`, equations (5.2)--(5.3) yield + +\[ +\tau_{4,p}(u) +\le L\log(1/p)-L H_2(a/L)+o(1), \tag{5.4} +\] + +matching (4.4) and proving (2.1). + +The auxiliary randomization causes no issue: for every fixed occupation configuration, algorithmic success implies the existence of an open path, so averaging the success probability is a valid lower bound on the percolation event. + +## 6. Matching adaptive path + +Assume `a>=b>=0`; the other sectors follow by square symmetry. Use columns as independent layers. At each successful step increase `x` by one and inspect the three sites at vertical offsets `-1,0,+1` in the next column. + +The survival probability is + +\[ +q_3(p)=1-(1-p)^3=3p-3p^2+p^3. \tag{6.1} +\] + +Conditional on survival, the selected vertical increment is exactly uniform on `{-1,0,+1}`. After `na` steps the endpoint is `(na,nb)` precisely when the increment sum equals `nb`. + +If `J_3(beta)` is the Cramer rate for the uniform three-step law, + +\[ +P\{S_{na}=nb\} +=\exp[-na J_3(b/a)+o(n)]. \tag{6.2} +\] + +Therefore + +\[ +\tau_{8,p}(a,b) +\le a[-\log q_3(p)+J_3(b/a)] \tag{6.3} +\] + +and, using `J_3=log3-h_3` and `-log q_3=-log(3p)+o(1)`, + +\[ +\tau_{8,p}(a,b) +\le a\log(1/p)-a h_3(b/a)+o(1). \tag{6.4} +\] + +## 7. Matching all-walk support function + +For king steps, + +\[ +M_8(t_x,t_y) +=(1+2\cosh t_x)(1+2\cosh t_y)-1. \tag{7.1} +\] + +When `a>b>=0`, the optimal `t_x->+infinity` while `t_y` stays at the finite Cramer dual parameter associated with `beta=b/a`. Uniformly for bounded `t_y`, + +\[ +M_8(t_x,t_y) +=e^{t_x}(1+2\cosh t_y)(1+o(1)). \tag{7.2} +\] + +The constraint `pM_8=1` therefore gives + +\[ +t_x=\log(1/p)-\log(1+2\cosh t_y)+o(1). \tag{7.3} +\] + +Substitution into `a t_x+b t_y` and optimization over `t_y` gives + +\[ +\sup_{pM_8<1}t\cdot(a,b) +=a\log(1/p) +-a\inf_t\{\log(1+2\cosh t)-(b/a)t\} ++o(1), \tag{7.4} +\] + +which is + +\[ +a\log(1/p)-a h_3(b/a)+o(1). \tag{7.5} +\] + +At the boundary `b=a`, the dual parameter tends to `+infinity`; direct optimization of (7.1) gives the same leading value `a log(1/p)+o(1)`, consistent with `h_3(1)=0`. Thus (3.4) matches the adaptive upper bound in every direction and proves (2.2). + +## 8. Large-d centres for a fixed rational direction + +Let + +\[ +r=\sqrt{a^2+b^2}, +\qquad e=(a,b)/r. +\] + +By homogeneity, + +\[ +\tau_p(e)=\tau_p(a,b)/r. +\] + +### NN + +Put `L=A+B` and `alpha=A/L`. Solving + +\[ +\tau_{4,p}(e)=d +\] + +with (2.1) gives + +\[ +\boxed{ +p_{4,e}(d) +=\exp[-H_2(\alpha)] +\exp[-(r/L)d]\,[1+o(1)].} \tag{8.1} +\] + +### Matching + +Put `M=max(A,B)`, `beta=min(A,B)/M`. Solving the matching equation gives + +\[ +\boxed{ +p_{8,e}(d) +=\exp[-h_3(\beta)] +\exp[-(r/M)d]\,[1+o(1)].} \tag{8.2} +\] + +These formulas are for `d->infinity` along a fixed rational direction. They do not require a directional OZ amplitude. + +## 9. Examples + +### Axis + +For `e=(1,0)`, + +\[ +H_2(1)=0, +\qquad h_3(0)=\log3. +\] + +Hence + +\[ +p_{4,e}(d)\sim e^{-d}, +\qquad +p_{8,e}(d)\sim\frac13e^{-d}, \tag{9.1} +\] + +recovering `dilute-directional-mass-centres-20260914.md`. + +### Diagonal + +For `e=(1,1)/\sqrt2`, + +\[ +H_2(1/2)=\log2, +\qquad h_3(1)=0. +\] + +Thus + +\[ +\boxed{ +p_{4,e}(d)\sim\frac12 e^{-d/\sqrt2}, +\qquad +p_{8,e}(d)\sim e^{-\sqrt2 d}.} \tag{9.2} +\] + +The matching centre is exponentially smaller than the NN centre in `d`; the graph-enhancement separation is far stronger here than the axial factor-three difference. + +### Generic genuinely tilted direction + +If `A>B>0`, then + +\[ +\frac{r}{M}>\frac{r}{A+B}. +\] + +Therefore + +\[ +\boxed{ +\frac{p_{8,e}(d)}{p_{4,e}(d)} +=\exp[-c(e)d+O(1)]\to0} \tag{9.3} +\] + +for an explicit `c(e)>0`. So at very large exponential aspect, matching enhancement creates exponentially separated occupation centres for every non-axial fixed rational direction. + +## 10. Interface to the homological free energy + +The leading dilute directional cost can be written + +\[ +\tau_{G,p}(u) +=d_G(u)\log(1/p)-s_G(u)+o(1), \tag{10.1} +\] + +where `d_G(u)` is graph distance per period and `s_G(u)` is geodesic entropy. In the conjectural homological free energy + +\[ +\Psi_\Lambda(p)=\min_u\left\{\tau_{G,p}(u)-\log\frac{N}{|u|}\right\}, +\] + +the `p->0` race between primitive slopes is therefore controlled first by graph-distance geometry and then by geodesic entropy, before any closed-component sewing amplitude enters. + +This gives a concrete asymptotic ordering of candidate homology classes in the large-`d` regime and a useful analytic control for #765-style tilted birth centres. diff --git a/docs/manuscripts/geometric-balance/dilute-directional-mass-centres-20260914.md b/docs/manuscripts/geometric-balance/dilute-directional-mass-centres-20260914.md new file mode 100644 index 000000000..e70bf6cb3 --- /dev/null +++ b/docs/manuscripts/geometric-balance/dilute-directional-mass-centres-20260914.md @@ -0,0 +1,311 @@ +# Elementary dilute directional masses and the large-d birth centres + +2026-09-14. A rigorous small-`p` comparison using only one adaptive forward path and an all-walk exponential-tilt bound. No Ornstein--Zernike theorem, component sewing, or prefactor assumption is used. + +## 1. Result + +Let `kappa_4(p)` and `kappa_8(p)` be the horizontal inverse correlation lengths of square-site NN and matching (king) adjacency. As `p downarrow 0`, + +\[ +\boxed{\kappa_4(p)=-\log p+O(p),} \tag{1.1} +\] + +\[ +\boxed{\kappa_8(p)=-\log(3p)+O(p).} \tag{1.2} +\] + +Consequently, for the exponential-aspect centres + +\[ +a(d)=\kappa_4^{-1}(d), +\qquad +c(d)=\kappa_8^{-1}(d)=1-b(d), +\] + +one has as `d->infinity`, + +\[ +\boxed{a(d)=e^{-d}+O(e^{-2d}),} \tag{1.3} +\] + +\[ +\boxed{c(d)=\frac13e^{-d}+O(e^{-2d}),} \tag{1.4} +\] + +and therefore + +\[ +\boxed{a(d)+b(d)-1 +=a(d)-c(d) +=\frac23e^{-d}+O(e^{-2d}).} \tag{1.5} +\] + +The separated birth gap also has + +\[ +\boxed{b(d)-a(d)=1-\frac43e^{-d}+O(e^{-2d}).} \tag{1.6} +\] + +Thus the strict centre asymmetry from the enhancement theorem has a concrete elementary extreme-elongation scale. + +## 2. A general all-walk exponential-tilt bound + +Let a finite-range graph have step set `S subset Z^2` and independent site density `p`. If `0` is connected to `ne_1`, there is a self-avoiding open path from `0` to `ne_1`. A path of `L` edges uses `L+1` open vertices, so a union bound followed by replacing self-avoiding paths by all walks gives, for every `t>0`, + +\[ +P_p(0\leftrightarrow ne_1) +\le p e^{-tn}\sum_{L\ge0} +\left[p\sum_{s\in S}e^{t s_x}\right]^L. \tag{2.1} +\] + +Whenever + +\[ +pM_S(t)<1, +\qquad +M_S(t)=\sum_{s\in S}e^{t s_x}, \tag{2.2} +\] + +this yields + +\[ +P_p(0\leftrightarrow ne_1) +\le\frac{p}{1-pM_S(t)}e^{-tn}. \tag{2.3} +\] + +Hence + +\[ +\boxed{\kappa(p)\ge t_*}, \tag{2.4} +\] + +where `t_*` is the positive solution of `pM_S(t_*)=1`. Taking `tinfinity` limit at the logarithmic level; it is not inferred from a finite path count alone. + +## 6. Inverting the masses + +If + +\[ +d=-\log p+O(p), \tag{6.1} +\] + +then + +\[ +\log(pe^d)=O(p), +\] + +so `p=e^{-d}(1+O(p))`; substituting once gives + +\[ +p=e^{-d}+O(e^{-2d}). \tag{6.2} +\] + +Applying this to (1.1) proves (1.3). + +Likewise if + +\[ +d=-\log(3p)+O(p), \tag{6.3} +\] + +then + +\[ +p=\frac13e^{-d}+O(e^{-2d}), \tag{6.4} +\] + +proving (1.4). Equations (1.5)--(1.6) follow algebraically. + +## 7. Consequences for the dual-even/odd coordinates + +The limiting separated-window coordinates are + +\[ +G_\infty(d)=b(d)-a(d), +\qquad +C_\infty(d)=\frac{a(d)+b(d)-1}{2}. +\] + +Hence + +\[ +\boxed{G_\infty(d)=1-\frac43e^{-d}+O(e^{-2d}),} \tag{7.1} +\] + +\[ +\boxed{C_\infty(d)=\frac13e^{-d}+O(e^{-2d}).} \tag{7.2} +\] + +Using the exact integrated identities from `neutral-gas-topological-charge-20260914.md`, + +\[ +\int_0^1P_1(p)\,dp +\to 1-\frac43e^{-d}+O(e^{-2d}), \tag{7.3} +\] + +and + +\[ +\boxed{ +\int_0^1M(p)\,dp +\to-\frac23e^{-d}+O(e^{-2d}).} \tag{7.4} +\] + +Thus even in the very elongated regime, where the unscaled birth mixture is nearly the endpoint split, the complement-odd area under the matching curve retains a calculable exponentially small asymmetry. + +## 8. Relation to the fixed-p dilute component crossover + +`dilute-winding-crossover.md` gives the more refined complete-component formula + +\[ +\nu_w(p)\sim p^w I_0(2wp) +\] + +in the joint regime `wp^2->0`. The present note has a different limit order: first define the infinite-plane mass by `n->infinity` at fixed `p`, then take `p->0`. Its elementary `O(p)` mass bracket is consistent with the Bessel exponent `-log p-2p+...` on the NN side but does not assert the `-2p` coefficient at fixed `p`. + +For matching, the factor `3` already appears at the shortest-path entropy level and is likewise compatible with the leading minimal matching-cycle coefficient in the existing dilute note. + +No interchange of those two limits is made here. diff --git a/docs/manuscripts/geometric-balance/dimension-21over4-resonance-20260914.md b/docs/manuscripts/geometric-balance/dimension-21over4-resonance-20260914.md new file mode 100644 index 000000000..fedc841a5 --- /dev/null +++ b/docs/manuscripts/geometric-balance/dimension-21over4-resonance-20260914.md @@ -0,0 +1,290 @@ +# The `x=21/4` resonance: scalar eight-arm versus spin-four thermal descendant + +Date: 2026-09-14 + +Status: conceptual reorganization of #768 after the oblique safe-transfer results. The equality of scaling dimensions below uses the standard percolation arm-exponent input; the lattice angular data are deterministic finite-width controls already committed on #771. No LCFT indecomposable structure is asserted merely from degeneracy. + +## 1. The two leading stories are exactly degenerate in total dimension + +The current square-lattice sector-odd candidates have often been described as competing mechanisms: + +1. a scalar alternating eight-arm / eight-leg field; +2. a spin `+/-4` descendant in the thermal family. + +But their total scaling dimensions are exactly the same. + +For critical percolation, the alternating `j`-arm exponent is + +```text +x_j = (j^2-1)/12. +``` + +Therefore + +```text +x_8 = 63/12 = 21/4. +``` + +The thermal field has + +```text +x_t = 5/4, +``` + +and a level-four chiral descendant has + +```text +x_t+4 = 21/4. +``` + +Thus the pseudo-critical exponent four cannot distinguish these mechanisms even in principle. + +This is stronger than saying that two unrelated fits happen to be numerically close: the continuum dimensions are algebraically identical. + +## 2. They differ in spin, not in cylinder energy + +A scalar eight-arm field has + +```text +(h,hbar)=(21/8,21/8), +spin=0. +``` + +The thermal spin-four descendant has, schematically, + +```text +(h,hbar)=(37/8,5/8) or (5/8,37/8), +spin=+/-4, +``` + +with reflection selecting the real combination. + +Both have + +```text +h+hbar=21/4. +``` + +Hence an axis-aligned cylinder energy sees the same `ell^-17/4` per-length power from both. The only clean discriminator is their transformation under rotation / map-sector data. + +On a square lattice, spin four is itself invariant under a `pi/2` microscopic rotation: + +```text +exp(i*4*pi/2)=1. +``` + +Therefore the square point group also does **not** distinguish scalar spin zero from continuum spin four. Both belong to the trivial `C4` lattice representation. + +The cylinder orientation relative to the microscopic lattice supplies the missing analyzer. + +## 3. The correct leading angular decomposition + +At the `x=21/4` scale, reflection and square symmetry allow + +```text +E_odd^phys(pc;ell,theta) + = ell^(-17/4) + [ B0 + B4 cos(4 theta) ] + + smaller terms, +``` + +where, schematically, + +```text +B0 : scalar eight-arm / any spin-0 contribution at this dimension, +B4 : spin-four thermal-family contribution. +``` + +More generally there can be multiple map/operator contributions inside each angular channel, but this is the minimal two-coordinate decomposition relevant to the current debate. + +The important consequence is: + +> `eight-arm` and `spin-four` should no longer be treated as mutually exclusive exponent hypotheses. They are two angular components of the same leading total-dimension shell unless an additional parity/map rule removes one of them. + +## 4. Existing oblique data already resolve most of the leading shell + +The current same-model oblique safe-transfer results give + +```text +E_odd^phys * ell^(17/4) / cos(4theta) ~= 1.0 +``` + +across axis, diagonal, `(2,1)`, `(3,1)`, `(3,2)` controls, with the expected sign changes. + +A near-equal-circumference Pell pair is already available: + +```text +axis w=7: ell=7, +diagonal n=5: ell=5 sqrt(2), +ell_diag^2-ell_axis^2=1. +``` + +The scaled critical mismatch gives + +```text +F_axis = +1.0358025515, +F_diag = -1.0206765597. +``` + +Hence + +```text +F_spin4-like = (F_axis-F_diag)/2 = 1.0282395556, +F_even = (F_axis+F_diag)/2 = 0.0075629959. +``` + +The angular-even leakage is only about `0.74%` of the angular-odd projector. Because the circumferences are not exactly equal, ordinary radial finite-size drift contributes to this `0.74%`; it is therefore an upper-scale diagnostic, not a clean estimate of `B0/B4`. + +The root projector independently gives essentially the same `~0.7%` leakage. + +Thus a scalar `x=21/4` component with amplitude comparable to the observed spin-four component is already strongly disfavored. + +## 5. Exact equal circumference is now a scalar-eight-arm measurement + +The proposed `(4,3), n=2` versus axis `w=10` pair has exactly equal physical circumference `ell=10`. + +Let + +```text +c4 = cos(4 theta_(4,3)) = -527/625. +``` + +At equal `ell`, define + +```text +E_axis = ell^(17/4) E^phys_axis, +E_43 = ell^(17/4) E^phys_43. +``` + +If only scalar and spin-four components are relevant at this order, + +```text +E_axis = B0 + B4, +E_43 = B0 + c4 B4. +``` + +Therefore one can solve directly + +```text +B4 = (E_axis-E_43)/(1-c4), +B0 = (E_43-c4 E_axis)/(1-c4). +``` + +This is a cleaner interpretation of the equal-circumference computation than merely calling it another spin-four check: it is an actual two-component projector on the degenerate `x=21/4` shell. + +The same algebra can be applied to the root response after dividing by the common leading thermal slope, with the usual caveat that subleading thermal anisotropy can contaminate the ratio. + +## 6. Pell near-node directions become especially decisive + +At the exact spin-four node + +```text +theta=pi/8, +cos(4theta)=0. +``` + +A **scalar** `x=21/4` component survives there at order + +```text +ell^-4 in the root shift. +``` + +A pure spin-four component does not. + +For Pell approximants with + +```text +cos(4theta)=O(ell^-2), +``` + +the spin-four `ell^-4` root contribution is geometrically demoted to `ell^-6`. + +Therefore: + +- if `B0 != 0`, a scalar eight-arm term eventually dominates the Pell sequence again as `ell^-4`; +- if `B0 = 0`, the leading node response is `ell^-6` or smaller and comes from spin-four geometric leakage / another angular sector. + +This is a much sharper use of the Pell experiment than simply “seeing the next correction power”. + +The existing `(5,2), n=2` point already lies close to the node and tracks the small `cos4theta` prediction rather than showing an obvious unsuppressed scalar floor. That is suggestive but not asymptotic evidence. + +## 7. Why exact degeneracy does not automatically imply a Jordan/logarithmic pair + +The equality + +```text +x_scalar = x_spin4 = 21/4 +``` + +is a degeneracy of **total scaling dimension**. + +Under the full emergent rotation group the two operators carry different spin, so they remain distinct representation sectors. Equality of `x` alone does not produce a Jordan cell, logarithm, or operator mixing. + +At the lattice `C4` level both appear in the same discrete irrep, so a microscopic observable may couple to both. This explains why exponent and point-group selection were insufficient. But any logarithmic/indecomposable mixing requires an additional LCFT/module statement and must be demonstrated separately. + +This prevents a repeat of the earlier Jordan over-interpretation elsewhere in the repository. + +## 8. Reframing the six-arm kill test + +The abundant theta/T3 simultaneous-birth geometry is matching-even pathwise and therefore does not by itself compete in the matching-odd one-point channel. + +The true ordering of the theory questions is now: + +1. **Lower-dimension exclusion:** does any six-arm / lower operator carry matching-odd map parity and a nonzero one-point sector difference? +2. **Leading-shell decomposition:** if lower channels cancel, what are `B0` and `B4` inside the degenerate `x=21/4` shell? +3. **Operator identity:** if `B4` dominates, is it specifically the level-four thermal quasiprimary after the actual percolation module/null quotient, or another spin-four field of the same dimension? + +Only step 3 is an operator-naming question. Steps 1--2 are observable/matrix-element questions and can be attacked directly. + +## 9. A new pivotal interpretation of the spin-four channel + +There is a plausible mechanism that makes the thermal-descendant interpretation more natural than a literal eight-arm event. + +The exact local transition decomposition has + +```text +alpha = rank 0->1 insertion mass, +beta = rank 0->2 insertion mass, +gamma = rank 1->2 insertion mass. +``` + +Pathwise matching exchanges `alpha <-> gamma` and leaves `beta` even. Hence + +```text +alpha-gamma +``` + +is a local matching-odd observable, whereas the direct jump-two / theta-spine six-arm channel sits primarily in the matching-even `beta` sector. + +Both `alpha` and `gamma` are naturally controlled at leading order by the same four-arm/thermal pivotal mechanism. Their common isotropic amplitude can cancel in the difference, leaving the first *anisotropic correction to the thermal pivotal amplitude*. A spin-four level-four thermal descendant has exactly the required dimension + +```text +x_4arm + 4 = 5/4 + 4 = 21/4. +``` + +This gives a concrete alternative to the slogan “the root is controlled by an eight-arm event”: + +> the `21/4` exponent may arise because the **difference between two four-arm thermal pivotal amplitudes** first appears in the spin-four anisotropic correction. + +The equality with the eight-arm exponent is then a resonance of dimensions, not evidence that eight geometrically alternating arms are the microscopic event being counted. + +This conjecture can be tested by the signed `alpha-gamma` output of #769 rather than by the absolute jump-two mass `beta`. + +## 10. Minimal next tests + +1. **Equal-circumference projector.** Use axis `w=10` and `(4,3),n=2` to extract `(B0,B4)` directly at the same `ell`. + +2. **Pivotal parity.** In #769, report topology-resolved contributions separately to `alpha`, `beta`, `gamma`, and especially `alpha-gamma`. A six-arm class that is large in `beta` but cancels from `alpha-gamma` is not a kill of the spin-four hypothesis. + +3. **Pell node only after step 1.** If `B0` is bounded tightly near zero, a second `(5,2)` width becomes an efficient probe of the next angular sector. If `B0` is nonzero, the node instead becomes the best way to measure the scalar eight-arm amplitude. + +4. **Module calculation.** Only after the observable shell is decomposed should one spend theory effort deciding whether the spin-four component is the thermal level-four quasiprimary, a logarithmic partner, or another map-resolved spin-four field. + +## 11. Claim boundary + +- `x_8=21/4` and `x_t+4=21/4` use standard percolation/CFT scaling inputs. +- spin zero versus spin four is exact representation bookkeeping. +- the oblique finite-transfer values and near-equal-length projector are existing deterministic controls. +- dominance of `B4`, vanishing of `B0`, the pivotal-amplitude mechanism, and the precise thermal-module identification remain conjectural. + +The main purpose of this note is to replace a false binary choice by a measurable two-coordinate leading shell. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/direction-uniform-mass-slope-20260914.md b/docs/manuscripts/geometric-balance/direction-uniform-mass-slope-20260914.md new file mode 100644 index 000000000..ac9147d00 --- /dev/null +++ b/docs/manuscripts/geometric-balance/direction-uniform-mass-slope-20260914.md @@ -0,0 +1,286 @@ +# A direction-uniform quantitative occupation slope for the correlation norm + +2026-09-14. Quantitative strengthening of the all-direction strict `p`-monotonicity proved in the directional centre theorem. + +The same Friedgut--Kalai constant that yielded the axial mass-slope inequality gives a multiplicative comparison **uniform in direction**. + +## 1. Statement + +Let `G` be NN or matching SITE percolation on `Z^2`. Let + +\[ +\tau_p(e),\qquad e\in S^1,\quad00` be a universal constant for the transitive Friedgut--Kalai sharp-threshold inequality in the convention used in `exponential-birth-centres.md`. + +**Theorem.** For every + +\[ +0=B>0`. Suppose for contradiction that + +\[ +A-B<\frac{q-p}{\rho_{FK}}B. \tag{2.2} +\] + +Then + +\[ +\rho_{FK}\frac{A-B}{B}0` sufficiently small, so that + +\[ +\boxed{ +\rho_{FK}\frac{A-d+\zeta}{d}0) +=\exp[-(A-d+o(1))\ell_n]. \tag{2.7} +\] + +Therefore for large `n`, + +\[ +P_p(r>0)> +\epsilon_n:=e^{-(A-d+\zeta)\ell_n}. \tag{2.8} +\] + +The event `r>0` is increasing and invariant under the transitive translation action on the `N_n` torus sites. Also + +\[ +\frac{\log N_n}{\ell_n}\to d, \tag{2.9} +\] + +because the extra `log ell_n/ell_n` vanishes. The Friedgut--Kalai threshold increment for probability `epsilon_n` therefore tends to + +\[ +\rho_{FK}\frac{A-d+\zeta}{d}, \tag{2.10} +\] + +which is strictly less than `q-p` by (2.5). + +Hence at parameter `q`, + +\[ +P_q(r>0)>1-\epsilon_n\to1. \tag{2.11} +\] + +But `d0)\to0. \tag{2.12} +\] + +Contradiction. Thus (1.3) holds. + +No differentiability of `tau` is used. + +## 3. Direction-uniform logarithmic slope + +Rearrange (1.4): + +\[ +\frac{\tau_p(e)}{\tau_q(e)} +\ge1+\frac{q-p}{\rho_{FK}}. \tag{3.1} +\] + +At a differentiability point in `p`, let `q downarrow p`. Then + +\[ +\boxed{ +-\partial_p\tau_p(e) +\ge\frac{\tau_p(e)}{\rho_{FK}}>0.} \tag{3.2} +\] + +Equivalently, + +\[ +\boxed{ +-\partial_p\log\tau_p(e)\ge\frac1{\rho_{FK}}.} \tag{3.3} +\] + +This is uniform in direction. + +The inequality supplies a positive lower bound on every regular directional Gumbel scale slope once the centre mass is fixed. + +## 4. Minimum period spectrum inherits the same inequality + +For any fixed period lattice `Lambda`, define + +\[ +\rho_1(p) +=\min_{\lambda\in\Lambda\setminus0}\tau_p(\lambda). \tag{4.1} +\] + +Since (1.4) holds for every period vector, + +\[ +\tau_p(\lambda) +\ge +\left(1+\frac{q-p}{\rho_{FK}}\right)\tau_q(\lambda). \tag{4.2} +\] + +Taking minima gives + +\[ +\boxed{ +\rho_1(p) +\ge +\left(1+\frac{q-p}{\rho_{FK}}\right)\rho_1(q).} \tag{4.3} +\] + +Thus the scalar period-spectrum cost has a quantitative strict decrease even when its minimizing projective direction switches. + +## 5. Consequence for normalized period-spectrum limits + +Let `N_n` be any period-lattice sequence and suppose on an interval `J` + +\[ +\frac{\rho_{1,n}(p)}{\log N_n}\to g(p) \tag{5.1} +\] + +pointwise (local uniform convergence is useful for centre trapping but not needed for this inequality). + +Passing to the limit in (4.3), + +\[ +\boxed{ +g(p) +\ge +\left(1+\frac{q-p}{\rho_{FK}}\right)g(q).} \tag{5.2} +\] + +Whenever `g(q)>0`, + +\[ +\boxed{g(p)>g(q)\quad(pd_2>0`, put + +\[ +p_i=a(d_i), \tag{6.2} +\] + +so `p_10` with + +\[ +\tau_{8,p}(e)\le\tau_{4,p+\delta_I}(e). \tag{7.1} +\] + +Apply (1.3) to `G4`: + +\[ +\tau_{4,p}(e)-\tau_{4,p+\delta_I}(e) +\ge +\frac{\delta_I}{\rho_{FK}}\tau_{4,p+\delta_I}(e). \tag{7.2} +\] + +Hence the same-parameter mass gap has the quantitative lower bound + +\[ +\boxed{ +\tau_{4,p}(e)-\tau_{8,p}(e) +\ge +\frac{\delta_I}{\rho_{FK}}\tau_{4,p+\delta_I}(e)>0.} \tag{7.3} +\] + +uniformly in direction on the compact parameter interval. + +This supplies an explicit conceptual source for the Wulff-body Minkowski buffer in `strict-wulff-body-separation-20260914.md`. + +## 8. Claim boundary + +The only quantitative constant is the universal Friedgut--Kalai constant already imported in the parent manuscript; it is not numerically optimized here. The theorem uses the author-level varying-direction winding-rate theorem on PR #771, but no OZ or p-analyticity input. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/directional-enhancement-sandwich-20260914.md b/docs/manuscripts/geometric-balance/directional-enhancement-sandwich-20260914.md new file mode 100644 index 000000000..b49cb1fdd --- /dev/null +++ b/docs/manuscripts/geometric-balance/directional-enhancement-sandwich-20260914.md @@ -0,0 +1,144 @@ +# Direction-uniform enhancement sandwich for the full inverse-correlation norm + +2026-09-14. Consequence of the two-terminal pivotal conversion in `matching-enhancement-mass-gap-20260914.md`. The axial strict inequality is already closed there. This note records the stronger direction-uniform comparison that does **not** require a separate angle-by-angle computation. + +## 1. Endpoint-independent pivotal conversion + +The bounded surgery in Lemma 4.1 of the mass-gap note uses only: + +1. a two-terminal connectivity event; +2. an original pivotal vertex lying on a shortest/induced open path joining the two terminals; +3. the Balister--Bollobas--Riordan square-lattice two-arm rerouting in a fixed ball; +4. insertion of the same facial-site gadget. + +Nothing in the local map uses that the target is `n e1`. Therefore, for every compact + +\[ +I\Subset(0,p_c(G8)), +\] + +there exists `delta_I>0` and a finite endpoint cutoff such that for **every sufficiently distant original vertex** `x in Z^2` and every `p in I`, + +\[ +\boxed{ +P_p^{G8}(0\leftrightarrow x) +\ge +P_{p+\delta_I}^{G4}(0\leftrightarrow x).} \tag{1.1} +\] + +The same `delta_I` works for all directions. Endpoint-near pivotal cases are still only finitely many local types and are absorbed into the same finite-energy constant. + +This is stronger than comparing a finite set of axial/diagonal cylinder masses. + +## 2. Full directional norm comparison + +Let `tau_{G,p}` denote the subcritical inverse-correlation norm, so along any sequence `x_n/|x_n| -> e`, + +\[ +-\frac1{|x_n|}\log P_p^G(0\leftrightarrow x_n) +\to \tau_{G,p}(e). +\] + +Taking logarithmic rates in (1.1) gives, uniformly in direction, + +\[ +\boxed{ +\tau_{8,p}(e) +\le +\tau_{4,p+\delta_I}(e), +\qquad p\in I,\ e\in S^1.} \tag{2.1} +\] + +Equivalently for homogeneous vectors `x`, + +\[ +\tau_{8,p}(x)\le\tau_{4,p+\delta_I}(x). \tag{2.2} +\] + +The existing graph-inclusion comparison only gives + +\[ +\tau_{8,p}(x)\le\tau_{4,p}(x). +\] + +Equation (2.2) is a genuine strict-parameter improvement: one full unit of local matching enhancement beats a positive ordinary-site sprinkling uniformly over direction. + +## 3. What is still needed for a same-p strict directional inequality + +If one has strict parameter monotonicity of the NN norm in every direction, + +\[ +\tau_{4,p+\delta}(e)<\tau_{4,p}(e), \tag{3.1} +\] + +then (2.1) immediately yields + +\[ +\boxed{\tau_{8,p}(e)<\tau_{4,p}(e)} \tag{3.2} +\] + +uniformly on compact `p` intervals and all directions. + +The present #739 branch proves a quantitative strict parameter inequality for the axial mass `kappa_4`; that is enough for the strict centre theorem already recorded. This note does **not** silently promote (3.1) to every direction without either: + +- a direct directional analogue of the branch's Friedgut--Kalai argument, or +- a precise site-percolation theorem giving strict `p` monotonicity of the full norm. + +The non-strict enhancement sandwich (2.1) itself is already rigorous at the same author-proof level as the two-terminal pivotal map and is useful without (3.1). + +## 4. Interface to #765 + +For a tilted exponential torus with shortest direction `e`, the centre equation should use `tau_{G,p}(e)`, not the axial mass. Equation (2.1) supplies a model comparison: + +\[ +\tau_{8,p}(e)=d +\quad\Longrightarrow\quad +\tau_{4,p+\delta_I}(e)\ge d. \tag{4.1} +\] + +Together with monotonicity in `p`, this constrains the relative black/matching directional centre locations before any numerical directional mass is estimated. + +Combined with the deterministic lattice-vector separation already proved in `structural-consequences-20260914.md`, the #765 architecture becomes: + +1. shortest Euclidean period selects the only relevant homology direction in exponential elongation; +2. `tau_{G,p}(e)` sets the birth centre in that direction; +3. the matching enhancement sandwich compares the two graph norms uniformly over `e`. + +The remaining hard step is then the actual SITE directional seam/connection estimate, not multi-direction competition. + +## 5. Interface to #766 + +The first-exit exponential-moment domain in #766 is designed to give rigorous finite inner approximations to the Wulff/correlation body. Equation (2.2) gives an independent inclusion check for any such certificates. + +Write the polar/Wulff body schematically as + +\[ +K_{G,p}=\{t:t\cdot x\le\tau_{G,p}(x)\ \forall x\}. +\] + +From (2.2), + +\[ +\boxed{K_{8,p}\subseteq K_{4,p+\delta_I}.} \tag{5.1} +\] + +Thus any certified first-exit inner body for `K_{8,p}` that exits a rigorous outer body for `K_{4,p+delta_I}` would signal a normalization/implementation error. Conversely, a directional certificate need not rediscover the graph-enhancement ordering numerically. + +## 6. Possible stronger theorem + +The most useful next theoretical closure is a direction-uniform strict parameter inequality + +\[ +\tau_{4,p}(e)-\tau_{4,q}(e) +\ge c_I(q-p)\tau_{4,q}(e), +\qquad p0. \tag{6.2} +\] + +Such a result would simultaneously strengthen #765 and provide a clean consistency constraint for #766's numerical Wulff-body certificates. diff --git a/docs/manuscripts/geometric-balance/double-null-spin4-vs-scalar-20260914.md b/docs/manuscripts/geometric-balance/double-null-spin4-vs-scalar-20260914.md new file mode 100644 index 000000000..a0e60d785 --- /dev/null +++ b/docs/manuscripts/geometric-balance/double-null-spin4-vs-scalar-20260914.md @@ -0,0 +1,238 @@ +# Double-null tomography for the `x=21/4` shell: orientation zero and hexagonal-modulus zero + +Date: 2026-09-14 + +Status: scaling-design note combining the current oblique-cylinder result with the existing hexagonal/Pell modular programme. The group-theory zeros are exact for a continuum spin-four one-point function; transfer to square-site finite-size amplitudes is a universality/scaling hypothesis with declared arithmetic leakage. + +## 1. Why one null is not enough + +The leading square matching-root correction currently lives in a degenerate total-dimension shell + +```text +x=21/4, +``` + +which can contain at least + +```text +spin 0 : scalar eight-arm-like contribution, +spin 4 : thermal-family anisotropic contribution. +``` + +An exponent-four root shift cannot separate them. + +The oblique cylinder now supplies one powerful spin-four projector through `cos(4theta)`. There is a second, logically independent projector already implicit in the hexagonal-modulus programme: the automorphism group of the equianharmonic torus. + +Using both gives a **double-null test** of whether an unsuppressed scalar `x=21/4` floor survives. + +## 2. Orientation null on a long cylinder + +For a square-lattice spin-four correction, + +```text +A4(theta) proportional to cos(4theta). +``` + +At + +```text +theta = pi/8, +``` + +the leading spin-four contribution vanishes. + +Primitive Pell directions satisfying + +```text +a^2-2ab-b^2 = +/-1 +``` + +approach that direction with + +```text +cos(4theta)=O(|u|^-2). +``` + +Therefore a spin-four root term that is normally `ell^-4` is demoted to + +```text +ell^-6 +``` + +along the Pell sequence. + +A scalar `x=21/4` contribution has no `cos4theta` zero and remains `ell^-4` unless a separate sector rule kills it. + +## 3. Modular null at the hexagonal torus + +Let + +```text +tau_hex = exp(i pi/3) +``` + +(up to the chosen equivalent fundamental-domain convention). + +The continuum torus has a 60-degree automorphism at this elliptic fixed point. A one-point amplitude of spin `s` transforms by + +```text +exp(i s pi/3). +``` + +For `s=4`, + +```text +exp(i4pi/3) != 1. +``` + +Therefore the scalar torus one-point coefficient of a genuine spin-four field must vanish exactly at `tau_hex`: + +```text +boxed: +F4(tau_hex)=0. +``` + +A spin-zero `x=21/4` field is not killed by this automorphism. + +Thus the finite-size charge/root correction at the `x=21/4` shell should have the local structure + +```text +V_21/4(tau,theta_lat) + = b0 F0(tau) + + b4 Re[e^{i4theta_lat} F4(tau)]. +``` + +At `tau=tau_hex`, the second term vanishes while the first generically does not. + +This is a different zero from the cylinder orientation node: here the microscopic lattice orientation can be fixed; the continuum torus modulus supplies the selection rule. + +## 4. Arithmetic approximants predict an extra two powers + +Existing Pell/Eisenstein-style integer-period approximants approach the elliptic modulus with a shape error of order + +```text +delta tau = O(N^-1) +``` + +for area `N~L^2`, i.e. + +```text +delta tau = O(L^-2). +``` + +If the spin-four one-point coefficient has a generic first-order zero in the local modular coordinate, + +```text +F4(tau_L) = O(delta tau), +``` + +then its ordinary `L^-4` root correction becomes + +```text +L^-4 * L^-2 = L^-6. +``` + +This mirrors the orientation Pell node, but the origin of the extra `L^-2` is completely different. + +A scalar `x=21/4` component would remain `L^-4` after subtracting the known continuum shape baseline. + +The order of the modular zero should be checked in the correct elliptic local coordinate; a higher-order zero only strengthens the suppression. + +## 5. The double-null decision table + +Suppose the ordinary square/axis sequence has an `L^-4` root correction. + +### Both null sequences become `L^-6` or smaller + +Strong evidence that the unsuppressed leading amplitude is spin four and that any scalar `x=21/4` component is small/zero in the rank-source channel. + +### Orientation node suppresses but hex modulus does not + +Then the long-cylinder `cos4theta` mechanism is real, but a scalar/map contribution can survive on finite-aspect tori. The leading shell is genuinely multidimensional. + +### Hex modulus suppresses but orientation node does not + +Then the observed cylinder angular law has likely mixed geometry/metric effects or a different modular representation; re-audit the physical direction normalization. + +### Neither suppresses + +A scalar `x=21/4` floor or another angular-even mechanism is present at comparable scale; the pure spin-four interpretation is incomplete. + +## 6. Why the tests are statistically and theoretically complementary + +The two nulls have different nuisance directions. + +### Orientation Pell + +Keeps the semi-infinite cylinder logic and changes the homology direction relative to the microscopic square lattice. Main nuisance: row-memory / finite-circumference corrections and the arithmetic approach to `theta=pi/8`. + +### Hexagonal modulus + +Keeps a finite-aspect torus and changes the modular shape. Main nuisance: continuum shape subtraction, integer-period approximation, and mixing with other torus solution sectors. + +A common scalar floor should survive both. A genuine spin-four one-point coefficient is constrained by both independent symmetries. + +## 7. Relation to #156 homology-character tomography + +At the exact hexagonal modulus, the **scalar** spin-four one-point amplitude vanishes, but spin information can still be present in nontrivial homology/map characters. This is precisely why #156's C3/projective-homology tomography is useful as a positive control rather than merely another scalar null. + +However, the matching root itself is an aggregate rank-charge observable. The positive character channel should not be substituted for the original root observable. Its role is to verify that the spin-four sector exists even where the scalar projection kills it. + +This yields the desirable pattern + +```text +scalar rank-charge H4 response: null at tau_hex, +nontrivial homology character: allowed positive spin signal. +``` + +subject to the exact lattice-to-character dictionary already being maintained in #156. + +## 8. Relation to modular tomography (#585) + +Any map-resolved torus basis proposed for the h-odd `x=21/4` correction must respect the elliptic selection rule: + +```text +spin-four basis coefficient vanishes at tau_hex, +spin-zero basis coefficient need not. +``` + +Together with the large-aspect cylinder boundary, this gives two nonlocal constraints on the same torus solution: + +```text +rho -> infinity : reproduce cos4theta cylinder amplitude, +tau -> tau_hex : vanish in the scalar spin-four channel. +``` + +A modular basis that satisfies only one is not an adequate physical candidate. + +## 9. Minimal production rule + +Do not start a generic modulus ladder. + +Use already-planned/available general-period infrastructure and ask only whether one carefully chosen sequence approaching `tau_hex` can distinguish + +```text +L^-4 scalar floor +vs +L^-6-or-smaller spin-four leakage. +``` + +The continuum critical rank/homology baseline must be evaluated at the **actual** finite modulus before forming the lattice residual, as already required by #156. + +Likewise, do not add a second orientation Pell width until the exact-equal-circumference `(4,3)` projector first constrains the scalar coordinate `B0`. + +## 10. Claim boundary + +Exact/group-theoretic: + +- `cos4theta` orientation zero; +- 60-degree elliptic stabilizer kills a scalar one-point spin-four amplitude at `tau_hex`; +- spin-zero is not symmetry-killed by either spin rule. + +Conditional/scaling: + +- `L^-4 -> L^-6` suppression along the declared arithmetic approximants; +- application to square-site matching-root finite-size corrections; +- dominance/absence of the scalar eight-arm coordinate. + +The double-null strategy is useful precisely because the two zeros arise from independent geometries while targeting the same spin representation. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/dual-odd-charge-scaling-function-20260914.md b/docs/manuscripts/geometric-balance/dual-odd-charge-scaling-function-20260914.md new file mode 100644 index 000000000..e74f2c569 --- /dev/null +++ b/docs/manuscripts/geometric-balance/dual-odd-charge-scaling-function-20260914.md @@ -0,0 +1,276 @@ +# A dual-odd universal thermal scaling function for the charge free energy + +2026-09-14. Exact cumulant identities plus a finite-width scaling conjecture. + +The first-derivative charge slope was already identified with safe-sector pivotal density. Going to higher logit derivatives reveals a stronger structure: the leading NN/matching thermal scaling function appears to be ODD under the complementary-sector exchange, so even derivatives cancel from the charge difference while odd derivatives add. + +## 1. Use the Bernoulli logit coordinate + +Put + +\[ +h=\log\frac p{1-p},\qquad p=\frac{e^h}{1+e^h}. \tag{1.1} +\] + +Complementation sends + +\[ +p\leftrightarrow1-p +\quad\Longleftrightarrow\quad +h\leftrightarrow-h. \tag{1.2} +\] + +For a safe transfer kernel, write the unnormalized row matrix as + +\[ +A_G(h)=\sum_{k=0}^w e^{hk}A_{G,k}, \tag{1.3} +\] + +and let `Lambda_G(h)` be its Perron root. The Bernoulli-normalized safe Perron root is + +\[ +\lambda_G(h)=\frac{\Lambda_G(h)}{(1+e^h)^w}, \tag{1.4} +\] + +so + +\[ +I_G(h)=-\log\lambda_G(h) +=w\log(1+e^h)-\log\Lambda_G(h). \tag{1.5} +\] + +The charge free-energy difference is + +\[ +\boxed{ +\Theta_w(h)=I_{4,w}(h)-I_{8,w}(-h).} \tag{1.6} +\] + +Its zero is the semi-infinite charge root. + +## 2. Exact Q-process cumulant identities + +The Perron transform of `A_G(h)` defines a stationary finite-state Q-process. Let `K_j` be the number of occupied sites added in row `j` under that process. + +Standard Perron pressure differentiation gives + +\[ +\partial_h\log\Lambda_G +=\bar K_G, \tag{2.1} +\] + +and + +\[ +\partial_h^2\log\Lambda_G +=\sigma_G^2, \tag{2.2} +\] + +where + +\[ +\sigma_G^2 +=\operatorname{Var}(K_0) ++2\sum_{j\ge1}\operatorname{Cov}(K_0,K_j) \tag{2.3} +\] + +is the Green--Kubo asymptotic variance per row. More generally higher derivatives are asymptotic cumulant rates of the additive row occupancy. + +Therefore + +\[ +\boxed{ +I_{G,h}=wp-\bar K_G,} \tag{2.4} +\] + +\[ +\boxed{ +I_{G,hh}=wp(1-p)-\sigma_G^2,} \tag{2.5} +\] + +and + +\[ +I_{G,hhh} +=wp(1-p)(1-2p)-\kappa_{3,G}^{\rm asym}. \tag{2.6} +\] + +Equation (2.4) is the already-used safe occupation deficit / pivotal identity. Equations (2.5)--(2.6) show that higher charge derivatives measure **deficits of thermal cumulants** under topological survival conditioning. + +## 3. Exact parity of charge derivatives + +Differentiate (1.6): + +\[ +\Theta_h +=I_{4,h}(h)+I_{8,h}(-h), \tag{3.1} +\] + +\[ +\Theta_{hh} +=I_{4,hh}(h)-I_{8,hh}(-h), \tag{3.2} +\] + +\[ +\Theta_{hhh} +=I_{4,hhh}(h)+I_{8,hhh}(-h). \tag{3.3} +\] + +Thus odd thermal cumulant responses ADD between the complementary sectors, while even responses SUBTRACT. + +This exact algebra is the natural place to look for a continuum dual-odd scaling function. + +## 4. Finite transfer evidence + +At the charge root `h_w`, the transparent transfer gives: + +| `w` | `w^(1/4) Theta_h` | `w^(1/2) Theta_hh` | `w^(-5/4) Theta_hhh` | +|---:|---:|---:|---:| +| 4 | 0.84215 | -0.08115 | 0.06078 | +| 5 | 0.83264 | -0.08124 | 0.06223 | +| 6 | 0.82724 | -0.08291 | 0.06306 | +| 7 | 0.82384 | -0.08499 | 0.06355 | +| 8 | 0.82156 | -0.08714 | 0.06385 | + +The half-step finite-difference controls in `fixed-width-charge-spectrum-derivatives-w4-w8-20260914.json` agree at the displayed precision. + +The individual second derivatives behave very differently from their difference: + +\[ +I_{4,hh}/\sqrt w +=0.1766,0.1765,0.1765,0.1766,0.1767, \tag{4.1} +\] + +while + +\[ +I_{8,hh}/\sqrt w +=0.1969,0.1927,0.1903,0.1887,0.1876. \tag{4.2} +\] + +Each sector separately has the expected thermal second-derivative scale `sqrt(w)`, but their difference is only about `w^-1/2`: + +\[ +\Theta_{hh}\sqrt w\approx-0.08\text{ to }-0.09. \tag{4.3} +\] + +So the leading `w^(+1/2)` even thermal response is cancelling very strongly between primal and complementary matching sectors. + +## 5. Dual-odd scaling-function conjecture + +The simplest continuum organization is + +\[ +\boxed{ +\Theta_w(h) +=\frac1w\,\mathcal F(X) ++\text{subleading sector-odd lattice corrections},} \tag{5.1} +\] + +with + +\[ +X=c_h(h-h_c)w^{3/4}, \tag{5.2} +\] + +and + +\[ +\boxed{\mathcal F(-X)=-\mathcal F(X).} \tag{5.3} +\] + +Here `c_h` is the thermal metric factor fixed by the matching/complement convention. At the level of this branch, the especially strong amplitude matching in the safe pivotal deficits suggests that the Bernoulli logit is already close to the natural common normalization. + +If (5.3) holds, then + +\[ +\Theta_h\asymp w^{-1/4}, \tag{5.4} +\] + +\[ +\Theta_{hh}=0\times w^{1/2}+\text{subleading}, \tag{5.5} +\] + +\[ +\Theta_{hhh}\asymp w^{5/4}. \tag{5.6} +\] + +These are exactly the three patterns visible in section 4. + +The semi-infinite root displacement `p_w-p_c=O(w^-4)` is far smaller than the thermal coordinate scale `w^-3/4`, so evaluating derivatives at `p_w` rather than exactly `p_c` does not affect the leading scaling function. + +## 6. First nonlinear coefficients + +Use the finite-width scaled derivatives as rough controls for + +\[ +\mathcal F(X)=a_1X+\frac{a_3}{6}X^3+O(X^5). \tag{6.1} +\] + +In the uncalibrated logit metric, the data suggest + +\[ +a_1\approx0.8\text{--}0.82, +\qquad +a_3\approx0.064. \tag{6.2} +\] + +The ratio + +\[ +\frac{a_3}{6a_1}\approx1.3\times10^{-2} \tag{6.3} +\] + +is small but clearly nonzero in the current widths. + +These numbers should not be called universal until the thermal metric factor is fixed. The **parity** prediction (5.3), and dimensionless ratios after a declared normalization, are the robust targets. + +## 7. Fixed-aspect charge crossover + +For a torus with aspect ratio + +\[ +\rho=m/w, \tag{7.1} +\] + +the periodic charge log-odds at leading scaling order should be + +\[ +\boxed{ +\theta_{w,m}(h) +\approx \rho\,\mathcal F(X)} \tag{7.2} +\] + +before exponentially small longitudinal trace corrections are added. + +This gives #767 a much sharper common-window object than two unconstrained colour intensities: + +- `rho` controls how strongly the dual-odd scaling function is amplified; +- `X` is the ordinary thermal coordinate; +- the matching root is `X=0` at leading universal order; +- sector-odd lattice anisotropy shifts that zero only at the much smaller `w^-4` scale on the square lattice. + +Thus the charge-neutral crossover coordinates and the fixed-width transfer now fit into one two-variable scaling picture. + +## 8. A hierarchy of derivative tests + +The conjecture predicts alternating behavior: + +\[ +\Theta^{(2j+1)}_h\asymp +w^{-1+(2j+1)3/4}, \tag{8.1} +\] + +while + +\[ +\Theta^{(2j)}_h +\] + +has its leading universal thermal contribution cancelled and is controlled by irrelevant/metric corrections. + +So the next discriminating transfer test is not another root fit but the fourth and fifth logit derivatives, with careful numerical differentiation or exact Perron perturbation. A fifth derivative at scale `w^(11/4)` and a strongly suppressed fourth derivative would be powerful evidence for (5.3). + +## 9. Claim boundary + +The Perron/cumulant identities and parity signs (3.1)--(3.3) are exact. The finite derivative table is a reproducible small-width control. The odd universal scaling function, the cancellation of all leading even derivatives, and the interpretation of `rho F(X)` as the full fixed-aspect charge scaling law are conjectural scaling statements requiring either a near-critical primal/matching scaling-limit proof or substantially wider transfer evidence. diff --git a/docs/manuscripts/geometric-balance/dual-odd-thermal-metric-20260914.md b/docs/manuscripts/geometric-balance/dual-odd-thermal-metric-20260914.md new file mode 100644 index 000000000..ec9354f6e --- /dev/null +++ b/docs/manuscripts/geometric-balance/dual-odd-thermal-metric-20260914.md @@ -0,0 +1,222 @@ +# Nonlinear thermal metric from the dual-odd charge scaling collapse + +2026-09-14. Deterministic small-width transfer analysis plus a scaling interpretation. This note refines `dual-odd-charge-scaling-function-20260914.md`: the leading even contamination of the raw logit-coordinate collapse has the exponent expected from an **analytic nonlinear thermal scaling field**, not from a new irrelevant CFT operator. + +## 1. Scaling collapse directly at the charge root + +Let + +\[ +h=\log\frac p{1-p}, +\qquad h_w=\log\frac{p_w^{ch}}{1-p_w^{ch}}, +\] + +and define + +\[ +\boxed{\mathcal F_w(X) +:=w\,\Theta_w\!\left(h_w+Xw^{-3/4}\right).} \tag{1.1} +\] + +The exponent `3/4` is the percolation thermal RG exponent `y_t=1/nu`. + +Using the transparent safe transfer for `w=4,...,9`, the curves already collapse strongly on `|X|<=1.5`. Selected values are + +| `X` | `w=4` | `w=6` | `w=8` | `w=9` | +|---:|---:|---:|---:|---:| +| `-1.5` | `-1.319583` | `-1.291718` | `-1.280354` | `-1.276937` | +| `-1` | `-0.862378` | `-0.844698` | `-0.837686` | `-0.835626` | +| `-0.5` | `-0.424881` | `-0.416666` | `-0.413478` | `-0.412561` | +| `0.5` | `0.419797` | `0.413189` | `0.410733` | `0.410060` | +| `1` | `0.841895` | `0.830532` | `0.826457` | `0.825384` | +| `1.5` | `1.273028` | `1.258942` | `1.254202` | `1.253042` | + +This is a substantially stronger check than comparing only the first and third derivatives at `X=0`. + +The machine-readable values are in `results/geometric-consistency/dual-odd-scaling-collapse-w4-w9-20260914.json`. + +## 2. Odd part and a fifth-derivative prediction + +Define + +\[ +\mathcal F_w^{odd}(X) +=\frac{\mathcal F_w(X)-\mathcal F_w(-X)}2. \tag{2.1} +\] + +A descriptive odd-polynomial fit on `|X|<=1.5`, + +\[ +\mathcal F_w^{odd}(X) +=a_{1,w}X+a_{3,w}X^3+a_{5,w}X^5, \tag{2.2} +\] + +is already stable: + +| `w` | `a1` | `a3` | `a5` | `120 a5` | +|---:|---:|---:|---:|---:| +| 4 | 0.8421553 | 0.0101276 | -0.00014588 | -0.01751 | +| 5 | 0.8326406 | 0.0103698 | -0.00013482 | -0.01618 | +| 6 | 0.8272367 | 0.0105087 | -0.00013059 | -0.01567 | +| 7 | 0.8238353 | 0.0105902 | -0.00012876 | -0.01545 | +| 8 | 0.8215595 | 0.0106400 | -0.00012791 | -0.01535 | +| 9 | 0.8199608 | 0.0106716 | -0.00012751 | -0.01530 | + +The fit is not a global functional ansatz; it is only a local summary of the deterministic transfer curve. It nevertheless gives a sharp next test of the odd-scaling-function conjecture: + +\[ +\boxed{w^{-11/4}\,\Theta_w^{(5)}(h_w) +\ \hbox{should approach a small negative constant, roughly }-1.5\times10^{-2}} +\tag{2.3} +\] + +in the present uncalibrated logit metric, modulo the analytic metric corrections described below. + +## 3. The even residual has thermal exponent `3/4` + +Define the even contamination + +\[ +S_w(X)=\mathcal F_w(X)+\mathcal F_w(-X). \tag{3.1} +\] + +It is small but highly structured. For example at `X=1`, + +\[ +S_w(1)= +-0.02048,-0.01657,-0.01417,-0.01249,-0.01123,-0.01024 +\] + +for `w=4,...,9`. + +Multiplying by `w^{3/4}` nearly freezes the sequence: + +\[ +w^{3/4}S_w(1) +=-0.05793,-0.05540,-0.05431,-0.05374,-0.05342,-0.05322. +\tag{3.2} +\] + +The same scaling occurs at `X=0.5` and `1.5`. + +Thus the leading failure of exact oddness in the **raw lattice logit coordinate** is consistent with + +\[ +\boxed{S_w(X)=O(w^{-3/4}).} \tag{3.3} +\] + +This exponent is exactly the one generated by an analytic nonlinear thermal scaling field. + +## 4. Analytic scaling-field explanation + +Suppose the true continuum thermal coordinate `Y` is related to the raw scaled logit coordinate by + +\[ +Y=X+c_2 X^2w^{-3/4}+O(w^{-3/2}). \tag{4.1} +\] + +Let the leading universal charge function be odd: + +\[ +\mathcal F(-Y)=-\mathcal F(Y). \tag{4.2} +\] + +Then + +\[ +\mathcal F(X+c_2X^2w^{-3/4}) ++\mathcal F(-X+c_2X^2w^{-3/4}) +=2c_2X^2\mathcal F'(X)w^{-3/4}+O(w^{-3/2}). \tag{4.3} +\] + +So a quadratic analytic metric term produces exactly the observed exponent and `X` dependence, without invoking an extra even continuum field. + +Using the odd fit to estimate `F'(X)`, the transfer curves give the effective coefficient + +\[ +c_{2,w}\approx +-0.03248,-0.03162,-0.03130,-0.03117,-0.03111,-0.03109 +\tag{4.4} +\] + +for `w=4,...,9`. + +The small-width sequence is compatible with a limiting value near `-0.031`, but no exact value is claimed. The numerical proximity to `-1/32` at intermediate widths should be treated as a mnemonic, not an identity. + +## 5. An improved thermal coordinate removes most of the even contamination + +As a deterministic diagnostic, take provisionally + +\[ +Y=X-\frac1{32}X^2w^{-3/4}. \tag{5.1} +\] + +Solving (5.1) for the raw `X` values corresponding to `+Y` and `-Y`, and reevaluating the exact transfer function, nearly eliminates the symmetric part. At `Y=1`, for example, + +\[ +\mathcal F_w(X_+(1))+\mathcal F_w(X_-(1)) +\] + +falls from about `-2.0e-2` at raw `X=±1` to + +\[ +-1.22\times10^{-3},\ -4.37\times10^{-4},\ -1.77\times10^{-4}, +-7.15\times10^{-5},\ -2.41\times10^{-5},\ -1.52\times10^{-6} +\] + +for `w=4,...,9`. + +This is not evidence that `-1/32` is exact. It is evidence that **one quadratic analytic reparameterization accounts for almost all of the leading even residual across a finite interval of X**, not merely at the origin. + +## 6. Consequence for interpreting even derivatives + +The earlier transfer table found + +\[ +\Theta_{hh}\sqrt w\approx-0.08\text{ to }-0.09. \tag{6.1} +\] + +while the individual sector second derivatives are `O(sqrt(w))`. + +Equation (4.1) explains how an odd universal function can coexist with a nonzero lattice `Theta_hh`: differentiating an odd function expressed in a nonlinear microscopic coordinate produces a subleading even derivative. + +Therefore the correct hierarchy is: + +1. leading universal charge function: dual-odd; +2. analytic thermal-field reparameterization: generates `w^{-3/4}` even contamination in the scaled curve; +3. genuine irrelevant fields: produce additional, usually different correction exponents; +4. square sector-odd spin-four correction: shifts the zero itself only at `w^{-4}`. + +These mechanisms should not be fitted with one generic correction polynomial. + +## 7. A practical route to a continuum charge function + +The transfer data suggest a clean extraction procedure. + +1. Locate `h_w` from the exact Perron crossing. +2. Measure `F_w(X)=w Theta(h_w+Xw^-3/4)` on a symmetric `X` grid. +3. Use the symmetric part to fit the analytic thermal metric coefficients (`c_2`, then higher orders if needed). +4. Re-express the data in the improved thermal coordinate. +5. Extrapolate the antisymmetric part to obtain the universal odd function. +6. Only after this subtraction fit sector-odd rotational corrections or compare with a Potts TBA/NLIE candidate. + +This avoids contaminating the universal function with a coordinate artifact and gives `integrable-potts-magnetic-charge-scaling-20260914.md` a sharper lattice target. + +## 8. Strong new falsification tests + +The analytic-metric interpretation predicts more than one exponent. + +- `w^(3/4) S_w(X)` should converge to a function proportional to `X^2 F'(X)`. +- The ratio + \[ + \frac{w^{3/4}S_w(X)}{2X^2F_w^{odd\,\prime}(X)} \tag{8.1} + \] + should approach an `X`-independent constant on a bounded thermal interval. +- After the quadratic coordinate correction, the residual symmetric part should drop to the next analytic/irrelevant scale. +- The fifth odd derivative should have the `w^{11/4}` scale predicted in (2.3). + +Failure of the first two tests would refute the simple nonlinear-metric explanation even if the raw curves continue to look visually collapsed. + +## 9. Claim boundary + +The values in sections 1--3 are deterministic outputs of the safe transfer. The odd polynomial is descriptive. The interpretation of the `w^-3/4` symmetric residual as a quadratic analytic thermal scaling field is a scaling hypothesis strongly supported by its exponent and `X` dependence. No exact value is assigned to `c_2`, and no claim is made that all even corrections are analytic coordinate effects. diff --git a/docs/manuscripts/geometric-balance/dual-safe-pivotal-amplitude-20260914.md b/docs/manuscripts/geometric-balance/dual-safe-pivotal-amplitude-20260914.md new file mode 100644 index 000000000..3ba1bb84b --- /dev/null +++ b/docs/manuscripts/geometric-balance/dual-safe-pivotal-amplitude-20260914.md @@ -0,0 +1,168 @@ +# A common primal/dual safe-sector pivotal amplitude + +2026-09-14. Strong scaling conjecture extracted from the exact Perron/Russo identities. + +The fixed-width charge slope is already exactly typed as a sum of two conditioned pivotal intensities. The new observation is that the NN-safe and complementary-matching-safe contributions appear to approach the SAME leading Bernoulli-score amplitude. + +## 1. Exact finite-width quantities + +At the charge-coexistence root `p_w`, put + +\[ +q_w=1-p_w. \tag{1.1} +\] + +Let + +\[ +\bar K^0_{4,w},\qquad \bar K^0_{8,w} \tag{1.2} +\] + +be the mean occupied sites in one row under the two Perron/Doob safe phases. Define their occupation deficits relative to an unconditioned Bernoulli row: + +\[ +\boxed{ +d_{4,w}=wp_w-\bar K^0_{4,w}, +\qquad +d_{8,w}=wq_w-\bar K^0_{8,w}.} \tag{1.3} +\] + +The exact score/pivotal identities give + +\[ +d_{4,w}=p_w r^0_{4,w}, +\qquad +d_{8,w}=q_w r^0_{8,w}, \tag{1.4} +\] + +where `r^0` is the safe-conditioned topological pivotal density per row. + +Also + +\[ +\boxed{ +p_wq_w\Theta'_w(p_w)=d_{4,w}+d_{8,w}.} \tag{1.5} +\] + +No scaling assumption enters (1.3)--(1.5). + +## 2. Transfer data + +For `w=4,...,8`: + +| `w` | `d4 w^(1/4)` | `d8 w^(1/4)` | `d4/d8` | +|---:|---:|---:|---:| +| 4 | 0.4152842 | 0.4268705 | 0.972858 | +| 5 | 0.4130767 | 0.4195635 | 0.984539 | +| 6 | 0.4116079 | 0.4156285 | 0.990327 | +| 7 | 0.4105738 | 0.4132612 | 0.993497 | +| 8 | 0.4098310 | 0.4117281 | 0.995392 | + +The approach to a common amplitude is substantially cleaner than either raw derivative alone. + +The corresponding conditional pivotal intensities satisfy + +| `w` | `r4 w^(1/4)` | `r8 w^(1/4)` | `r4/r8` | +|---:|---:|---:|---:| +| 4 | 0.7021850 | 1.0447587 | 0.672103 | +| 5 | 0.6974869 | 1.0289366 | 0.677872 | +| 6 | 0.6946883 | 1.0199656 | 0.681090 | +| 7 | 0.6928117 | 1.0144357 | 0.682953 | +| 8 | 0.6914963 | 1.0108043 | 0.684105 | + +while + +\[ +\frac{1-p_c}{p_c} +=0.6870631169\ldots. \tag{2.1} +\] + +## 3. Common-amplitude conjecture + +The natural scaling statement is + +\[ +\boxed{ +d_{4,w}w^{1/4}\to C_{\rm piv}, +\qquad +d_{8,w}w^{1/4}\to C_{\rm piv}.} \tag{3.1} +\] + +Equivalently, + +\[ +\boxed{ +\frac{r^0_{4,w}}{r^0_{8,w}} +\longrightarrow\frac{q_c}{p_c}.} \tag{3.2} +\] + +The finite data suggest `C_piv` around `0.40--0.41`; no precision claim is made from these widths. + +Then (1.5) forces one thermal-slope amplitude: + +\[ +\boxed{ +\Theta'_w(p_w)w^{1/4} +\longrightarrow +\frac{2C_{\rm piv}}{p_cq_c}.} \tag{3.3} +\] + +For `C_piv≈0.41`, the right side is about `3.40`, matching the direct transfer sequence. + +## 4. Continuum interpretation + +The two safe sectors converge to the same magnetic/topological primary. Their derivatives with respect to complementary Bernoulli thermal coordinates probe opposite sides of the same thermal perturbation. + +The occupation deficit `wp-Kbar` is the Bernoulli score response of that sector. Unlike the raw pivotal count, it already includes the microscopic parameter metric factor. Therefore a common leading amplitude in (3.1) is exactly what one expects if the complement map `q=1-p` fixes the relative normalization of the two thermal coordinates in the scaling limit. + +This is stronger than merely saying both sides have exponent `1/4`: it asserts a primal/dual amplitude relation. + +## 5. Four-arm reading + +The exact identity + +\[ +d_{G,w}=p\,r^0_{G,w} \tag{5.1} +\] + +and the critical four-arm mechanism suggest + +\[ +r^0_{G,w}\asymp C_G w^{-1/4}. \tag{5.2} +\] + +Equation (3.1) is then the amplitude relation + +\[ +p_c C_4=q_c C_8=C_{\rm piv}. \tag{5.3} +\] + +So the two raw pivotal amplitudes are NOT predicted equal; their Bernoulli-weighted amplitudes are. + +This is an important distinction for any direct pivotal sampling experiment. + +## 6. A stronger diagnostic than numerical differentiation + +Future wider transfer calculations should report all three objects: + +\[ +d_4w^{1/4},\qquad d_8w^{1/4},\qquad +p q\Theta'_w w^{1/4}. \tag{6.1} +\] + +The exact identity demands the third equal the sum of the first two at every width. The scaling conjecture further demands the first two coalesce. + +A direct Q-process pivotal sampler can test the same statement without taking eigenvalue finite differences. + +## 7. Relation to the sector-odd correction + +This common thermal amplitude is a **sector-even normalization statement about the derivative**. It does not imply the critical sector energies are equal at finite width. Their tiny difference is the separate sector-odd irrelevant correction responsible for the `w^-4` pseudo-critical shift. + +Thus the picture is: + +- thermal response: common primal/dual amplitude, order `w^-1/4`; +- critical mismatch: much smaller rotational sector-odd amplitude, order `w^-17/4` on the square lattice. + +## 8. Claim boundary + +Equations (1.3)--(1.5) are exact. The convergence to a common amplitude and the ratio (3.2) are scaling conjectures strongly supported by the transparent widths. A rigorous proof would require a near-critical primal/matching scaling limit with the relative thermal metric fixed by complement duality, or an equivalent arm-amplitude theorem. diff --git a/docs/manuscripts/geometric-balance/dual-surface-excess-slope-20260914.md b/docs/manuscripts/geometric-balance/dual-surface-excess-slope-20260914.md new file mode 100644 index 000000000..3f3f583da --- /dev/null +++ b/docs/manuscripts/geometric-balance/dual-surface-excess-slope-20260914.md @@ -0,0 +1,228 @@ +# Dual surface excess equals the inverse-mass slope + +2026-09-14. Exact finite-width identity plus a regular-point asymptotic consequence of the component-activity derivative theorem already on #739. + +This note links three quantities that previously appeared separately: + +1. the black subcritical complete-component logit score; +2. the exponentially large complementary white component's boundary--volume cancellation; +3. the mass slope `v=-kappa'(p)` that sets the median-centred Gumbel scale. + +## 1. Setup + +Fix black NN occupation probability + +\[ +pp_c(G8). +\] + +Let + +\[ +\nu_w(p)=\nu_w^4(p)=\nu_w^8(q) \tag{1.1} +\] + +be the exact complementary complete-winding-component intensity. + +For a black component selected under component Palm, write + +\[ +n=|C|,\qquad b=|\partial_4 C| \tag{1.2} +\] + +with **distinct external boundary sites**. + +For the complementary white matching component selected under its component Palm at density `q`, write + +\[ +N=|C^*|,\qquad B=|\partial_8 C^*|. \tag{1.3} +\] + +The two Palm laws are very different: black components have `O(w)` mean volume at fixed subcritical `p`, whereas the alternating-barrier theorem gives the white component an exponentially large longitudinal scale of order `1/nu_w`. + +## 2. Exact black logit score + +Use black logit coordinate + +\[ +z=\log\frac{p}{q},\qquad \partial_z=pq\partial_p. \tag{2.1} +\] + +A complete black component has activity + +\[ +p^nq^b. +\] + +Therefore the exact finite-width component-Palm derivative identity is + +\[ +\boxed{ +\partial_z\log\nu_w^4(p) +=E_{4,p}[q n-p b].} \tag{2.2} +\] + +This is the logit version already underlying the semiconvexity proof in `poisson-birth-windows.md`. + +## 3. Exact white dual score in the **same black coordinate** + +The natural white occupation parameter is `q`, whose own logit is + +\[ +z_8=\log\frac q p=-z. \tag{3.1} +\] + +At white density `q`, the component activity is + +\[ +q^N p^B. +\] + +Its natural white-logit score is + +\[ +\partial_{z_8}\log\nu_w^8(q) +=E_{8,q}[pN-qB]. \tag{3.2} +\] + +But `z_8=-z` and the exact complementary intensity identity says + +\[ +\nu_w^8(q)=\nu_w^4(p). +\] + +Thus differentiating with respect to the **black** coordinate `z` gives + +\[ +\partial_z\log\nu_w^4(p) +=E_{8,q}[qB-pN]. \tag{3.3} +\] + +Combining (2.2)--(3.3): + +\[ +\boxed{ +E_{4,p}[q n-p b] +=E_{8,1-p}[q B-p N] +=\partial_z\log\nu_w(p).} \tag{3.4} +\] + +This is an exact finite-`w` dual surface-excess identity. + +## 4. Regular-point limit gives the mass slope + +The existing author-level component-intensity theorem gives + +\[ +\frac1w\log\nu_w(p)\to-\kappa_4(p), \tag{4.1} +\] + +and at every differentiability point of `kappa_4`, uniformly for moving `p_w->p`, + +\[ +\frac1w\partial_p\log\nu_w(p_w) +\to-\kappa_4'(p)=:v_4(p)>0. \tag{4.2} +\] + +Since `partial_z=pq partial_p`, equation (3.4) yields + +\[ +\boxed{ +\frac1wE_{4,p}[q n-p b] +\longrightarrow pq\,v_4(p),} \tag{4.3} +\] + +and, far more strikingly on the giant complementary side, + +\[ +\boxed{ +\frac1wE_{8,q}[q B-p N] +\longrightarrow pq\,v_4(p).} \tag{4.4} +\] + +The white variables `N` and `B` are individually exponentially large in `w`, but their leading bulk pieces cancel in `qB-pN`, leaving an `O(w)` surface excess whose coefficient is exactly the black inverse-mass slope. + +## 5. Bulk ratio and surface excess are two orders of the same identity + +The reciprocal white-span theorem gives `E N >= E L ~ 1/nu_w`, so `E N` is exponentially larger than `w`. Dividing the exact identity (3.4) by `E N` gives + +\[ +\frac{E B}{E N}=\frac p q+O(w\nu_w), \tag{5.1} +\] + +recovering + +\[ +\boxed{\frac{E B}{E N}\to\frac p q.} \tag{5.2} +\] + +Equation (4.4) supplies the **next order** after this bulk cancellation: + +\[ +\boxed{ +E B +=\frac p q E N ++p\,v_4(p)w+o(w).} \tag{5.3} +\] + +Here we used `pq v w / q = p v w` when solving `q E B-p E N = pq v w+o(w)` for `E B`. + +Thus the same score identity predicts both: + +- the leading giant-component boundary/volume ratio `p/q`; +- the subleading `+p v_4 w` boundary excess. + +No Brownian range assumption is needed. + +## 6. An independent route to the Gumbel scale + +The median-centred first-birth Gumbel variable on the black side uses + +\[ +v_4(p)=-\kappa_4'(p). \tag{6.1} +\] + +Usually one would estimate this from nearby masses or from the derivative of the log intensity itself. Equation (4.4) gives a geometrically different estimator: + +\[ +\boxed{ +v_4(p) +=\lim_{w\to\infty} +\frac{E_{white}[qB-pN]}{pq\,w}.} \tag{6.2} +\] + +Similarly, (4.3) gives the black-side score estimator + +\[ +v_4(p) +=\lim_{w\to\infty} +\frac{E_{black}[q n-p b]}{pq\,w}. \tag{6.3} +\] + +The two are exact dual finite-width views of the same derivative. Agreement between them is a strong implementation/Palm-normalization check; neither requires numerical finite differences in `p`. + +## 7. Why this is useful for #762 + +A joint morphology computation already interested in occupation count and **distinct** external boundary sites can report + +\[ +q n-p b +\] + +for black component Palm essentially for free. + +On the complementary white side, the raw `N,B` values are huge and strongly correlated. The residual + +\[ +qB-pN \tag{7.1} +\] + +is the meaningful surface quantity: the leading bulk terms are supposed to cancel. Measuring `B/N` alone checks only the leading order; measuring (7.1)/`w` additionally tests the derivative/Gumbel normalization. + +An edge-incidence boundary count cannot be substituted for `B`; the exact component activity uses distinct external sites and the cancellation coefficient would change. + +## 8. Second-derivative warning + +Differentiating once more gives exact variance/curvature identities, but on the giant white side `N+B` is exponentially large and must cancel against an equally large score variance to leave the scaled semiconvex curvature. This is a delicate second-order object and should not be inferred from a modest finite covariance table. + +The first-derivative surface-excess identity (3.4) is the robust target. diff --git a/docs/manuscripts/geometric-balance/equal-circumference-spin4-projector-20260914.md b/docs/manuscripts/geometric-balance/equal-circumference-spin4-projector-20260914.md new file mode 100644 index 000000000..0dd3b24cf --- /dev/null +++ b/docs/manuscripts/geometric-balance/equal-circumference-spin4-projector-20260914.md @@ -0,0 +1,305 @@ +# Equal-circumference angular projector for the sector-odd correction + +Date: 2026-09-14 + +Status: analysis/design note motivated by `docs/research-compass-beyond-exactness-20260914.md`. It uses only already committed safe-transfer outputs for the numerical compression below. The proposed `(4,3)` computation is a discriminating next experiment, not an automatic width/angle scan. + +## 1. Question after the oblique spin-four result + +The current oblique safe-transfer evidence is already substantially stronger than an exponent fit: + +- the critical charge-sector mismatch changes sign with `cos(4 theta)`; +- after physical metric normalization the free-energy amplitude collapses across axis, diagonal and several oblique directions; +- the charge-root shift shows the same angular sign/magnitude law. + +The useful remaining question is therefore no longer simply + +> is the leading correction compatible with spin four? + +but rather + +> after the leading spin-four piece is projected out, is the first residual still the same angular sector with an ordinary radial `ell^-2` dressing, or is there a genuinely angular-orthogonal lower competitor (scalar / spin eight / another map sector)? + +This distinction is directly aligned with the research compass: identify what breaks the common sector, rather than extending a solved finite model for its own sake. + +## 2. Angular Fourier bookkeeping + +Let + +```text +F(ell,theta) = ell^(17/4) E_charge^phys(pc;ell,theta), +``` + +where `E_charge^phys=|u| Theta_row` for primitive circumference direction `u` and `ell=n|u|`. + +Square `C4` symmetry plus reflection allows the angular expansion + +```text +F(ell,theta) + = A0(ell) + + A4(ell) cos(4 theta) + + A8(ell) cos(8 theta) + + A12(ell) cos(12 theta) + ... . +``` + +The current leading hypothesis is `A4(ell)->B != 0`, with the other harmonics subleading after the `ell^(17/4)` rescaling. + +For axis and diagonal orientations, + +```text +theta_axis = 0, +theta_diag = pi/4, +``` + +so + +```text +F_odd = [F(axis)-F(diag)]/2 = A4 + A12 + ..., +F_even = [F(axis)+F(diag)]/2 = A0 + A8 + A16 + ... . +``` + +Thus a scalar `x=21/4` competitor belongs to the angular-even projector, while the proposed spin-four term belongs to the angular-odd projector. + +The same construction can be applied to the scaled root response + +```text +R(ell,theta) = -(p_root-pc) ell^4. +``` + +## 3. Existing no-new-compute Pell projector + +There is already a near-equal-circumference axis/diagonal pair: + +```text +axis: w=7, ell_a=7, +diagonal: n=5, ell_d=5 sqrt(2), +ell_d^2-ell_a^2 = 1. +``` + +The relative circumference mismatch is only about one percent and is forced by the Pell relation `7^2-2*5^2=-1`. + +### Critical free-energy mismatch + +From the committed axis and diagonal controls, + +```text +F_axis(w=7) = +1.0358025515351343, +F_diag(n=5) = -1.0206765596904341. +``` + +Therefore + +```text +F_odd = 1.0282395556127844, +F_even = 0.0075629959223501, +|F_even/F_odd| = 0.0073553. +``` + +So the angular-even leakage is only about `0.74%` of the angular-odd component at this already-available pair. + +Because the physical circumferences are not exactly equal, this number must **not** be called a scalar-amplitude measurement: ordinary radial finite-size drift can itself generate an `O(1%)` mismatch between the two members. It is nevertheless a strong indication that a same-order scalar term is not competing at comparable amplitude. + +### Root response + +Using the same pair, + +```text +R_axis(w=7) = +0.3035281536402238, +R_diag(n=5) = -0.2991554482270620, +``` + +hence + +```text +R_odd = 0.3013418009336429, +R_even = 0.0021863527065809, +|R_even/R_odd| = 0.0072554. +``` + +The free-energy and root projectors independently put the even leakage at essentially the same `~0.7%` level. + +This is more informative than another fit of the exponent four: it says that the leading angular-orthogonal contamination is already small before any new production. + +## 4. A stronger compression already visible: one spin-four tower with an `ell^-2` dressing + +For three directions with two usable circumferences each, fit only the two-term form + +```text +A_est(ell) = A_inf + A_2 / ell^2, +``` + +where `A_est=-(p_root-pc)ell^4/cos(4theta)`. + +The resulting two-point coefficients are + +| direction | `A_inf` | `A_2` | +|---|---:|---:| +| axis `(1,0)` from `w=8,9` | `0.2899430` | `0.656237` | +| diagonal `(1,1)` from `n=4,5` | `0.2863246` | `0.641543` | +| `(2,1)` from `n=3,4` | `0.2921425` | `0.690466` | + +The agreement is unexpectedly tight for such small widths: the inferred asymptotic amplitude varies only at the percent level, and the `ell^-2` coefficient varies by only a few percent. + +The direct critical free-energy amplitude shows the same pattern. Writing + +```text +B_est(ell)=B_inf+B_2/ell^2, +``` + +for the corresponding two-width pairs gives approximately + +| direction | `B_inf` | `B_2` | +|---|---:|---:| +| axis `(1,0)` | `0.9774` | `2.830` | +| diagonal `(1,1)` | `0.9653` | `2.769` | +| `(2,1)` | `0.9847` | `2.965` | + +The axis thermal-slope controls (`w=4..8`) are also well summarized by + +```text +C(ell)=C_inf+C_2/ell^2, +C_inf ~= 3.377, +C_2 ~= 1.74, +``` + +where `C` is the physical `p` derivative coefficient in `partial_p E_charge^phys ~ C ell^-1/4`. + +The independent leading amplitudes then satisfy + +```text +B_inf/C_inf ~= 0.289, +``` + +matching the root-amplitude estimates above. + +This motivates a sharper, falsifiable hypothesis: + +> **Factorized spin-four tower.** Over the first two visible correction orders, the critical sector-odd response is dominated by one `cos(4 theta)` angular sector with ordinary radial `ell^-2` dressing, while the leading thermal response is scalar. The observed root ladder `ell^-4, ell^-6, ...` can therefore arise without assigning a new angular mechanism to every even power. + +Schematically, + +```text +E_charge^phys(pc;ell,theta) + = cos(4theta) ell^(-17/4) + [B0 + B2 ell^-2 + O(ell^-4)] + + R_perp(ell,theta), + +partial_p E_charge^phys(pc;ell,theta) + = ell^(-1/4) + [C0 + C2 ell^-2 + O(ell^-4)] + + S_perp(ell,theta), +``` + +and therefore + +```text +p_root-pc + = -cos(4theta) ell^-4 + [A0 + A2 ell^-2 + O(ell^-4)] + + Q_perp(ell,theta). +``` + +This is a data compression / conjecture, not an operator theorem. The `ell^-2` dressing could arise from nonlinear scaling fields, descendant mixing, geometry, or another correction with the same angular character. It should not yet be named as a specific Virasoro descendant. + +## 5. One decisive next computation: exact equal circumference + +The cleanest next experiment is **not** another angle ladder. Use a primitive Pythagorean direction + +```text +u=(4,3), |u|=5, +n=2, +ell=10, +cos(4theta)= -527/625 = -0.8432, +``` + +and compare it with the ordinary axis transfer at + +```text +u=(1,0), w=10, ell=10. +``` + +These two cylinders have **exactly the same physical circumference** in the same microscopic square-site model. Radial finite-size corrections therefore cancel from the leading angular ratio instead of being estimated or interpolated. + +Compute only three objects at the fixed diagnostic `pc_ref` and at the charge root: + +1. physical critical mismatch + +```text +E = |u| [I4(pc)-I8(1-pc)]; +``` + +2. physical thermal derivative + +```text +D = |u| partial_p[I4(p)-I8(1-p)] at pc, +``` + +preferably by left/right Perron Feynman--Hellmann differentiation rather than a coarse finite difference; + +3. charge root `p_root`. + +The factorized hypothesis predicts the same-length ratios + +```text +E_(4,3) / E_axis ~= -527/625, +D_(4,3) / D_axis ~= 1, +(p_root_(4,3)-pc)/(p_root_axis-pc) ~= -527/625, +``` + +up to genuinely angular-orthogonal residuals and numerical error. + +The most useful reported quantities are therefore the **residuals**, not another fitted amplitude: + +```text +r_E = E_(4,3) - (-527/625) E_axis, +r_D = D_(4,3) - D_axis, +r_p = (p_root_(4,3)-pc) + - (-527/625)(p_root_axis-pc). +``` + +Normalize them by the corresponding leading axis signal and propagate Perron/eigensolver error into the differences. + +### Diagnostic numerical scale, not an acceptance target + +The existing two-width compression gives roughly + +```text +A(ell=10) ~ 0.296, +B(ell=10) ~ 1.00, +C(ell=10) ~ 3.39. +``` + +So one expects root shifts of order + +```text +axis ell=10: p_root-pc ~ -2.96e-5, +(4,3), n=2: p_root-pc ~ +2.50e-5, +``` + +if the factorized description is approximately right. These estimates are only sanity checks; the test is the same-length angular ratio, not agreement with the extrapolated digits. + +## 6. Compute boundary and stop rule + +The generic oblique safe-transfer code already handles arbitrary primitive directions. For `(4,3)` the matching row memory is larger than the diagonal case, but `n=2` is deliberately chosen so the first task is bounded. + +Execution order: + +1. build `(4,3), n=2` state spaces and report exact state counts / memory before any extension; +2. reproduce a previously committed oblique direction with the same executable environment; +3. compute `E,D,p_root` and certified/residual numerical errors; +4. compare only to axis `w=10` at the same physical circumference; +5. **stop** after this pair unless the residual is large enough to require identification. + +Decision: + +- if `r_E,r_D,r_p` are small at the percent level or below, do **not** open an angle ladder; the next high-value work is the module/matrix-element origin of the already isolated spin-four sector; +- if a residual is clearly larger than solver error and the known radial correction scale, identify its angular character before adding widths; +- only if a real angular-orthogonal residual is found should a near-node direction such as `(5,2)` receive a second width. + +This experiment directly tests two explanations and therefore meets the research-compass criterion for additional compute. + +## 7. Claim boundary + +The existing numerical projector and two-width coefficient compression are algebraic re-analyses of committed deterministic safe-transfer outputs. They are not asymptotic fits with enough widths to establish correction exponents. + +The factorized spin-four tower is a new working conjecture. The equal-circumference `(4,3)` design is proposed because it can falsify that conjecture while controlling the principal radial/metric confounder in one computation. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/erratum-common-x4-is-angular-scalar-20260914.md b/docs/manuscripts/geometric-balance/erratum-common-x4-is-angular-scalar-20260914.md new file mode 100644 index 000000000..886be4853 --- /dev/null +++ b/docs/manuscripts/geometric-balance/erratum-common-x4-is-angular-scalar-20260914.md @@ -0,0 +1,165 @@ +# Erratum: the observed common `x≈4` correction is angular-scalar, not established H4/KdV + +Date: 2026-09-14 + +Status: explicit correction to `kdv-identity-family-cancellation-target-20260914.md`. That earlier note over-assigned the angular character of the common `x≈4` correction and should not be cited as the current mechanism. The radial-dimension observation remains; the H4/KdV identification is withdrawn. + +## 1. What was wrong + +The earlier note took the large common `x≈4` correction in the average magnetic/safe gap and treated it as an H4/spin-four identity-family KdV correction. + +The already committed oblique physical-gap data do not support that assignment. + +For a direction `u=(a,b)`, use + +```text +ell=n|u|, +E_phys=|u| I0, +2 pi x_m = 0.6544984694978736... +``` + +and form the dimension-four radial diagnostic + +```text +C_common(ell,theta) + = [E_phys - (2 pi x_m)/ell] ell^3. (1.1) +``` + +Using existing committed values gives approximately + +```text +diagonal (1,1), n=4 : 0.2900 +diagonal (1,1), n=5 : 0.3010 +(2,1), n=3 : 0.3089 +(2,1), n=4 : 0.3269 +(3,1), n=3 : 0.3405 +(3,2), n=2 : 0.3065 +``` + +These values stay **positive and near one common radial amplitude** across orientations with both positive and negative `cos(4theta)`. + +Dividing by `cos(4theta)` destroys the collapse and changes signs. Thus the visible common correction is much more naturally angular H0/scalar than H4. + +## 2. Corrected leading interpretation + +The current hierarchy should be written as + +```text +common average gap: + angular H0, + x≈4, + candidate identity-family scalar such as T Tbar / scalar descendant mixture; + +sector difference / root numerator: + angular H4, + x≈21/4, + first visible noncommon block. +``` + +The ordinary common `x=4` correction and the leading noncommon H4 correction are therefore different angular sectors. + +## 3. What survives from the “same representation” idea + +A weaker conditional cancellation route remains valid in spirit. + +If rank0 and rank2 critical sectors are two copies of the same Virasoro highest-weight representation, then a **scalar identity-family perturbation** whose diagonal matrix element is fixed by that representation has the same first-order shift in both copies, provided the microscopic coupling is common and no map/multiplicity operator acts differently. + +For a `T\bar T`-type scalar perturbation the diagonal primary matrix element factorizes through the cylinder stress-tensor zero modes and depends only on `(h,\bar h,c)` under the standard assumptions. Equal sector weights then imply equal scalar shift. + +This is the corrected theorem template: + +```text +same continuum primary copies ++ same scalar identity-family coupling ++ no nontrivial map-space matrix +=> common x=4 H0 shift cancels from the sector difference. +``` + +It is **not** a KdV/H4 theorem. + +## 4. The actual null to prove is `D_0^(4)=0`, not `D_4^(4)=0` + +In the difference-tangent notation, the visible common block is now typed as + +```text +angular index a=0, +radial dimension x≈4. +``` + +Therefore the structural cancellation needed by the root is + +```text +boxed: +D_0^(4)=0. (4.1) +``` + +not the previously written `D_4^(4)=0`. + +The first visible nonzero angular H4 difference coefficient remains + +```text +D_4^(21/4) != 0 +``` + +as the empirical/scaling target. + +This distinction is important because it separates ordinary isotropic finite-size corrections from the anisotropic Matching-One root mechanism. + +## 5. Consequence for the radial-dressing story + +An H0 `x=4` scalar can still dress the H4 `x=21/4` coupling multiplicatively and generate an `ell^-2` radial correction **inside the H4 coefficient**: + +```text +P4(ell) + = a4 ell^-4 [1+c2 ell^-2+...]. +``` + +That earlier dressing idea remains possible. + +But, as already corrected elsewhere, an exact H4 angular projector removes the entire `P4(ell)H4(theta)` term including such dressing. It cannot explain a post-H4 angular-orthogonal residual. + +Thus two separate statements are now retained: + +```text +common H0 x=4 field may radially dress P4; +post-H4 residual requires H0/H8/... or nonlinear H4^2 response. +``` + +## 6. Relation to Ward theory + +The present high-value Ward question is no longer “is the common x4 H4 block a KdV charge?” + +It is instead: + +1. identify the actual scalar `x≈4` correction in the common magnetic/topological gap; +2. show why its matrix element is equal in rank0/rank2 extreme sectors; +3. separately derive the H4 `x=21/4` difference response. + +A Virasoro identity-family argument may still solve item 2, but the relevant scalar quasiprimary/operator must be typed correctly before applying a charge formula. + +## 7. Supersession statement + +`kdv-identity-family-cancellation-target-20260914.md` is superseded as a Matching-One mechanism note because it assigned H4 character to the wrong observed block. + +The standard KdV eigenvalue formula quoted there is correct CFT algebra; its application to the common `x≈4` lattice correction was unsupported by the angular data. + +Do not use that note as evidence for `D_4^(4)=0`. + +## 8. Claim boundary + +Data-level correction: + +- oblique common-gap `ell^-3` residual is approximately orientation-independent rather than proportional to `cos4theta` over existing controls. + +Working interpretation: + +- common block: H0, `x≈4`; +- leading noncommon block: H4, `x≈21/4`. + +Open: + +- exact continuum identity of the common H0 block; +- proof of equal rank0/rank2 matrix element; +- possible map-resolved scalar multiplicity. + +The purpose of this erratum is to keep angular typing ahead of field naming, as required by the reverse audit. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/euclidean-vs-correlation-systole-counterexample-20260914.md b/docs/manuscripts/geometric-balance/euclidean-vs-correlation-systole-counterexample-20260914.md new file mode 100644 index 000000000..1e9157124 --- /dev/null +++ b/docs/manuscripts/geometric-balance/euclidean-vs-correlation-systole-counterexample-20260914.md @@ -0,0 +1,199 @@ +# Euclidean systole need not minimize the correlation norm + +2026-09-14. Explicit period-lattice family showing that the warning in #765 is genuine outside the exponentially elongated regime. The example uses the rigorous dilute directional asymptotics from `dilute-directional-geodesic-entropy-20260914.md`. + +## 1. The lattice family + +Let + +\[ +\Lambda_n=\operatorname{span}_{\mathbb Z} +\{u_n,v_n\}, +\] + +with + +\[ +u_n=(10n,0), +\qquad +v_n=(5n,9n). \tag{1.1} +\] + +Its area is + +\[ +N_n=|\det(u_n,v_n)|=90n^2. \tag{1.2} +\] + +Every lattice vector is + +\[ +a u_n+b v_n +=n(10a+5b,9b). \tag{1.3} +\] + +If `b=0`, the shortest nonzero vector has length `10n`. If `|b|>=1`, minimizing over `a` gives + +\[ +|10a+5b|\ge +\begin{cases} +0,&b\text{ even},\\ +5,&b\text{ odd}, +\end{cases} \tag{1.4} +\] + +so + +\[ +|a u_n+b v_n|^2/n^2 +\ge +\begin{cases} +81b^2,&b\text{ even},\\ +25+81b^2,&b\text{ odd}. +\end{cases} \tag{1.5} +\] + +For `b=+/-1` this is at least `106>100`; for `|b|>=2` it is at least `324`. Therefore + +\[ +\boxed{\ell(\Lambda_n)=10n,\qquad +\text{the unique Euclidean shortest line is }\mathbb R u_n.} \tag{1.6} +\] + +## 2. Matching dilute correlation costs + +The fixed-direction dilute theorem gives, for every fixed integer vector `x`, + +\[ +\tau_{8,p}(x) +=\|x\|_\infty\log(1/p)+O_x(1), +\qquad p\downarrow0, \tag{2.1} +\] + +where the `O(1)` term is the geodesic-entropy correction. + +For the Euclidean systole, + +\[ +\tau_{8,p}(u_n) +=10n\log(1/p)+O(n). \tag{2.2} +\] + +But + +\[ +v_n=(5n,9n), +\qquad +v_n-u_n=(-5n,9n), \tag{2.3} +\] + +so both have + +\[ +\|v_n\|_\infty=\|v_n-u_n\|_\infty=9n. \tag{2.4} +\] + +Hence + +\[ +\tau_{8,p}(v_n) +=9n\log(1/p)+O(n), \tag{2.5} +\] + +\[ +\tau_{8,p}(v_n-u_n) +=9n\log(1/p)+O(n). \tag{2.6} +\] + +For all sufficiently small fixed `p`, the `log(1/p)` advantage beats the bounded-per-`n` entropy constants, giving + +\[ +\boxed{ +\tau_{8,p}(v_n)<\tau_{8,p}(u_n), +\qquad +\tau_{8,p}(v_n-u_n)<\tau_{8,p}(u_n).} \tag{2.7} +\] + +The two cheaper lines are nonparallel and form another basis of the same lattice: + +\[ +|\det(v_n,v_n-u_n)|=90n^2=N_n. \tag{2.8} +\] + +Thus the matching correlation norm has **two nonparallel cheapest primitive classes** while the Euclidean norm has a unique shortest class. + +## 3. NN behaves differently in the same family + +For NN site percolation, the dilute directional theorem instead gives + +\[ +\tau_{4,p}(x) +=\|x\|_1\log(1/p)+O_x(1). \tag{3.1} +\] + +Here + +\[ +\|u_n\|_1=10n, +\qquad +\|v_n\|_1=\|v_n-u_n\|_1=14n. \tag{3.2} +\] + +Therefore the Euclidean axis systole `u_n` is also the cheapest NN class at sufficiently small `p`. + +The same finite period lattice can therefore have different rare winding directions for NN and matching connectivity. + +## 4. Why this does not contradict the exponential-elongation separation lemma + +For this family, + +\[ +\frac{N_n/\ell_n}{\ell_n} +=\frac{9n}{10n}=0.9, \tag{4.1} +\] + +so the transverse height is only of the same order as the systole. The deterministic direction-separation result in `structural-consequences-20260914.md` assumes + +\[ +h/\ell\to\infty, \tag{4.2} +\] + +which holds in the fixed-positive exponential-aspect regime but **not** here. + +Under (4.2), every nonparallel period has Euclidean length at least `h`, and norm equivalence makes its correlation cost diverge relative to the shortest vector. That special geometry legitimately restores the Euclidean shortest direction. + +So the two statements fit together: + +- general period shape: minimize the **correlation norm**, not Euclidean length; +- exponential elongation: determinant geometry forces every nonparallel class to be so long that the Euclidean systole automatically wins. + +## 5. Consequence for #765 theorem statements + +A correct general directional centre theorem should be phrased using + +\[ +\boxed{ +\min_{v\in\Lambda\setminus\{0\}}\tau_{G,p}(v),} \tag{5.1} +\] + +with the projective minimizer set retained when it is nonunique. + +Only after a geometric condition such as `h/ell -> infinity` has been imposed may this be reduced to + +\[ +\tau_{G,p}(u_{euclidean\ systole}). \tag{5.2} +\] + +The counterexample shows that this distinction is mathematical, not merely conservative wording. + +## 6. Multi-direction implication + +At small matching `p`, the present family has two equal-cost nonparallel candidate slopes. The exact projective-homology classification says the rank-one sector may carry only **one** line at a time; a second nonparallel essential class forces rank two. + +Therefore this lattice family is a natural finite-shape testbed for the projective hard-core-gas picture in `projective-homology-gas-20260914.md`: + +- first birth: a race between the two cheap projective lines; +- rank-one plateau: selected line remains frozen; +- second nonparallel winding: immediate transition to rank two. + +No independent two-species Poisson model can represent all three statements simultaneously. diff --git a/docs/manuscripts/geometric-balance/euler-critical-site-two-birth-process-20260914.md b/docs/manuscripts/geometric-balance/euler-critical-site-two-birth-process-20260914.md new file mode 100644 index 000000000..e42d04479 --- /dev/null +++ b/docs/manuscripts/geometric-balance/euler-critical-site-two-birth-process-20260914.md @@ -0,0 +1,266 @@ +# Euler-critical insertion sites: a discrete-Morse view of the two-birth process + +Date: 2026-09-14 + +Status: exact finite consequence of the square-site Euler identity and rank monotonicity. The “Morse” terminology is descriptive; no smooth Morse-theory theorem is being imported. + +## 1. Local Euler increment + +For a monotone insertion permutation, just before a white site `v` turns black define + +```text +c_b(v) : number of distinct black NN components touched by v, +c_w(v) : number of distinct white matching components adjacent through v + after v is removed / equivalently merged when v is added in reverse, +d_b(v) : number of occupied NN neighbours of v, +f_b(v) : number of elementary squares completed all-black by adding v. +``` + +The configuration Euler identity gives exactly + +```text +boxed: +Delta_v X + = 1 - c_b(v) - c_w(v) + d_b(v) - f_b(v), (1.1) +``` + +where + +```text +X=r-1. +``` + +Because ambient homology rank is monotone under adding occupied sites, + +```text +Delta_v X in {0,1,2}. (1.2) +``` + +Thus (1.1) implies the nontrivial digital-topology inequality + +```text +1-c_b-c_w+d_b-f_b >=0 +``` + +for every actual insertion state, although the separate terms have no obvious sign. + +## 2. Only one or two sites in the whole permutation change ambient rank + +The empty configuration has + +```text +X=-1, +``` + +and the full configuration has + +```text +X=+1. +``` + +Therefore along every strict insertion order + +```text +sum_(v in insertion order) Delta_v X = 2. (2.1) +``` + +Together with (1.2), there are only two possible topological histories: + +### Two simple critical sites + +```text +Delta X=1 at J1, +Delta X=1 at J2, +all other insertions Delta X=0. +``` + +This is the ordinary rank path + +```text +0 -> 1 -> 2. +``` + +### One double critical site + +```text +Delta X=2 at one insertion, +all other insertions Delta X=0. +``` + +This is the direct + +```text +0 -> 2 +``` + +birth with zero rank-one persistence gap. + +Hence the entire two-birth process is exactly a process of one or two **Euler-critical insertion sites** inside an otherwise Euler-neutral permutation. + +## 3. Forward/reverse union-find computes the whole rank path without lifted homology + +For a fixed permutation `pi` of the torus sites: + +### Forward black sweep + +Add sites in the order `pi_1,...,pi_N` on the NN graph and store for every `k`: + +```text +k4(k), +E(k), +F0(k), +K=k. +``` + +`k4` is updated by an ordinary union-find; `E,F0` are local motif counts. + +### Reverse white sweep + +Start from no white sites at `k=N` and add sites in reverse permutation order on the matching graph. This gives + +```text +k8_white(k) +``` + +for the complement of the first `k` black sites, again by an ordinary union-find with no homology gains. + +Then at every cardinality + +```text +boxed: +X_k + = k4(k)-k8_white(k)-k+E(k)-F0(k), +r_k=X_k+1. (3.1) +``` + +Thus the **entire rank path and birth times `(J1,J2)` can be recovered using two ordinary connectivity sweeps**, with the Euler identity supplying the homology rank. + +This is an independent algorithmic route to threshold-rank trajectories on honest square tori. It does not replace lifted homology when projective slope or actual winding class is required. + +## 4. Direct birth is a local/global Euler imbalance, not a bare arm count + +A `0->2` birth occurs exactly when + +```text +1-c_b-c_w+d_b-f_b=2. (4.1) +``` + +This formula clarifies why both theta/T3 and four-germ/rose geometries can realize the same rank jump. + +For example, three black attachment germs can belong to **one** external black component, so `c_b` need not equal the number of visible black arms. The local degree/germs, black-component merging, white-component splitting and completed-face term jointly determine the index. + +Therefore + +```text +arm count != Euler critical index. +``` + +Arm geometry is a continuum/local-separation refinement of the critical site, not its finite definition. + +## 5. A natural classification for #769 + +For every critical insertion, store at minimum + +```text +(rank_before, rank_after), +d_b, +f_b, +c_b, +c_w, +number/cyclic order of black attachment germs, +number/cyclic order of white separating germs, +black/white ambient lift data before/after. +``` + +The tuple `(d_b,f_b,c_b,c_w)` fixes the Euler index through (1.1); the germ/lift data distinguish theta, rose, split-component and small-torus degeneracies. + +Then report topology classes separately for + +```text +index 1 first birth, +index 1 second birth, +index 2 direct birth. +``` + +This is more informative than classifying only all `Delta_v X=2` configurations by a guessed arm label. + +## 6. Relation to `alpha,beta,gamma` + +At a fixed site / parameter, let + +```text +alpha : 0->1 transition mass, +beta : 0->2 transition mass, +gamma : 1->2 transition mass. +``` + +Then the local Euler index gives + +```text +E[Delta_v X] + = alpha + 2 beta + gamma. +``` + +The rank-one occupation derivative gives the signed combination + +```text +P1'/N = alpha-gamma. +``` + +So the two natural lattice channels are + +```text +activity/index channel : alpha+2beta+gamma, +birth-asymmetry channel: alpha-gamma. +``` + +A large theta/T3 contribution to `beta` is a statement about index-2 activity. It does not by itself imply a large contribution to the signed birth-asymmetry channel relevant to the noncommon root correction. + +This recovers the recent parity correction without assigning a continuum field sign. + +## 7. Matching reflection of critical sites + +Under complement and reversed insertion order, the first and second births exchange. The set of Euler-critical sites is transported accordingly: + +```text +index-1 first <-> index-1 second, +index-2 direct <-> index-2 direct. +``` + +This is the pathwise source of the matching-even persistence-gap/direct-birth law and the odd midpoint/birth-asymmetry coordinate. + +Again this is a finite path relation, not an OPE automorphism. + +## 8. Research consequence + +The useful continuum question can now be phrased: + +> what is the scaling geometry of the one or two Euler-critical insertion sites, and which signed combination of their local environments survives in the rank0/rank2 free-energy difference? + +This is sharper than asking which arm event “is” the root correction. + +Potential mechanisms become: + +```text +theta/6-arm geometry dominates index-2 activity beta; +spin-four anisotropic correction lives in the difference between the two index-1 critical-site environments; +scalar/post-H4 corrections alter the common critical-site law or its higher connected responses. +``` + +These are falsifiable decompositions on actual lattice events. + +## 9. Claim boundary + +Exact: + +- local Euler increment (1.1); +- nonnegative increments / total increment 2; +- one-or-two critical-site classification; +- forward/reverse connectivity reconstruction of the rank path. + +Programme: + +- use critical-site environment classes as the lattice interface to arm/CFT fields; +- identify which signed environment statistics control the observed H4 and post-H4 blocks. + +The conceptual compression is substantial: a length-`N` rank process contains only one or two topologically active insertion events. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/euler-localization-of-rank-source-20260914.md b/docs/manuscripts/geometric-balance/euler-localization-of-rank-source-20260914.md new file mode 100644 index 000000000..d0b3a1a0c --- /dev/null +++ b/docs/manuscripts/geometric-balance/euler-localization-of-rank-source-20260914.md @@ -0,0 +1,317 @@ +# Euler localization of the torus rank source: a symmetric primal--dual cluster gas + +Date: 2026-09-14 + +Status: exact finite embedded-graph identity for a cellular graph on the torus, plus a conjectural loop/TL interpretation. This sharpens the Krushkal source dictionary and may provide a direct route toward the modified trace sought in #782. + +## 1. Setup + +Let `G=(V,E)` be a connected cellular graph embedded on `T^2`. Let `A subset E` be a spanning FK subgraph and let `A*` denote the complementary dual subgraph: a dual edge is present exactly when the corresponding primal edge is absent. + +Write + +```text +k(A) = number of connected components of A, +k(A*) = number of connected components of A*, +r(A) = rank im[H1(A)->H1(T^2)] in {0,1,2}. +``` + +The claim below is configurationwise and contains no probabilistic/scaling input. + +## 2. Euler derivation + +Let `N(A)` be a regular neighbourhood of the primal subgraph. It retracts to `A`, so + +```text +chi(N(A)) = |V|-|A|. +``` + +Let + +```text +g = genus carried by N(A), +g* = genus carried by the closure of T^2\N(A), +b = number of common boundary components. +``` + +Then + +```text +|V|-|A| = 2 k(A) - 2g - b. (2.1) +``` + +The complementary region retracts to the complementary dual ribbon subgraph `A*`. Since `G` is cellular on the torus, + +```text +|V|-|E|+|F|=0, +``` + +and therefore + +```text +chi(T^2\N(A)) + = |F|-|E\A| + = |A|-|V| + = 2 k(A*) - 2g* - b. (2.2) +``` + +Equating the two expressions for `b` gives + +```text +k(A)-k(A*)+|A|-|V| = g-g*. (2.3) +``` + +For the torus, Krushkal's topological exponents satisfy + +```text +s=2g, +s_perp=2g*, +``` + +and the rank-source identity from the preceding note gives + +```text +r(A)-1 = g-g*. +``` + +Hence the exact configurationwise formula is + +```text +boxed: +r(A)-1 = k(A)-k(A*)+|A|-|V|. (2.4) +``` + +Checks: + +- empty primal subgraph: `k(A)=|V|`, `k(A*)=1`, `|A|=0` -> `r-1=-1`; +- full primal subgraph: Euler duality gives `r-1=+1`; +- rank-one states give zero charge. + +## 3. The topological source becomes a two-sided cluster fugacity + +Start from the ordinary FK weight + +```text +W_Q,v(A)=Q^{k(A)} v^{|A|}. +``` + +Insert the bounded rank source: + +```text +W_Q,v,h(A) + = Q^{k(A)} v^{|A|} exp[h(r(A)-1)]. +``` + +Using (2.4), + +```text +W_Q,v,h(A) + = e^{-h|V|} + (Q e^h)^{k(A)} + (e^{-h})^{k(A*)} + (v e^h)^{|A|}. (3.1) +``` + +Thus, up to the harmless global factor `e^{-h|V|}`, the rank source is exactly a **primal--dual two-cluster-fugacity random-cluster model**: + +```text +Q_primal = Q e^h, +Q_dual = e^-h, +v_source = v e^h. +``` + +A global surface-topology source has been converted into ordinary component and edge counts of the primal/dual pair. + +This is stronger than merely knowing that the Krushkal polynomial contains the source: it supplies an explicit finite transfer bookkeeping rule. + +## 4. Product invariant + +The two cluster fugacities obey + +```text +boxed: +Q_primal * Q_dual = Q, +``` + +independent of `h`. + +Therefore varying the topological charge redistributes cluster weight between primal and dual sides while keeping their product fixed. + +This strongly suggests that in a medial-loop representation the ordinary bulk loop weight `sqrt(Q)` can remain fixed while `h` is represented by an orientation/face/background-charge bias rather than by changing the loop fugacity itself. + +That loop statement is a conjectural representation interpretation; equation (3.1) is exact. + +## 5. The graph-polynomial source is the symmetric point + +The generic-Q graph-polynomial balance source is already known exactly: + +```text +h_Q = -1/2 log Q. +``` + +Substitute it into the two cluster fugacities: + +```text +Q_primal = Q e^{h_Q} = sqrt(Q), +Q_dual = e^{-h_Q} = sqrt(Q). +``` + +Therefore + +```text +boxed: +the graph-polynomial topological source is exactly the point where +primal and dual cluster fugacities become equal. +``` + +The sourced edge fugacity becomes + +```text +v_source = v/sqrt(Q). +``` + +At the square-lattice self-dual FK point `v=sqrt(Q)`, this is simply + +```text +v_source=1. +``` + +Thus the sourced graph-polynomial condition has an unexpectedly symmetric form: + +```text +primal cluster fugacity = dual cluster fugacity = sqrt(Q), +rescaled edge fugacity = 1. +``` + +This is a plausible structural explanation for why the periodic Temperley--Lieb eigenvalue criterion is so natural at the graph-polynomial root. + +It is not yet a derivation of Jacobsen's complete transfer formula; seam/through-line and closure normalizations still have to be matched. + +## 6. A candidate route to the missing modified trace + +Equation (3.1) suggests replacing the abstract question + +```text +"how do we insert e^{h(r-1)} into a periodic TL trace?" +``` + +by the more concrete problem + +```text +"how do we assign distinct closure fugacities to primal and dual clusters, +while keeping Q_primal Q_dual=Q?" +``` + +A connectivity transfer can do this locally in time: whenever a primal or complementary dual component closes, multiply by its declared cluster fugacity. The only additional care is for components that survive around the periodic direction / noncontractible sectors. + +The medial-loop formulation should encode the same asymmetry through the alternating primal/dual faces separated by the loops. Because the product fugacity is fixed, an oriented-loop/height background charge is a natural candidate language. + +This narrows the #782 representation problem substantially: + +```text +finite state-sum projector : solved by (2.4)/(3.1), +local primal-dual transfer : apparently constructible, +periodic TL/Markov trace closure : still to derive, +massive/TBA continuation : still open. +``` + +## 7. Rank-one neutral source remains independent + +The charge source `h` changes the relative weight of rank0 and rank2 while leaving rank1 with unit topological factor. + +The existing affine-TL seam `alpha` instead acts inside rank one: + +```text +z=alpha^2/Q, +``` + +weighting the number `K` of parallel essential clusters. + +Thus the two finite source directions are genuinely complementary: + +```text +h : primal--dual cluster fugacity imbalance / topological charge, +alpha : neutral essential-count fugacity. +``` + +A combined periodic transfer should carry both. + +## 8. Relation to the h-odd sector and Matching One + +Differentiate (3.1) at `h=0`: + +```text +r-1 = k(A)-k(A*)+|A|-|V|. +``` + +So the matching-odd topological observable is an **Euler imbalance** between primal clusters, dual clusters and occupied edges. + +This offers another microscopic interpretation of the first noncommon correction: + +> the relevant continuum field must survive after the extensive/local primal--dual Euler pieces cancel down to the bounded torus topology charge. + +It also explains why low-order ordinary thermal/geometric corrections can be large in individual sectors yet disappear from the rank-source response. + +## 9. Possible relation to the thermal spin-four mechanism + +Under the sourced representation, a microscopic anisotropy can affect + +```text +primal cluster closure statistics, +dual cluster closure statistics, +edge density. +``` + +The matching-root differential response is their specific Euler-signed combination. + +A sector-even identity-family anisotropy can contribute almost equally to the primal and dual cluster gas and cancel. The first anisotropic mismatch of the thermal/pivotal structure can then survive as the observed spin-four response. + +This is qualitative at present but gives a more concrete cluster-language target than an abstract `x=21/4` label. + +## 10. Relation to original graph-polynomial algebra + +The previous exact result + +```text +Z_2-Q Z_0=0 +``` + +was reinterpreted as a topological charge source `h_Q=-1/2 log Q`. + +The new Euler localization shows that this same source is the **primal-dual symmetric cluster-gas point**. Hence two seemingly different explanations of the graph-polynomial criterion coincide: + +```text +topological sector balance +<=> +primal/dual cluster-fugacity symmetry after Euler localization. +``` + +This equivalence may be useful for proving which transfer eigenvalue sectors are compared by the periodic TL criterion. + +## 11. Site-model boundary + +Everything above is exact for an edge/FK spanning-subgraph model on a cellular torus. + +Square-site Matching One has an analogous digital-Alexander rank charge but does not literally have the edge-complement dual subgraph used in (2.4) without passing to an incidence/decorated cell representation. Therefore the two-cluster-fugacity localization is presently a Potts/FK continuum/interface tool, not an asserted exact square-site identity. + +The distinction is important for #275 and for any direct numerical source implementation. + +## 12. Claim boundary + +Exact: + +```text +r-1=k(A)-k(A*)+|A|-|V|, +W_Q,v,h=e^{-h|V|}(Qe^h)^k(e^{-h})^{k*}(ve^h)^{|A|}, +Q_primal Q_dual=Q, +h_Q=-1/2 log Q -> Q_primal=Q_dual=sqrt(Q). +``` + +Conjectural/interface: + +- an oriented-loop/height realization with fixed bulk loop fugacity `sqrt(Q)`; +- direct identification with the needed periodic affine-TL modified trace; +- massive/TBA continuation; +- application as an exact site-percolation transfer identity. + +This turns the topological source from a closure-only label into an explicit primal--dual cluster-gas deformation. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/euler-source-rg-tangent-program-20260914.md b/docs/manuscripts/geometric-balance/euler-source-rg-tangent-program-20260914.md new file mode 100644 index 000000000..65bbb6dd2 --- /dev/null +++ b/docs/manuscripts/geometric-balance/euler-source-rg-tangent-program-20260914.md @@ -0,0 +1,200 @@ +# From the exact Euler source to an RG tangent map + +Date: 2026-09-14 + +Status: programme built around an exact finite sourced-family involution. It partially answers the structural concern of #61 without claiming a local CFT/OPE automorphism. + +## 1. An exact two-parameter sourced family exists on the doubled site model + +For the black NN / white matching pair define the exact topological source + +```text +X=r4-1=k4-k8-K+E-F0. +``` + +The sourced finite weight is + +```text +P_{p,h}(omega) + proportional to + p^K (1-p)^(N-K) exp[hX]. +``` + +On the doubled model family, complement/matching gives exactly + +```text +(G4,p,h,omega) + <-> +(G8,1-p,-h,omega^c). +``` + +Thus, unlike an assumed local CFT parity, this sourced finite-family exchange is an actual involution. + +## 2. Center the thermal coordinate + +Let `p_c^4` and `p_c^8=1-p_c^4` be the paired critical points. Use a local thermal coordinate `t` chosen so complement sends + +```text +t -> -t. +``` + +At linear order one can take `t=p-p_c` up to a positive metric factor, provided the two members use paired normalizations. + +Then on the exact two-dimensional sourced family + +```text +(t,h) -> (-t,-h). (2.1) +``` + +So the doubled matching action on the span of these two finite tangent directions is explicit. + +This is a small but genuine piece of the RG-tangent map requested by #61. + +## 3. What this does and does not establish + +It establishes the microscopic transformation of two **source coordinates**: + +```text +thermal complement t, +topological Euler source h. +``` + +It does not establish that a local scaling field with a given dimension/spin is an eigenvector of matching exchange. Under RG, the microscopic `h` direction may mix with every continuum field allowed by the same geometric/source quantum numbers. + +The correct question is therefore to measure/derive the image of the `h` tangent under coarse graining. + +## 4. Add local residualized sources to enlarge the tangent space + +Choose a finite local source basis `g_a` after normalization and thermal residualization. Bernoulli chaos gives a convenient exact microscopic grading: + +```text +degree 2 even control, +degree 3 odd-under-complement control, +selected D4 spatial representations. +``` + +The full finite source coordinates are + +```text +u=(t,h,g_2,g_3,...). +``` + +The exact pair exchange acts linearly on these microscopic source functions at the baseline (with the known chaos signs after graph/color pairing). + +The RG question becomes: + +```text +what is the large-scale linear map from this microscopic source basis +into the low-dimensional scaling-response space? +``` + +No OPE algebra is needed for this first stage. + +## 5. Observable response matrix + +For a declared set of finite observables / safe free energies `O_i`, compute + +```text +R_ia(L)=partial_(g_a) O_i +``` + +with normalized physical source weights. + +Useful rows include + +```text +Euler/rank charge response, +fixed-b rank shape response, +safe-sector free-energy difference, +angular H4/H0 projected root response. +``` + +Useful columns include + +```text +thermal t, +topological source h, +degree-2 residualized motif, +degree-3 residualized motif, +controlled anisotropy source. +``` + +The singular vectors / stable column relations of `R(L)` as `L` grows are empirical RG tangent directions. Only after a stable block appears should one attach continuum field labels. + +## 6. The topological source is especially informative because it is bounded + +Although its Euler representation contains extensive cluster/local terms, configurationwise + +```text +X in {-1,0,+1}. +``` + +Thus `h` couples directly to the exact root-information bottleneck rather than to a generic bulk density. + +The mixed response + +```text +partial_g partial_h log Z |_(h=g=0) + = Cov(X,H_g) +``` + +is exactly the numerator controlling root motion under source `g`. + +At the root, with thermal score `S_p`, + +```text +T_g=-Cov(X,H_g)/Cov(X,S_p). +``` + +Therefore the entire #802 root-tangent programme can be viewed as measuring mixed susceptibilities with the exact Euler source. + +## 7. A clean interpretation of “first noncommon correction” + +Instead of saying a continuum field is “matching odd,” define the observable fact: + +```text +a perturbation/source has a nonzero mixed susceptibility with X +(or the corresponding safe-phase Euler/topological free-energy difference). +``` + +Then ask which angular/radial scaling block carries that mixed susceptibility. + +For the current square model the leading block appears H4-like with root exponent four. A later angular-scalar block may have exponent seven. These are statements about the response of the exact `h`-sourced observable, not parity assignments of abstract fields. + +## 8. Relation to generic-Q / bond-FK source + +The bond/FK Krushkal/Euler source provides an analogous exact topological tangent, with its own local cluster-gas dictionary. At Q=1 the continuum limits of the bond and site realizations may belong to the same topological source class, but that is a universality/interface statement, not an exact lattice equality. + +This suggests a future high-value cross-model test: compare normalized mixed `h-g` response ratios between square-site and a rigorously controlled bond/FK realization after matching thermal metrics and modulus. + +## 9. Acceptance levels for #61-style parity language + +Use three levels: + +```text +Level 1: exact microscopic source exchange + (established for t,h and Bernoulli-chaos source functions); + +Level 2: RG tangent block / scaling-response exchange + (to be inferred/proved from response matrix or transfer algebra); + +Level 3: local OPE/interchiral automorphism + (not required for finite-size selection and currently unproved). +``` + +Most current Matching-One conclusions only need Level 2, not Level 3. + +## 10. Claim boundary + +Exact: + +- finite sourced-family involution `(t,h)->(-t,-h)` on the doubled site pair; +- mixed susceptibility `Cov(X,H)`; +- residualized root response formula. + +Programme: + +- construct the empirical/theoretical RG tangent map from a small source-response matrix; +- identify continuum blocks only after stable angular/radial/source structure emerges. + +This is a constructive replacement for prematurely assigning a scalar matching parity to continuum fields. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/euler-source-vs-common-q-tangent-20260914.md b/docs/manuscripts/geometric-balance/euler-source-vs-common-q-tangent-20260914.md new file mode 100644 index 000000000..b71e7d5b1 --- /dev/null +++ b/docs/manuscripts/geometric-balance/euler-source-vs-common-q-tangent-20260914.md @@ -0,0 +1,196 @@ +# Euler charge tangent versus the common Potts `Q` tangent + +Date: 2026-09-14 + +Status: source-direction correction to the generic-Q/log-collision narrative. The finite Euler-source decomposition is exact; continuum branch projections remain to be derived. + +## 1. The source directions are not automatically the same + +For square-site Matching One, the exact Euler source is + +```text +X=r4-1=k4-k8-K+E-F0. +``` + +Exponentiating gives + +```text +e^{hX} + = e^{+h k4} + e^{-h k8} + e^{-h K+hE-hF0}. +``` + +Thus, at the level of black/white cluster fugacities, the `h` tangent contains + +```text +boxed: +(delta log Q_black, delta log Q_white) + = (+1,-1) delta h, +``` + +plus explicit local site/edge/plaquette counterterms. + +This is an **antisymmetric two-colour cluster-fugacity tangent**. + +By contrast, the ordinary generic Potts `Q` derivative changes the common cluster fugacity of a declared random-cluster/Potts model. In a two-colour extension its cluster-fugacity component is naturally common/symmetric rather than the Euler difference direction. + +Therefore + +```text +Euler h tangent != generic common-Q tangent +``` + +unless an additional model-specific map proves otherwise. + +## 2. Why this matters for V14/W22 collision + +The exact generic-Q Kac collision + +```text +x_Kac(Q)-x_W(Q) + = c_Q (Q-1)+... +``` + +with + +```text +c_Q=-15/(2 pi sqrt(3)) +``` + +is a statement about branch splitting under the **standard Potts Q deformation**. + +If an observable is a Q derivative of a colliding pair, pole-cancelled amplitudes can generate a logarithmic size term through + +```text +partial_Q L^-x(Q) + = -x_Q' log L * L^-x + ... . +``` + +But the Matching-One root is an `h` response of the Euler-sourced paired site family. The Q-splitting velocity only predicts its log coefficient if the actual `h` tangent has a nonzero projection onto the same generic-Q branch-splitting direction. + +That projection has not been established. + +Therefore the current V14/W22 note should be read as + +```text +representation-theory resonance + possible generic-Q log mechanism, +``` + +not as a direct prediction that the Matching-One Euler source must show the corresponding log. + +## 3. A two-colour deformation space is the natural missing object + +Introduce formal local coordinates near percolation such as + +```text +q_b = log Q_black, +q_w = log Q_white, +t = thermal coordinate, +... local counterterm coordinates ... +``` + +and define + +```text +q_common = (q_b+q_w)/2, +q_diff = (q_b-q_w)/2. +``` + +Then the Euler source has + +```text +partial_h q_diff = 1, +partial_h q_common = 0, +``` + +before the declared local counterterms are included. + +The standard Potts Q tangent is primarily a `q_common` direction, with its own thermal/representation compensation. + +The continuum RG problem is therefore to determine the tangent map + +```text +(q_common,q_diff,t,...) + -> scaling-field/block couplings. +``` + +The V14/W22 branch splitting supplies information along `q_common`; Matching One needs the `q_diff` column. + +## 4. Bond/FK canonical coordinates provide an analogy, not the site proof + +For toroidal FK the earlier exact canonical coordinates were + +```text +x=v/sqrt(Q), +eta=h+(1/2)log Q. +``` + +There, changing Q at fixed `eta=0` requires a compensating `h` shift, while `eta` is an independent topological-charge direction. This already shows algebraically that “Q” and “topological charge” are distinct tangent coordinates. + +The square-site Euler source has a parallel conceptual structure, but its site/edge/face counterterms make the precise generic extension different. Do not import the FK coordinate formula as an exact site identity. + +## 5. A targeted generic-extension test + +Before a large generic-Q production, construct the smallest finite two-colour extension in which: + +1. black NN clusters receive fugacity `Q_b`; +2. white matching clusters receive fugacity `Q_w`; +3. local site/edge/plaquette weights are explicit; +4. `Q_b=Q_w=1` recovers ordinary site percolation; +5. the Euler source path is exactly reproduced by + +```text +Q_b=e^h, +Q_w=e^-h, +``` + +with the local Euler counterterms. + +Then compute a small response matrix of the candidate post-H4 block to + +```text +partial_(q_common), +partial_(q_diff), +partial_t. +``` + +This is much more informative than differentiating only standard Q and assuming the answer transports. + +## 6. Possible outcomes for the log collision + +### Common-Q only + +If the collision/Jordan logarithmic residue couples only to `q_common` while the Euler `q_diff` column is regular/small, the V14/W22 collision is structurally real but mostly irrelevant to Matching-One root response. + +### Shared block + +If both common and difference tangents project onto the colliding block, the generic-Q splitting velocity can be used after multiplying by the measured tangent-projection coefficient. + +### Difference-specific structure + +The Euler tangent may excite a different partner/combination than the standard Potts Q tangent. Then the correct log coefficient must be derived from the two-colour extension, not inherited from `d_Q Delta x`. + +## 7. Relation to the source-normalization audit + +The local counterterms `-K+E-F0` are not optional decorations. They are part of the exact Euler source. Dropping them changes the tangent direction, just as omitting the row normalizer in #802 changed the physical source response. + +Therefore any generic extension must match the **whole source path**, not only the cluster-fugacity signs. + +## 8. Claim boundary + +Exact: + +- Euler source decomposition into `(+ black cluster, - white cluster)` plus local counterterms; +- distinction between common and difference fugacity tangent coordinates. + +Known external algebra: + +- V14/W22 generic-Q dimension collision and Q splitting velocity. + +Open: + +- RG projection of the Euler difference tangent onto that colliding block; +- whether the Matching-One post-H4 residual therefore carries a compulsory/visible logarithmic partner. + +The main correction is simple: **a generic-Q collision is only relevant after the physical source tangent has been projected onto it.** \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/exploration-noise-and-random-information-20260914.md b/docs/manuscripts/geometric-balance/exploration-noise-and-random-information-20260914.md new file mode 100644 index 000000000..3c190c9c8 --- /dev/null +++ b/docs/manuscripts/geometric-balance/exploration-noise-and-random-information-20260914.md @@ -0,0 +1,375 @@ +# 探索噪声、巨大簇的联合涨落与随机信息 + +2026-09-14。直接续写 #764 / #772，接在 #771 上。研究对象仍是同一批独立 site 标签：黑色 NN 概率 p 固定在 (0,pc_NN)，白色 matching 概率 q=1-p。每个完整白色 essential 簇计数一次，而不是按顶点、切口或某次成功配置计数。 + +本记录有三个层次：第 1–2 节是独立的精确探索恒等式；第 3–5 节给出利用已有黑色笼罩/稀有屏障输入的联合弱极限论证；第 6–8 节是其推论与明确未证的更远猜想。有限计算检查的是精确接口，不能证明宽度极限。没有使用固定-p OZ 前因子、单位缝合振幅或质量的 p 解析性。 + +## 0. 本轮究竟增加了什么 + +此前已经得到完整白簇的尺度律及一个评分投影： + + (nu L, nu K/w, nu B/w) -> (E, theta E, (p/q) theta E), + sqrt(nu/w) (K/q-B/p) -> sqrt(c E) Z, + E~Exp(1), c=theta/(p q^2). + +其中 nu=nu_w 是每行完整簇密度，K 是白色站点数，B 是不同外部黑站点数。一个黑站点若邻接两个不同白簇，分别计入两个簇的 B。 + +本轮的核心连接是： + +1. 从固定源探索一个簇，查询评分在不同站点之间精确正交，甚至所有互异站点的非空多线性矩都为零。它们仍然可以依赖。 +2. 在平面超临界白色模型中，体积/边界的一个特定组合因此是精确二阶白噪声。它不是原始占据场，也不是连接概率。 +3. 指数局部的 bulk 场与指数稀有的屏障可以通过同一分块同时取极限：得到相互独立的 Brownian 过程和 Poisson 屏障。随机屏障间隔再把 Brownian 增量变成混合高斯。 +4. 整个 (K,B) 剩余波动的候选协方差不再是任意 2×2 矩阵。除 theta、theta' 外，只剩一个平面协方差和 Gamma(q) 需要确定。 +5. 空间评分的正交模式共享一个指数时钟：协方差为零，平方的相关系数却趋向 1/5。除去实际观察信息后才恢复高斯。 +6. 同一簇的局部概率反演 K/(K+B) 出现 Student-t_2 极限。其均方误差不能用弱极限的表面尺度直接估计。 + +已有输入的详细论证在所有者交付 `black-white-gap-law.md`、`giant-white-bulk-and-response.md`，以及 #764 评论 5659233515 / 5659520483。#771 的 `supercritical-white-slab-bulk-20260914.md` 独立写出了相同的边界密度恒等式。重合恒等式不是第二份新结果。 + +## 1. 有限固定源的精确探索恒等式 + +取任意有限图，固定一个非空 open 源集合。其余站点 v 的标签 n_v 独立，Pr(n_v=1)=q_v，p_v=1-q_v。探索所有与源相连的 open 站点，以及它们不同的外部 closed 邻接站点。固定源不计入随机评分。 + +令 A_v 表示 v 被查询，等价于删去 v 后，v 至少有一个邻点可以通过 open 路径连接源。A_v 不依赖 n_v。定义 + + xi_v = A_v (n_v-q_v)/(p_v q_v). + +查询为 open 时贡献 1/q_v；查询为 closed 时贡献 -1/p_v；未查询时为零。 + +### 定理 1：多线性归一化 + +对任意有限参数 h_v，只要 q_v+h_v 合法， + + E prod_v (1+h_v xi_v) = 1. (1.1) + +两侧为多项式，所以恒等式也代数延拓到任意 h_v。 + +证明一：完整源簇及其外边界的概率恰为查询站点的独立标签权重。prod(1+h_v xi_v) 是将 q_v 改成 q_v+h_v 后，这个完整观察结果的似然比。对所有完整结果求和仍为一。没有查询的站点已被积分掉，不能再对它们追加概率因子。 + +证明二：按任意固定探索规则依次查询。每次查询前的位置和是否查询已由历史决定，下一标签仍是独立 Bernoulli(q_v)。每个新乘子条件期望为一；有限次迭代即得 (1.1)。 + +比较系数给出，对任意非空互异站点集合 T， + + E prod_(v in T) xi_v = 0. (1.2) + +特别地， + + E xi_v=0, + Cov(xi_u,xi_v)=0 (u!=v), + Var(xi_v)=Pr(A_v)/(p_v q_v). (1.3) + +这不是独立性。某个站点是否被查询，可以由前一个站点的标签决定；平方评分之间通常有相关。 + +### 精确响应公式 + +若 F 是完整源簇/边界的固定函数，不显含 q，则 + + partial_(q_v) E F = E(F xi_v). (1.4) + +这是被查询数据的真实评分，不是把全图 Bernoulli 评分不加区别地替换成较小的和。未查询部分的条件评分为零。 + +本轮在三个有回路/分支的有限源图、两个有理 q 上，枚举了 320 个配置；检查 314 个非空多线性零矩、194 个协方差等式和194个源响应等式，并检查有限非均匀参数变化的正规化恰为一。 + +## 2. 平面无限簇中的精确白噪声方向 + +在无限白色 matching 图上，定义 + + eta_v = 1{v 属于无限白簇}, + beta_v = 1{v 是黑色且邻接至少一个无限白簇}. + +这里 beta_v 是一次性 presence 指标，不是多个有限 essential 簇的边界重数。令 A_v^infty 表示删去 v 后，某个邻点仍可连接到无穷。 + +有限度保证：添加一个站点不能把有限个有限分量变成无限分量。因此 + + eta_v=n_v A_v^infty, + beta_v=(1-n_v) A_v^infty, + xi_v=eta_v/q-beta_v/p. + +取逐渐增大的 wired 方框，固定其中任意有限组内点，第 1 节的 xi_v 收敛到这里的 xi_v。变量有界，故 (1.2)–(1.3) 传到无限体积。记 theta=E eta_0，则 + + E beta_0=(p/q)theta, + E xi_0=0, + Cov(xi_0,xi_z)=c 1{z=0}, + c=theta/(p q^2). (2.1) + +所以这个由无限簇占据与外边界相减得到的场，具有完全平坦的二阶谱。原始 eta、beta 场各自并不独立。 + +### 2.1 二维 bulk 协方差只剩一个自由标量 + +令 X_v=(eta_v,beta_v)，并定义绝对收敛的平面协方差和 + + Sigma = sum_(z in Z^2) Cov(X_0,X_z), + Gamma(q)=Sigma_11. + +第 3 节的局部笼罩界给出指数局部近似，从而保证上述绝对收敛。这个 Gamma 是无限簇成员指标的协方差和，不是 sum P(0<->z)；后者在超临界平面不是同一个有限对象。 + +在可逐项使用 Russo 公式的参数上，有 + + Sigma (1/q,-1/p)^T = (theta', beta')^T, + beta'=(p/q)theta'-theta/q^2. (2.2) + +理由是，若改变站点 u 能改变某个固定站点是否无限连通，则在删去 u 的外部配置中，u 必须邻接一个无限分量。因此源评分 xi_u 与全 Bernoulli 评分给出同一个 pivotal 贡献。距离 r 处的 pivotal 会在 u 关闭时留下延伸至少 r-O(1) 的有限白簇；黑色笼罩界把其概率压成 C exp(-cr)。影响和在紧亚临界黑色参数区间上一致可和，有限方框 Russo 公式可用积分和支配收敛传递。无需假设质量函数 kappa 的参数解析性。 + +展开 (2.2) 得到 + + Sigma = [ Gamma, + (p/q)Gamma-p theta'; + (p/q)Gamma-p theta', + (p^2/q^2)Gamma-(2p^2/q)theta'+p theta/q^2 ]. (2.3) + +并有 + + (1/q,-1/p) Sigma (1/q,-1/p)^T=c, + Gamma >= theta'^2/c. (2.4) + +定义剩余幅度 + + J(q)=Gamma(q)-theta'(q)^2/c >=0. (2.5) + +评分噪声是其中一个已知方向；J 衡量去掉该方向后真正剩下的 bulk 波动。是否处处 J>0、怎样有效计算它，仍然值得研究。本文没有从有限宽度矩阵拟合出一个无限 Gamma。 + +## 3. 一个足够强的局部性引理 + +对白色圆柱模型，eta_w(v) 表示 v 属于任意 essential 白簇；beta_w(v) 表示 v 黑且邻接至少一个 essential 白簇，仍然不计重数。它们是平移不变的有界场。 + +已有黑色 root-volume 输入给，对所有圆柱宽度一致， + + Pr(|C_black(v)|>=n) <= C exp(-cn). + +已有 cage 几何给：一个非essential白簇若从 v 走到距离 r，其可缩外边界对应一个大小至少 (r-C)/C 的黑簇，且该黑簇距 v 至多其大小乘一个固定常数。圆柱半径 k 的球至多 O(k^2) 个站点，因此对可能外边界的位置作分层求和，仍得到 C exp(-cr)。 + +本轮需要的局部事件不是仅仅“找到一条很长白路径”。在圆柱图距离 r 的球内，令 g_(w,r)(v) 为：v 的局部白分量含 essential 圈，或到达该球的边界。则对所有 r（包括 r>w） + + g_(w,r)>=eta_w, + Pr(g_(w,r)!=eta_w)<=C exp(-cr). (3.1) + +若全分量 essential，但局部未找到圈，它必已离开球；反方向的误差只能来自达到球边界的非essential白簇。这正是 cage 界控制的对象。r>w 时必须保留“局部圈”一项：很短的白色 essential 簇没有必要纵向延伸 r。 + +将邻点的 g 代入 beta 的定义，得到相同的指数局部近似。两个距离超过 2r+O(1) 的近似场依赖不交标签。故 + + |Cov(X_w(u),X_w(v))|<=C exp[-c dist(u,v)] (3.2) + +对宽度一致。对固定距离先取 w→∞，局部盒可注入平面；再利用 (3.2) 求和，得到圆柱协方差密度 Sigma_w→Sigma。局部场的任何有限组矩也可用同样的截断处理。 + +这一步没有对超临界白簇体积声称指数尾。指数尾施加于黑色笼罩；有限白簇体积可只有伸展指数尾。 + +## 4. 同一批标签仍能产生独立的 Brownian bulk 与 Poisson 屏障 + +记 s_w=sqrt(nu/w)。在纵向坐标 t=nu y 上考察 + + Z_w(t)=s_w sum_(0<=y (N,B_Sigma), (4.1) + +其中 N 是率一 Poisson 点过程，B_Sigma 是协方差密度 Sigma 的二维 Brownian 过程，二者独立。这里给出实现独立性所需的估计，而不把不同物理对象预设为独立。 + +取局部化半径 r_w=w^2、主体块长度 ell_w=w^6，并在主体块之间留大于 4r_w 的缓冲。所有尺度都远小于 nu^(-1)。 + +- 场局部化误差：在 O(1/nu) 行上，归一化 L1 误差不超过 C sqrt(w/nu) exp(-c w^2)，趋零。 +- 锚点局部化误差：已有完整黑簇体积尾使整个宏观窗口的错误概率为 polynomial(w)/nu 乘 exp(-c w^2)，趋零。 +- 主体块中的局部标签集合互不相交，因此块对“bulk奖励、锚点事件”相互独立。块内二者不要求独立。 +- 一个居中奖励块 A 的绝对值至多 C w ell_w。故 s_w |A|<=C sqrt(nu w) ell_w→0，Lindeberg 条件成立。其方差除以 w ell_w 趋向 Sigma，来自绝对可和协方差以及边缘占比趋零。 +- 单块多锚点的总误差由原先的局部 BK/Poisson 二阶界控制；单锚点概率为 nu ell_w(1+o(1))。 +- 对固定 t、z，块内混合项满足 + + |E[(exp(i t s_w A)-1) 1{块内有锚点}]| + <= C |t| s_w w ell_w Pr(块内有锚点). + + 在 O(1/(nu ell_w)) 个块上求和，总误差至多 C sqrt(nu w) ell_w，仍趋零。因此联合对数变换分成 Gaussian 和 Poisson 两项。 +- 缓冲区的锚点期望为 O(r_w/ell_w)。居中 bulk 方差占比也是 O(r_w/ell_w)，由 (3.2) 控制。局部化后的缓冲可以再次分成互不相交的块，最大偏差用独立增量不等式控制。主体块内的最大跳动已由 s_w w ell_w→0 控制。 + +这些估计给出有限维联合收敛及紧性，不仅是一个协方差为零的论断。固定几个有界横向/纵向测试函数时，同样的块论证给 Gaussian 随机测度。 + +### 为什么 Palm 转换没有除以一个未控制的小概率 + +在长度 T/nu 的平稳窗口中，锚点数量的均值为 T。对这个窗口内所有锚点求和，再除以其平均数量，是 Campbell/Palm 变换。局部 Poisson 二阶界给锚点计数的一致可积性。先对有界连续标记取极限，再除去窗口端点，就得到锚点 Palm 下的下一段间隔及对应 Brownian 增量。没有先证明一个 o(1) 误差，再除以单行锚点概率 nu。 + +因此，令 D 为相邻黑锚点间距， + + (nu D, Z_w(a_next)-Z_w(a)) -> (E,sqrt(E)G), (4.2) + +G~N(0,Sigma)，独立于 E~Exp(1)。连续有限个间隔也给独立的对应对。 + +## 5. 把 bulk 场真正转回一个完整白簇 + +eta_w 的总和会把同一高度内的其他 essential 白簇也算进去；beta_w 是 presence 而 B 是逐簇边界。因此还需要逐簇的误差控制。 + +令 N_W(y)、N_B(y) 为该行被白/黑 essential 簇跨度覆盖的次数。已有交替拓扑给 N_W<=1+N_B，并有 E N_B(0)=nu E_B L=O(nu w)。 + +Campbell 计数于是给 + + E_W |sum_(v in span C) eta_w(v)-K(C)| <= C w^2. (5.1) + +对边界把跨度扩大上下各一行。一个黑站点若邻接两个不同 essential 白簇，其黑分量必须 essential：非essential黑分量的外邻白边界只有一个 essential 外侧，其洞内白分量可缩。因此边界重数与 presence 的平均差至多常数乘黑色 essential 顶点密度。加上端点行，得到 + + E_W |sum_(v in enlarged span C) beta_w(v)-B(C)|<=C w^2. (5.2) + +两式乘 s_w 后趋零。白簇端点与配对黑锚点的距离也只有 O_P(w)；对任意固定多项式上界截断黑端部，其归一化贡献都趋零。 + +写 + + rho_K,w=E_W K/(w E_W L), + rho_B,w=E_W B/(w E_W L). + +由 nu E_W L=1+O(nu w)、精确质量搬运与边界重数界，rho_K,w、rho_B,w 与对应平稳场均值之差是 O(nu w)。乘以 wL 再乘 s_w，仍为 o_P(1)。 + +综上得到完整的二维弱极限： + + (nu L, + sqrt(nu/w) (K-w rho_K,w L, B-w rho_B,w L)) + -> (E,sqrt(E)G), G~N(0,Sigma), G independent of E. (5.3) + +这是本轮对 #772 的主要推进。Sigma 由平面的局部可和协方差给出，并受 (2.3) 限制；不是从有限宽度的三个矩拟合出来。 + +必须保留的边界：本节通过 L1 几何缺陷得到的是联合弱极限。它本身不自动认证所有残差高阶矩收敛。实际有限残差协方差趋向 Sigma，还应补出缺陷的相应高阶矩界。对纯评分投影，前一份物理倾斜证明已经有独立的指数可积性与固定矩控制；不要把它直接移植给全部二维残差。 + +### 更一般的空间评分 + +对有界列剖面 f_j(x)，令 G_ij,w=(1/w)sum_x f_i(x)f_j(x)→G_ij，并定义 + + S_f(C)=sum_(v in C) f(x_v)/q - sum_(v in boundary C) f(x_v)/p. + +平面评分场的精确白噪声协方差 (2.1)，加上相同的局部化/分块/Palm 转换，给出 + + (nu L, s_w S_f1,...,s_w S_fd) + -> (E,sqrt(cE) Z_G), Z_G~N(0,G), Z_G independent of E. (5.4) + +有限 d、统一有界剖面是这里的范围，不宣称维数可以任意快地随 w 增长。对纯列剖面，还可用非均匀列概率 q_x=q+s_w sum_j t_j f_j(x) 独立验证：互补黑簇大小 O(w) 保证 log(nu(q_x)/nu(q))=O(w s_w)→0；真实概率倾斜的二次项给 cE G。它不需要 homogeneous 的 23 态以下压缩对非均匀场仍成立。 + +若 f_j 还依赖 s=nu(y-min C)，则极限条件协方差是 + + c integral_0^E integral_0^1 f_i(x,s) f_j(x,s) dx ds. + +因此可以把它理解成定义在随机区间 [0,E] 上的 Gaussian 白噪声。非恒定纵向剖面的边缘分布一般不再是简单 Laplace。 + +## 6. 零协方差不等于独立；观察信息可以去掉共同长度 + +取正交、单位均方的两个列模式。极限写为 + + Y_1=sqrt(cE) Z_1, Y_2=sqrt(cE) Z_2. + +于是 + + Cov(Y_1,Y_2)=0, + E(Y_1^2 Y_2^2)=2c^2, + E Y_j^4=6c^2, + Corr(Y_1^2,Y_2^2)=1/5. (6.1) + +共同的随机长度可以完全不出现在协方差里，却清楚地出现在四阶量里。多个独立 Laplace 变量不是这里的正确联合模型。 + +定义实际观察信息矩阵 + + V_ij(C)=sum_(v in C) f_i f_j/q^2 + +sum_(v in boundary C) f_i f_j/p^2. + +由体积/边界填充，s_w^2 V→cE G。若 G 正定， + + V^(-1/2) (S_f1,...,S_fd)^T -> N(0,I), (6.2) + +且该标准高斯与 E 独立。有限矩阵偶尔奇异时可单列；其概率趋零。这里不用先知道 theta、nu 或另拟合一个振幅。评分有限均值 O(w)，在此尺度上可忽略；需要有限样本精确中心时仍应保留它。 + +本轮的全高度精确有理数计算：white matching q=3/4，f0=1，f1=(-1)^x，w=2,4。两模式的协方差恰为零；归一化混合四阶矩分别为 2.0299522762421、2.0188871689142；平方相关系数分别约 0.186431434307、0.200087918605。后一个相关系数的平方根评价是浮点显示，其原始矩全部为分数。两个宽度不构成极限证明。 + +## 7. 随机信息使一个概率反演产生 Student 分布 + +考虑只用一个完整白簇的 K、B 作诊断性反演 + + q_tilde=K/(K+B). + +这不是建议从模拟中重新估计本来已知的 q。它是一个明确逆问题，用来揭示拓扑选择之后的信息结构，也可检查采样是否错误地按体积加权。 + +令 S=K/q-B/p，T=K+B。逐配置有 + + q_tilde-q = p q S/T. (7.1) + +由先前评分/体积联合极限， + + (q_tilde-q)/sqrt(nu/w) + -> Z/sqrt(cE) = t_2/sqrt(c). (7.2) + +这里 t_2 是自由度二的 Student 分布。其密度为 + + f(x)=sqrt(c)/(2+c x^2)^(3/2). + +它的方差无穷。因此即使每个有限宽度的 q_tilde 都有界，Portmanteau 对 min(x^2,M) 的应用仍给 + + E[(q_tilde-q)^2]/(nu/w) -> +infinity. (7.3) + +同时 q_tilde→q，所以未缩放均方误差仍趋零。弱极限给出的典型尺度不是均方误差尺度。 + +若改用实际暴露量标准化， + + sqrt(T/[q_tilde(1-q_tilde)])(q_tilde-q) -> N(0,1). (7.4) + +没有删除罕见的短簇；随机信息本身解释了重尾。 + +### 独立簇汇总与观察协议 + +若从 n 个独立配置中各取一个 component-Palm 白簇，并汇总 K、B，保持 n 固定，则总时钟 E_n~Gamma(n,1)， + + (q_tilde_n-q)/sqrt(nu/w) -> t_(2n)/sqrt(c n). (7.5) + +n>=2 时右侧方差为 1/[c(n-1)]；这是极限分布的方差，不在缺少一致可积性时声称等于有限宽度方差的极限。相邻簇可使用联合过程结论另行处理，不把有限相邻样本默认为独立。 + +随机一行所见的白簇是长度偏倚，时钟变为 Gamma(2,1)，对应 t_4/sqrt(2c)，而不是 component-Palm 的 t_2/sqrt(c)。 + +### 两个参数尺度 + +簇内部标签的局部信息尺度是 sqrt(nu/w)，而黑屏障密度显著改变通常需要 O(1/w) 的参数变化。前者远小于后者。一个参数变化可以在几乎不改变宏观屏障过程的情况下，被巨大簇的内部信息分辨。 + +这不是免费超精度：一个簇约含 w/nu 个站点，读取或生成这些标签本身付出相应成本。也不把 q_tilde 称为完整 Palm 似然的全局最大值；完整似然还含 nu(q) 正规化。 + +## 8. 新猜想：第二个 bulk 模来自“未被查询的孔洞” + +令 A_v=eta_v+beta_v=1{删去 v 后，其邻域通向无穷}。精确地 + + eta_v=q A_v+p q xi_v. + +因此 + + J(q)=q^2 [sum_z Cov(A_0,A_z) - (a'(q))^2/c], + a(q)=theta(q)/q. (8.1) + +第二个模式不是另一种拓扑长度，而是查询集合本身的几何波动。p=1-q 很小时，最简单的未查询事件是 v 的八个 matching 邻点全部黑色，概率 p^8。这个事件不要求 v 本身黑色。 + +对仅由该八点图案定义的局部指示 D_v，重叠图案协方差和可以精确计算： + + sum_z Cov(D_0,D_z) + =p^8+4p^12+4p^13+12p^14+4p^15-25p^16. (8.2) + +(8.2) 是明确局部图案的恒等式，不是实际无限簇 J 的公式。更大的有限白岛会修正它。 + +据此提出可检验的低黑密度猜想： + + J(1-p) ~ p^8, p -> 0. (8.3) + +若成立，白色很稠密时二维 Gaussian bulk 波动的第二个方向虽真实存在，却比热评分方向小很多。只看低精度协方差矩阵，会把它误判为精确 rank-one。 + +可行推进：对有限白岛及其黑边界作低-p 聚类展开，先计算 theta、查询集合协方差和及 J 的首个非零系数；另一条路是在 wired 方框中精确比较 Var(查询数) 与它在评分方向上的投影。要保持源/边界正规化，不能把裸八点图案永久替代完整无限簇。 + +另两个保留猜想： + +- J(q)>0 对整个严格超临界 matching 区间成立。可尝试用分离白色围栏中的可切换有限白岛，构造评分方向之外的正方差下界。 +- 单簇 q_tilde 的均方误差可能有额外对数因子，约为 (nu/(c w)) log(1/nu)。本文只证明 (7.3)，没有证明该精确渐近；需要控制指数间隔的非常短端，而不仅是固定缩放区间。 + +## 9. 已执行的计算与直接写入边界 + +新增脚本 `scripts/giant_cluster_random_information.py` 使用未修改的 `tagged_winding_span.py`，固定 Git blob 为 `52f3611990ce2b1331d9e5296e0262f5e402e0d7`。它重新构造带奇偶列占据/不同边界站点标记的 direct activity，不使用只对均匀参数成立的旧小商。 + +w=2,3,4 的 colored 活动商分别为 5、9、23 态（两种邻接均如此）。每一步结算一行的占据与不同外边界，下方源边界和上方退出边界都保留。用有理 resolvent 的逐阶导数取得所有高度的评分矩，不作高度截断。 + +独立物理 lifted-BFS 枚举 10,054 个非空候选，逐系数核对其中 2,956 个完整绕行形状；六个非均匀列参数案例核对 nu_white(q_even,q_odd)=nu_black(1-q_even,1-q_odd)。精确 Hessian 核对 Cov(S)-E V 在两色之间一致。另有第 1 节的 wired 归一化与局部评分检查。12 项本地测试通过。 + +全体有理输出在 `results/geometric-consistency/giant-cluster-random-information.json`。此文件的有限结果不认证第 4–5 节的全尺寸论证；第 4–5 节明确依赖之前的 cage、锚点 Poisson 和交替拓扑输入。没有运行团队的大计算、Monte Carlo 或完整仓库测试套件。 + +## 来源与范围 + +- 内部输入：#764 评论 5659233515（交替与 gap），5659520483（bulk/score）；#771 `supercritical-white-slab-bulk-20260914.md` 与 `structural-consequences-20260914.md`。当前记录把使用的输入逐项写出，不把旧计算当新数据。 +- Antunovic–Veselic, arXiv:0707.1089v3, Theorems 2–3 与 Section 3：独立 site 亚临界体积尾、Russo/Harris/BK。相关 HTML 模型和定理段已读。https://arxiv.org/html/0707.1089v3 +- Mertens–Ziff, arXiv:1603.07289v2, Section II：matching 边界/绕行分类背景。不是本文随机信息或联合 CLT 的现成定理。https://arxiv.org/html/1603.07289v2 +- Kozubowski–Podgorski, Gaussian Mixture Representation of the Laplace Distribution Revisited, The American Statistician 74 (2020), 407–412, DOI 10.1080/00031305.2019.1630000。出版社摘要及书目信息已读；用于说明 Gaussian–exponential mixture 是已有概率结构，未声称读过付费全文。本文中的变换和 Student 推论均直接推导。 diff --git a/docs/manuscripts/geometric-balance/exponential-aspect-root-locking-20260914.md b/docs/manuscripts/geometric-balance/exponential-aspect-root-locking-20260914.md new file mode 100644 index 000000000..388455988 --- /dev/null +++ b/docs/manuscripts/geometric-balance/exponential-aspect-root-locking-20260914.md @@ -0,0 +1,192 @@ +# Exponential aspect should lock the finite balance root to the semi-infinite charge sequence + +2026-09-14. Conditional finite-size prediction combining the directional birth theory with the fixed-width charge-sector transfer. + +The conclusion is deliberately counterintuitive: + +> at fixed positive exponential aspect parameter `d`, the two individual homology births move to `d`-dependent macroscopic centres, but the finite matching/balance root should have the SAME leading `w^-4` width correction as the semi-infinite Jacobsen sequence, independent of `d`. + +## 1. Two different centres coexist + +For axial square tori + +\[ +C_w\times C_m, +\qquad +\frac{\log m}{w}\to d>0, \tag{1.1} +\] + +the individual NN rank births satisfy + +\[ +T_1\to a(d), +\qquad +T_2\to b(d), \tag{1.2} +\] + +with + +\[ +\kappa_4(a(d))=d, +\qquad +\kappa_8(1-b(d))=d. \tag{1.3} +\] + +The two centres can stay a fixed positive distance apart and move strongly with `d`. + +The matching root, however, is the zero of + +\[ +P_2(p)-P_0(p), \tag{1.4} +\] + +or equivalently of the charge fugacity. At fixed width and `m->infinity`, that root tends to + +\[ +p_w^{ch}:\quad +\lambda^0_{4,w}(p_w^{ch}) +=\lambda^0_{8,w}(1-p_w^{ch}). \tag{1.5} +\] + +This is the Jacobsen semi-infinite-cylinder eigenvalue sequence. + +## 2. Longitudinal finite-size correction + +For the periodic torus, the two charged topological sectors are transfer traces. At the semi-infinite crossing their leading eigenvalue contributions cancel. If + +\[ +\Delta_w +=\min\log\frac{\lambda_1}{|\lambda_2|} \tag{2.1} +\] + +is the relevant within-sector relaxation gap, then the finite-length root correction has the form + +\[ +|p^*_{w,m}-p_w^{ch}| +\lesssim +\frac{e^{-m\Delta_w}} +{m\Theta'_w(p_w^{ch})} \tag{2.2} +\] + +up to subleading trace amplitudes and possible extra cancellations. + +The transparent safe spectrum gives strong evidence for + +\[ +\Delta_w\sim\frac{2\pi}{w}, \tag{2.3} +\] + +while the thermal slope obeys + +\[ +\Theta'_w\asymp w^{-1/4}. \tag{2.4} +\] + +## 3. Exponential aspect annihilates longitudinal corrections + +Under (1.1), + +\[ +\frac mw +=\frac{e^{dw+o(w)}}w. \tag{3.1} +\] + +Equations (2.2)--(2.4) then give schematically + +\[ +|p^*_{w,m}-p_w^{ch}| +\le +\exp[-c e^{dw+o(w)}] \tag{3.2} +\] + +up to polynomial factors in `w`. + +In particular the finite-length error is smaller than every algebraic power of `w`: + +\[ +\boxed{ +p^*_{w,m}=p_w^{ch}+o(w^{-K}) +\quad\text{for every fixed }K.} \tag{3.3} +\] + +This is conditional on a uniform `Delta_w>=c/w` spectral-gap bound (or an equivalent periodic-transfer mixing statement). The current small-width spectrum strongly supports that hypothesis but does not prove it. + +## 4. Consequence: the root shift forgets d at leading order + +The semi-infinite square-site sequence has + +\[ +p_c-p_w^{ch}\asymp A_\square w^{-4}, \tag{4.1} +\] + +with the published Jacobsen widths and modern `p_c` reference giving a coefficient near `0.29` after the leading `1/w^2` extrapolation of `(p_c-p_w)w^4`. + +Combining with (3.3), + +\[ +\boxed{ +p_c-p^*_{w,m} +\sim A_\square w^{-4}} \tag{4.2} +\] + +for **every fixed `d>0`**, with the same leading amplitude as the semi-infinite cylinder. + +Thus the aspect parameter is absent from the leading balance-root correction even though it completely controls the separated birth centres in (1.2)--(1.3). + +## 5. A striking two-scale statement + +At fixed `d>0`, the finite system should simultaneously exhibit + +\[ +T_1=a(d)+O_P(w^{-1}), +\qquad +T_2=b(d)+O_P(w^{-1}), \tag{5.1} +\] + +at regular mass points (Gumbel windows), while + +\[ +p^*_{w,m}=p_c-A_\square w^{-4}+o(w^{-4}). \tag{5.2} +\] + +So the typical birth fluctuations are `1/w`, the birth-centre separation is `O(1)`, and the balance-root displacement is only `w^-4`. + +This is a particularly sharp version of **balance without concentration**: + +```text +birth locations : geometry-sensitive, d-dependent, O(1) apart +birth fluctuations : O(1/w) +balance root correction: d-independent to leading order, O(w^-4) +``` + +These are three different spectral/probability mechanisms. + +## 6. Why d=0 is singular for this statement + +The claim is for fixed positive `d`. At fixed macroscopic aspect ratio (`m=O(w)`), longitudinal trace corrections are only `e^{-O(1)}` and the root lies in the ordinary `w^-3/4` near-critical thermal window before the much smaller semi-infinite `w^-4` correction can be isolated. + +Therefore one must not take the `d>0` prediction continuously to `d=0` without resolving the simultaneous aspect crossover. + +The order/scale distinction is exactly the one encoded in `finite-aspect-charge-root-crossover-20260914.md`. + +## 7. Falsification test using existing-style archives + +No new large critical simulation is needed in principle. For several fixed `d>0` values and available widths, compute the exact/MC balance roots and form + +\[ +R_w(d)=w^4[p_c-p^*_{w,m(d)}]. \tag{7.1} +\] + +The prediction is + +\[ +\boxed{R_w(d_1)-R_w(d_2)\to0} \tag{7.2} +\] + +for any two positive fixed `d_1,d_2`, and all should approach the same semi-infinite coefficient. + +By contrast the lower/upper birth medians should visibly separate by different `d`-dependent amounts. Measuring both in the same data block would be an especially clean control. + +## 8. Claim boundary + +The fixed-`d` birth centres are author-level results already on the branch. The semi-infinite root sequence and its `w^-4` correction are established numerical/literature facts of the eigenvalue method. Equation (4.2) additionally assumes a periodic topological-sector relaxation gap of order at least `1/w`; the transparent safe spectrum points strongly to the specific constant `2 pi`, but a uniform proof is still missing. diff --git a/docs/manuscripts/geometric-balance/exponential-homology-class-selection-20260914.md b/docs/manuscripts/geometric-balance/exponential-homology-class-selection-20260914.md new file mode 100644 index 000000000..4f4381464 --- /dev/null +++ b/docs/manuscripts/geometric-balance/exponential-homology-class-selection-20260914.md @@ -0,0 +1,177 @@ +# Exponential aspect selects one projective homology class + +2026-09-14. Consequence of the varying-direction centre theorem and the full-period first-exit theta bound. + +In the genuine shortest-period exponential-aspect regime, the lower homology birth does **not** involve random competition among several projective directions. The period geometry itself forces one line to dominate with probability tending to one. + +## 1. Setup + +Let `Lambda_n`, `u_n`, `ell_n`, `h_n`, and `e_n` be as in `varying-direction-exponential-centres-20260914.md`: + +\[ +|u_n|=\ell_n\to\infty, +\qquad +u_n/\ell_n\to e, +\qquad +h_n=N_n/\ell_n, +\qquad +\frac{\log h_n}{\ell_n}\to d>0. \tag{1.1} +\] + +Let `[u_n]` denote the projective homology line generated by the shortest period. + +For a configuration with ambient rank one, let `L_n` be its unique projective homology line. + +## 2. Nonparallel periods have superlinear correlation-norm cost + +Complete `u_n` to a lattice basis `(u_n,v_n)`. Every nonparallel period has the form + +\[ +\lambda=a u_n+b v_n, +\qquad b\ne0. \tag{2.1} +\] + +The determinant identity gives + +\[ +|\lambda|\ge |b|h_n. \tag{2.2} +\] + +Fix a compact subcritical parameter interval `I`. Uniform norm equivalence on `I` gives `c_I>0` such that + +\[ +\tau_{G,p}(x)\ge c_I|x|, +\qquad p\in I. \tag{2.3} +\] + +Hence every nonparallel period satisfies + +\[ +\boxed{ +\tau_{G,p}(\lambda)\ge c_I h_n, +\qquad p\in I.} \tag{2.4} +\] + +Since `h_n=exp(d ell_n+o(ell_n))`, this is superlinear in `ell_n`. + +By contrast, + +\[ +\tau_{G,p}(u_n)=\ell_n\tau_{G,p}(e_n)=O_I(\ell_n). \tag{2.5} +\] + +## 3. Uniform upper bound for any nonparallel winding + +Use the first-exit torus bound with a fixed `epsilon>0`, uniformly for `p in I`. Restrict the period theta sum to nonparallel classes. Equation (2.4) gives + +\[ +P_p(\text{there is a nonparallel nonzero-homology cycle}) +\le +N_n C_{I,\epsilon}e^{-c_{I,\epsilon}h_n}. \tag{3.1} +\] + +Because + +\[ +\log N_n=O(\ell_n)+\log\ell_n +\ll h_n, \tag{3.2} +\] + +we obtain the superexponential-in-`ell_n` estimate + +\[ +\boxed{ +\sup_{p\in I} +P_p(\text{nonparallel winding}) +\le e^{-c_I' h_n}} \tag{3.3} +\] + +for all large `n`, after changing constants. + +Any winding whose lift class is a nonzero multiple of `u_n` has projective line `[u_n]`. Therefore, uniformly on `I`, + +\[ +\boxed{ +P_p(r=1,L_n\ne[u_n])\le e^{-c_I'h_n}.} \tag{3.4} +\] + +Likewise a rank-two configuration requires a nonparallel homology direction and hence is bounded by the same type of estimate throughout a compact interval strictly below the upper birth. + +## 4. The first-birth slope is deterministic asymptotically + +For NN let `a_e(d)` be the lower birth centre. Choose a compact interval + +\[ +I=[a_e(d)-\eta,a_e(d)+\eta] +\Subset(0,p_c(G4)) \tag{4.1} +\] + +with fixed small `eta>0`. + +The varying-direction theorem gives + +\[ +P(T_{1,n}\in I)\to1. \tag{4.2} +\] + +On the event that `T_1 in I`, the first rank-one homology line can differ from `[u_n]` only if a nonparallel winding exists at some parameter in `I`. By monotonicity it is enough to check the upper endpoint of `I`, so (3.3) gives + +\[ +P(L_{T_1}\ne[u_n],\ T_1\in I) +\le e^{-c h_n}. \tag{4.3} +\] + +Therefore + +\[ +\boxed{ +P(L_{T_1}=[u_n])\to1.} \tag{4.4} +\] + +If `u_n/ell_n -> e`, the embedded projective slope converges to the deterministic limiting direction `[e]`. + +## 5. The whole rank-one plateau inherits the same line + +`persistent-slope-marked-birth-20260914.md` proves that once the first rank is born, its one-dimensional homology image is frozen until the second birth. + +Hence with probability tending to one, + +\[ +\boxed{ +A_4(p)=\operatorname{span}(u_n) +\quad\text{for every }p\in(T_{1,n},T_{2,n}).} \tag{5.1} +\] + +In words: **the entire rank-one plateau carries the shortest-period projective line.** + +This is stronger than concentration of a directional wrapping marginal at one fixed parameter. + +## 6. Consequence for projective slope harmonics + +Let `Z_s^{(tau_n)}([u_n])` be any even embedded projective harmonic. On a compact parameter interval in the interior of the rank-one plateau, + +\[ +E[Z_s(L_n)\mid r=1] +=Z_s([u_n])+o(1). \tag{6.1} +\] + +Thus the slope harmonic in an exponentially elongated torus is primarily a **geometry marker**, not an additional fluctuating field. + +The nontrivial modular/critical continuum slope distribution from `projective-slope-modular-covariance-20260914.md` belongs to balanced two-dimensional shapes where several primitive homology sectors remain visible, not to the present exponential-aspect limit. + +## 7. Where multi-direction hard-core competition is actually relevant + +The projective Poisson hard-core closure on this branch remains useful, but only when several nonparallel period classes have comparable correlation-norm cost and comparable opportunity entropy. + +That cannot happen under (1.1): two independent `O(ell_n)` periods would force `N_n=O(ell_n^2)` and destroy exponential transverse height. + +Therefore: + +- #765 exponential shortest-period geometry -> deterministic slope selection; +- near-isotropic/modular or deliberately degenerate geometries -> possible projective hard-core direction competition. + +This separation prevents over-parameterizing the exponential birth window with a direction-mixture model that topology/geometry have already collapsed. + +## 8. Claim boundary + +The result uses the author-level first-exit theta upper bound and varying-direction centre theorem on PR #771. The superexponential estimate is uniform only on compact subcritical parameter intervals; no claim is made through the critical point or in geometries without exponential transverse separation. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/finite-aspect-charge-root-crossover-20260914.md b/docs/manuscripts/geometric-balance/finite-aspect-charge-root-crossover-20260914.md new file mode 100644 index 000000000..fe5261e7c --- /dev/null +++ b/docs/manuscripts/geometric-balance/finite-aspect-charge-root-crossover-20260914.md @@ -0,0 +1,218 @@ +# From the thermal w^-3/4 window to the semi-infinite w^-4 charge root + +2026-09-14. Corrected two-parameter scaling note for #767. + +A first draft treated the two charged sectors as generic boundary-to-boundary Perron amplitudes and obtained an `O(1/m)` root displacement. That is a valid generic transfer situation but is **not the right default for the periodic torus matching observable**. At `q=1`, the graph-polynomial/topological sectors are periodic transfer traces. A simple Perron eigenvalue appears in a trace with coefficient one, so at the semi-infinite eigenvalue crossing the two leading trace terms cancel exactly. Finite-`m` convergence is then controlled by subleading spectral gaps and is exponential in `m` at fixed `w`, as already observed in Jacobsen's finite-`m` calculations. + +This correction materially changes the aspect scale needed to see the intrinsic `w^-4` pseudo-critical shift. + +## 1. Semi-infinite charge crossing + +For circumference `w`, + +\[ +\Theta_w(p)=I^0_{4,w}(p)-I^0_{8,w}(1-p), \tag{1.1} +\] + +and + +\[ +p_w^{ch}:\quad\Theta_w(p_w^{ch})=0. \tag{1.2} +\] + +The transparent site transfer reproduces the Jacobsen `w x infinity` root sequence. Critical scaling gives + +\[ +\Theta'_w(p_w^{ch})\asymp w^{-1/4}, \tag{1.3} +\] + +while + +\[ +p_c-p_w^{ch}\asymp w^{-4}. \tag{1.4} +\] + +## 2. Why a generic amplitude formula would give 1/m + +For two arbitrary open-boundary transfer observables one can have + +\[ +Z_i(m)=A_i\lambda_i^m[1+o(1)]. \tag{2.1} +\] + +At `lambda_1=lambda_2`, unequal `A_i` produce a root displacement + +\[ +\delta p\sim\frac{\log(A_2/A_1)}{m\Theta'_w}. \tag{2.2} +\] + +This remains a useful warning for nonperiodic boundary conditions or modified observables. It was the source of the discarded `m~w^(17/4)` estimate in the first draft. + +## 3. Periodic torus: trace coefficients cancel + +For the actual periodic topological matching observable, the graph-polynomial/eigenvalue formulation decomposes the periodic transfer into open and closed topological blocks. At `q=1`, schematically + +\[ +Z_{2D}=\operatorname{Tr}(T_{open}^m),\qquad +Z_{0D}=\operatorname{Tr}(T_{closed}^m), \tag{3.1} +\] + +up to the common local normalization converting Boltzmann weights to Bernoulli probabilities. + +Let the dominant eigenvalues be `lambda_o,lambda_c`, and write the next spectral moduli as `mu_o,mu_c`. For simple Perron roots, + +\[ +\operatorname{Tr}(T_i^m) +=\lambda_i^m\left[1+O\left((|\mu_i|/\lambda_i)^m\right)\right].\tag{3.2} +\] + +The leading coefficient is **one**, not an arbitrary boundary overlap. + +At `p=p_w^{ch}`, + +\[ +\lambda_o=\lambda_c=\lambda_w, \tag{3.3} +\] + +so the leading terms in `Z_{2D}-Z_{0D}` cancel exactly. Define + +\[ +\Delta_w +=\min_i\log\frac{\lambda_w}{|\mu_i|}>0. \tag{3.4} +\] + +Then the finite-`m` mismatch at the semi-infinite root is + +\[ +Z_{2D}-Z_{0D} +=\lambda_w^m O(e^{-m\Delta_w}). \tag{3.5} +\] + +Differentiating the leading eigenvalue difference contributes the factor + +\[ +m\Theta'_w(p_w^{ch}). \tag{3.6} +\] + +Therefore the natural periodic-root correction is + +\[ +\boxed{ +|p^*_{w,m}-p_w^{ch}| +=O\left(\frac{e^{-m\Delta_w}} +{m\Theta'_w(p_w^{ch})}\right),} \tag{3.7} +\] + +provided the subleading eigenvalues remain separated and no equal-modulus oscillatory degeneracy changes the prefactor. This is exponential in `m` for every fixed `w`. + +The exact site-rank transfer should be audited directly if (3.7) is promoted as a theorem; the key correction here is that periodic trace structure removes the generic `O(1/m)` amplitude term. + +## 4. Critical scaling of the subleading gap + +At criticality a transfer spectral gap within a fixed CFT sector has the form + +\[ +\Delta_w\sim\frac{g}{w}, \tag{4.1} +\] + +where `g=2 pi Delta x` times the lattice velocity/geometry factor for the relevant next state. Put + +\[ +\rho=m/w. \tag{4.2} +\] + +Combining (1.3), (3.7), and (4.1), + +\[ +\boxed{ +|p^*_{w,m}-p_w^{ch}| +\lesssim +\frac{e^{-g\rho}}{\rho\,w^{3/4}}} \tag{4.3} +\] + +at the level of critical finite-size scaling. + +This is the corrected aspect crossover. + +## 5. Fixed aspect recovers the ordinary thermal window + +If `rho` is fixed, + +\[ +|p^*_{w,m}-p_w^{ch}| +=O(w^{-3/4}) \tag{5.1} +\] + +with an aspect-dependent coefficient `e^{-g rho}/rho`. + +Since the intrinsic semi-infinite displacement is only `w^-4`, the finite-aspect root is governed at leading order by the standard near-critical thermal scale. Thus a square or fixed-aspect torus should not be expected to reveal the Jacobsen `w^-4` shift directly. + +## 6. Growing aspect: only logarithmic growth may be needed + +To resolve the semi-infinite width shift before finite-length corrections dominate, require + +\[ +\frac{e^{-g\rho}}{\rho w^{3/4}}\ll w^{-4}, \tag{6.1} +\] + +or + +\[ +\boxed{ +\frac{e^{-g\rho}}\rho\ll w^{-13/4}.} \tag{6.2} +\] + +Thus a logarithmically growing aspect ratio + +\[ +\rho=C\log w \tag{6.3} +\] + +is already sufficient when + +\[ +gC>13/4 \tag{6.4} +\] + +(up to logarithmic factors and the actual sector gap constant). + +This replaces the discarded polynomial requirement `rho >> w^(13/4)`. The semi-infinite regime is reached much sooner because periodic traces cancel their leading amplitudes. + +## 7. Exponential aspect + +When + +\[ +\log m/w\to d>0, \tag{7.1} +\] + +`rho` itself is exponentially large, so finite-length transfer corrections to the charge root are beyond all algebraic orders in `w`. The fixed-width charge crossing is then effectively exact long before the lower/upper winding-component birth windows are considered. + +This cleanly coexists with the main geometric phenomenon: the matching root can track a very sharp charge-sector eigenvalue crossing while the two individual rank births converge to macroscopically separated parameters. + +## 8. Interface to #767 + +The corrected picture gives three compatible resolutions of the same charge field. + +1. **Fixed aspect `rho=O(1)`:** root motion occurs on the standard `w^-3/4` near-critical scale. +2. **Growing aspect:** finite-length corrections are additionally suppressed by the transfer factor `e^{-g rho}`. +3. **Semi-infinite / exponential aspect:** the intrinsic charge root is `p_w^{ch}=p_c+O(w^-4)` and finite-length corrections are negligible. + +The crossover is controlled by a **spectral gap in aspect ratio**, not by an arbitrary Perron amplitude ratio. + +## 9. A new concrete target + +The next useful computation is not a larger threshold table. It is the subleading spectrum of the transparent safe/topological transfer at `p_w^{ch}`: + +\[ +g_w=w\Delta_w. \tag{9.1} +\] + +If `g_w` approaches a nonzero limit, then (4.3) becomes quantitatively testable. Comparing that limit with CFT candidate gaps will identify which descendant/excitation controls approach to the semi-infinite charge criterion. + +## 10. Claim boundary + +- The generic amplitude formula (2.2) is correct for arbitrary boundary overlaps but is **not** used as the periodic-torus conclusion. +- Exponential fixed-`w` convergence follows from periodic trace blocks with simple separated Perron eigenvalues; Jacobsen's eigenvalue formulation supplies the corresponding graph-polynomial structure and finite-`m` controls. +- The simultaneous scaling `Delta_w~g/w` and `Theta'_w~w^-1/4` is CFT/critical-scaling input, not yet a rigorous square-site theorem. +- The earlier `m~w^(17/4)` crossover claim is withdrawn for the periodic rank observable. diff --git a/docs/manuscripts/geometric-balance/finite-reflection-dominance-20260914.md b/docs/manuscripts/geometric-balance/finite-reflection-dominance-20260914.md new file mode 100644 index 000000000..fea3e720c --- /dev/null +++ b/docs/manuscripts/geometric-balance/finite-reflection-dominance-20260914.md @@ -0,0 +1,236 @@ +# Finite reflection dominance from graph inclusion plus persistent Alexander duality + +2026-09-14. Exact finite theorem for every honest torus. No sharpness, RSW, mass, OZ or asymptotic limit is used. + +The result says that the NN two-birth law is intrinsically shifted to the right of its reflection about `1/2`. The strict large-`d` centre theorem on this branch is a quantitative asymptotic strengthening, not the source of the sign. + +## 1. Rank monotonicity under matching enhancement + +For the **same occupied vertex set** `omega`, the matching graph contains all NN edges and adds diagonals. Therefore the ambient homology image can only grow: + +\[ +\boxed{r_8(\omega)\ge r_4(\omega).} \tag{1.1} +\] + +Under the same continuous labels `U_v`, let + +\[ +T_1^G\le T_2^G +\] + +be the first parameters at which the ambient rank in graph `G` reaches one and two. Equation (1.1) gives pathwise + +\[ +\boxed{T_j^8(U)\le T_j^4(U),\qquad j=1,2.} \tag{1.2} +\] + +## 2. Persistent duality turns graph inclusion into reflection dominance + +The persistent digital-Alexander identity proved in `structural-consequences-20260914.md` gives, for reflected labels `V=1-U`, + +\[ +T_1^8(V)=1-T_2^4(U), +\qquad +T_2^8(V)=1-T_1^4(U). \tag{2.1} +\] + +Since `V` has the same iid uniform law as `U`, + +\[ +T_1^8\ \overset d=\ 1-T_2^4, +\qquad +T_2^8\ \overset d=\ 1-T_1^4. \tag{2.2} +\] + +Combine (1.2) and (2.2). In stochastic order, + +\[ +\boxed{1-T_2^4\preceq_{st}T_1^4,} \tag{2.3} +\] + +\[ +\boxed{1-T_1^4\preceq_{st}T_2^4.} \tag{2.4} +\] + +These are exact finite inequalities for the NN birth pair. + +## 3. Endpoint-sector form + +Write + +\[ +f_1(p)=P(T_1\le p)=1-P_0(p), +\qquad +f_2(p)=P(T_2\le p)=P_2(p). \tag{3.1} +\] + +Equation (2.3) is equivalent to + +\[ +f_1(p)+f_2(1-p)\le1. \tag{3.2} +\] + +Substituting (3.1) gives the particularly simple topological inequality + +\[ +\boxed{P_2(1-p)\le P_0(p),\qquad0\le p\le1.} \tag{3.3} +\] + +Reflecting `p` gives also + +\[ +P_2(p)\le P_0(1-p). \tag{3.4} +\] + +There is an even shorter proof of (3.3): digital Alexander gives + +\[ +P_2^4(1-p)=P_0^8(p), \tag{3.5} +\] + +while graph inclusion gives `P_0^8(p)<=P_0^4(p)`. + +## 4. Matching-function reflection inequality + +Recall + +\[ +M(p)=P_2(p)-P_0(p). \tag{4.1} +\] + +Adding (3.3) and (3.4) yields + +\[ +\boxed{M(p)+M(1-p)\le0.} \tag{4.2} +\] + +In particular + +\[ +\boxed{M(1/2)\le0.} \tag{4.3} +\] + +Since `M` is strictly increasing on every honest nontrivial torus, its unique zero obeys + +\[ +\boxed{p_\Lambda\ge1/2.} \tag{4.4} +\] + +Whenever the matching enhancement is strict at the rank level with positive probability—for example on the ordinary `L x L` square torus with `L>2`, where occupying the diagonal orbit gives a matching essential cycle but no NN edge—the inequalities are strict in the interior and + +\[ +p_\Lambda>1/2. \tag{4.5} +\] + +For arbitrary unusual period quotients, (4.4) is the unconditional statement; strictness only needs one configuration of positive product probability with `r_8>r_4`. + +## 5. Birth-mixture reflection dominance + +The fair birth mixture has CDF + +\[ +F(p)=\frac12[f_1(p)+f_2(p)]=\frac12[1+M(p)]. \tag{5.1} +\] + +Equation (4.2) is exactly + +\[ +\boxed{F(p)+F(1-p)\le1.} \tag{5.2} +\] + +For continuous birth labels, the reflected random variable `1-T` has CDF + +\[ +F_{1-T}(p)=1-F(1-p). \tag{5.3} +\] + +Thus + +\[ +\boxed{T\succeq_{st}1-T.} \tag{5.4} +\] + +The full finite mixture law, not merely its median, is reflection-shifted toward the high-`p` side. + +## 6. Quantile and moment consequences + +Let `Q` be the inverse CDF of the fair mixture. From (5.2), for every `u in (0,1)`, + +\[ +\boxed{Q(u)+Q(1-u)\ge1.} \tag{6.1} +\] + +In particular + +\[ +Q(1/2)\ge1/2, \tag{6.2} +\] + +\[ +Q(1/4)+Q(3/4)\ge1. \tag{6.3} +\] + +Stochastic dominance also gives + +\[ +\boxed{E T\ge1/2,} \tag{6.4} +\] + +and, in the dual-odd coordinate + +\[ +C=\frac{T_1+T_2-1}{2}, +\] + +\[ +\boxed{E C\ge0.} \tag{6.5} +\] + +Using the exact area identity + +\[ +E C=-\frac12\int_0^1M(p)\,dp, \tag{6.6} +\] + +we obtain the finite integral sign + +\[ +\boxed{\int_0^1M(p)\,dp\le0.} \tag{6.7} +\] + +Equivalently, in permutation birth indices, + +\[ +\boxed{E[K_1+K_2]\ge N+1.} \tag{6.8} +\] + +This last inequality can also be read directly from +`E K_1^8=N+1-E K_2^4` and `K_1^8<=K_1^4` in stochastic order. + +## 7. Relationship to the strict mass-gap theorem + +The finite reflection theorem gives the **sign** of the complement-odd centre for every size but not a uniform positive limiting gap. + +In the fixed-positive exponential-aspect regime, `matching-enhancement-mass-gap-20260914.md` proves the stronger deterministic-centre statement + +\[ +a(d)+b(d)>1. \tag{7.1} +\] + +The dilute directional calculation further gives an explicit large-`d` asymptotic for that positive displacement. These are quantitative refinements of the exact finite reflection dominance, not independent sign coincidences. + +## 8. A useful finite-data control + +Any exact/Monte-Carlo rank-birth archive on an honest torus should satisfy, within its declared statistical errors, + +\[ +Q(u)+Q(1-u)\ge1 \tag{8.1} +\] + +for every symmetric quantile pair, and + +\[ +E(K_1+K_2)\ge N+1. \tag{8.2} +\] + +Violations indicate a rank dictionary, matching-complement, weighting, or quantile-reconstruction error before they indicate new physics. diff --git a/docs/manuscripts/geometric-balance/first-exit-renormalization-hierarchy-20260914.md b/docs/manuscripts/geometric-balance/first-exit-renormalization-hierarchy-20260914.md new file mode 100644 index 000000000..6d97924ea --- /dev/null +++ b/docs/manuscripts/geometric-balance/first-exit-renormalization-hierarchy-20260914.md @@ -0,0 +1,246 @@ +# First-exit certificates as a convergent block-renormalization hierarchy + +2026-09-14. This note identifies the one-site first-exit certificate with the elementary all-walk exponential-tilt bound, and places the larger finite-box calculations of #761/#766 in one rigorous hierarchy converging to the true Wulff/exponential-moment domain. + +## 1. The smallest block reproduces the random-walk Green-function bound + +Take + +\[ +S=\{0\}. +\] + +For every external neighbour `v`, the first-exit event in the definition of `b_S(v;p)` is simply “the origin is occupied”, because the origin is the only internal neighbour available and the external site `v` itself is excluded from the event. + +Thus + +\[ +\boxed{b_{\{0\}}(v;p)=p.} \tag{1.1} +\] + +The one-site first-exit polynomial is therefore + +\[ +B_{\{0\}}(t;p) +=p\sum_{v\in S_G}e^{t\cdot v}, \tag{1.2} +\] + +where `S_G` is the graph step set. + +The theorem in `vector-first-exit-domain-20260914.md` says + +\[ +B_{\{0\}}(t;p)<1 +\quad\Longrightarrow\quad +\sum_xP(0\leftrightarrow x)e^{t\cdot x}<\infty. \tag{1.3} +\] + +But (1.2) is exactly the denominator condition in the all-walk Green-function/exponential-tilt bound used in the dilute directional analysis. The two arguments are the same certificate written in path-skeleton and random-walk language. + +## 2. Explicit NN and matching one-site domains + +### NN + +For steps `(+/-e_1,+/-e_2)`, + +\[ +\boxed{ +B^{(4)}_{\{0\}}(t_x,t_y) +=2p(\cosh t_x+\cosh t_y).} \tag{2.1} +\] + +Hence the certified domain is + +\[ +\mathcal C_0^{(4)}(p) +=\{t:2p(\cosh t_x+\cosh t_y)<1\}. \tag{2.2} +\] + +On the horizontal ray `(t,0)`, the boundary solves + +\[ +2p(\cosh t+1)=1, +\] + +so + +\[ +\boxed{ +t=\operatorname{arcosh}\left(\frac{1/p-2}{2}\right),} \tag{2.3} +\] + +which is exactly the elementary lower bound on `kappa_4(p)` in `dilute-directional-mass-centres-20260914.md`. + +### Matching / king graph + +The eight-neighbour step generating function factorizes: + +\[ +\sum_{v\in S_8}e^{t\cdot v} +=(1+2\cosh t_x)(1+2\cosh t_y)-1. \tag{2.4} +\] + +Therefore + +\[ +\boxed{ +B^{(8)}_{\{0\}}(t_x,t_y) +=p[(1+2\cosh t_x)(1+2\cosh t_y)-1].} \tag{2.5} +\] + +The horizontal intercept solves + +\[ +p(6\cosh t+2)=1, +\] + +or + +\[ +\boxed{ +t=\operatorname{arcosh}\left(\frac{1/p-2}{6}\right),} \tag{2.6} +\] + +again exactly the dilute mass lower bound already derived. + +Thus #766's vector certificate contains those earlier scalar estimates as its `S={0}` base case. + +## 3. The dilute geodesic entropy is the support function of the one-site body + +Let + +\[ +h_{\mathcal C_0}(u)=\sup_{t\in\mathcal C_0}t\cdot u. \tag{3.1} +\] + +The first-exit theorem gives + +\[ +\tau_p(u)\ge h_{\mathcal C_0(p)}(u). \tag{3.2} +\] + +As `p->0`, the support function of the NN body (2.2) satisfies + +\[ +h_{\mathcal C_0^{(4)}}(a,b) +=(|a|+|b|)\log(1/p) +-(|a|+|b|)H_2\left(\frac{|a|}{|a|+|b|}\right)+o(1). \tag{3.3} +\] + +The matching body gives + +\[ +h_{\mathcal C_0^{(8)}}(a,b) +=M\log(1/p)-Mh_3(m/M)+o(1), \tag{3.4} +\] + +with `M=max(|a|,|b|)` and `m=min(|a|,|b|)`. + +The adaptive-path lower-probability constructions in `dilute-directional-geodesic-entropy-20260914.md` prove matching upper bounds on `tau`. Therefore the true Wulff support function is asymptotic to the **smallest-block first-exit certificate** in the dilute limit. + +This explains why graph distance and shortest-path entropy are the first two dilute terms: before larger blocks matter, the exponential-moment domain is controlled by the bare step generating function. + +## 4. Larger S resums local connectivity before the next exit + +For a nontrivial finite `S`, the coefficient + +\[ +b_S(v;p) +\] + +already sums every way the origin can connect inside `S` to an internal neighbour of `v`. Thus moving from `S={0}` to a larger block replaces a single bare step by an **exact local connected passage**. + +The first-exit skeleton then concatenates those passages using BK. In renormalization language: + +```text +one-site certificate = bare walk kernel, +finite S certificate = exact local connected block kernel, +large S = increasingly complete local resummation, +S -> infinity = true exponential-moment/Wulff domain on compact interiors. +``` + +No equality between successive block kernels is asserted; different shapes can outperform each other in different directions. + +## 5. A monotone accumulated certificate even though individual boxes need not nest + +If `S_1 subset S_2`, it is **not** necessary or generally safe to assume + +\[ +\mathcal C_{S_1}\subseteq\mathcal C_{S_2}. \tag{5.1} +\] + +The finite first-exit polynomials use different decompositions and may cross. + +The true domain is convex, however, and every individual `C_S` is certified. Therefore define the accumulated body + +\[ +\boxed{ +K_R^{cert} +=\operatorname{conv}\left( +\bigcup_{S\in\mathfrak S_R}\mathcal C_S +\right),} \tag{5.2} +\] + +where `mathfrak S_R` is any increasing family of finite block shapes/budgets. Then + +\[ +K_R^{cert}\subseteq K_{R'}^{cert}\subseteq\mathcal D_p +\quad(R0`. + +Together these turn finite first-exit boxes from an ad hoc numerical trick into a convergent block-renormalization scheme with a solved base case and a clear shape-selection principle. diff --git a/docs/manuscripts/geometric-balance/first-exit-torus-winding-upper-20260914.md b/docs/manuscripts/geometric-balance/first-exit-torus-winding-upper-20260914.md new file mode 100644 index 000000000..e31eb3577 --- /dev/null +++ b/docs/manuscripts/geometric-balance/first-exit-torus-winding-upper-20260914.md @@ -0,0 +1,297 @@ +# First-exit Wulff certificates give a full-period torus winding upper bound + +2026-09-14. This note is the rigorous bridge from `vector-first-exit-domain-20260914.md` to the directional/exponential-torus questions in #765. It replaces a tempting but unsafe direct comparison between quotient paths and full-plane two-point events. + +The result is general: on any sufficiently large honest integer-period torus, positive homology is bounded by a theta sum over **complete period vectors**, with their full planar correlation-norm cost. + +## 1. Setup + +Work with a translation-invariant finite-range independent site model `G` on `Z^2` at a fixed subcritical parameter `p`. + +Let + +\[ +\tau_p(x) +\] + +be the homogeneous inverse-correlation norm and let + +\[ +K_p=\{t:t\cdot x\le\tau_p(x)\ \forall x\} \tag{1.1} +\] + +be its polar/exponential-moment body. By the first-exit theorem on this branch, + +\[ +\mathcal D_p=K_p^\circ \tag{1.2} +\] + +and every compact `T subset K_p^circ` is contained in one sufficiently large finite first-exit certificate + +\[ +\mathcal C_S=\{t:B_S(t)<1\}. \tag{1.3} +\] + +Let `Lambda` be a rank-two period lattice and + +\[ +T_\Lambda=\mathbb Z^2/\Lambda, +\qquad +N=[\mathbb Z^2:\Lambda]. \tag{1.4} +\] + +Assume the shortest period is large enough that every translate of the chosen finite set `S`, enlarged by the finite interaction range, injects into the quotient. + +Write + +\[ +f_\Lambda(p)=P_p(r>0). \tag{1.5} +\] + +## 2. Every positive-rank configuration contains a simple nonzero-homology cycle + +If `r>0`, choose an occupied closed walk with nonzero ambient homology and minimum edge length. It cannot repeat a torus vertex other than its start/end. A repeated vertex would split the walk into two shorter closed walks whose homology classes sum to the original nonzero class; at least one shorter piece would still have nonzero homology. + +Hence a minimum witness is a graph-simple occupied cycle. + +Lift it from a starting vertex `z` to a self-avoiding path in `Z^2` from a chosen lift of `z` to + +\[ +z+\lambda, +\qquad +\lambda\in\Lambda\setminus\{0\}. \tag{2.1} +\] + +The lifted endpoint is a different plane vertex but the same torus vertex. + +## 3. Local first-exit skeletons remain BK-disjoint on the quotient + +Apply the first-exit decomposition of `vector-first-exit-domain-20260914.md` to the lifted simple cycle path using translates of `S`. + +Starting from `y_0=z`, obtain exit displacements + +\[ +v_1,\ldots,v_k\in\partial_{ext}S \tag{3.1} +\] + +and a final residual `s in S` such that + +\[ +\lambda=v_1+\cdots+v_k+s. \tag{3.2} +\] + +The `i`th first-exit witness uses only the actual cycle vertices from `y_{i-1}` through the internal neighbour immediately preceding `y_i`; the exit site itself is excluded and becomes the starting site of the next witness. + +Because the torus cycle is vertex-simple, these witness sets are disjoint **as quotient site variables**, not merely as plane lifts. This is the point that fails for a generic pair of quotient connection events but holds for the canonical simple-cycle skeleton. + +Each witness is local in one translate of the fixed finite set `S`; injectivity of that translate means its probability is exactly the planar coefficient + +\[ +b_S(v_i;p). \tag{3.3} +\] + +Repeated site BK on the quotient therefore gives, for any fixed skeleton, + +\[ +P(\text{that skeleton is witnessed}) +\le\prod_i b_S(v_i;p). \tag{3.4} +\] + +No event is imposed on the final residual, so the fact that the terminal torus vertex equals the initial one creates no shared witness variable. + +## 4. Pointwise renewal bound for a homology vector + +Define the first-exit renewal majorant + +\[ +R_S(x)= +\sum_{k\ge0} +\sum_{v_1,\ldots,v_k} +\sum_{s\in S} +1_{\{x=v_1+\cdots+v_k+s\}} +\prod_i b_S(v_i;p). \tag{4.1} +\] + +Section 3 proves that, for a fixed start vertex, the probability that a canonical simple winding witness has lift class `lambda` is at most + +\[ +R_S(\lambda). \tag{4.2} +\] + +For any `t in C_S`, the same generating-function calculation as in the planar first-exit theorem gives + +\[ +\sum_xR_S(x)e^{t\cdot x} +=\frac{\sum_{s\in S}e^{t\cdot s}}{1-B_S(t)}. \tag{4.3} +\] + +Hence pointwise + +\[ +R_S(\lambda) +\le C_S(t)e^{-t\cdot\lambda}, +\qquad +C_S(t)=\frac{\sum_{s\in S}e^{t\cdot s}}{1-B_S(t)}. \tag{4.4} +\] + +Let `T` be any compact subset of `C_S`. The denominator stays uniformly away from zero, so + +\[ +C_T:=\sup_{t\in T}C_S(t)<\infty. \tag{4.5} +\] + +Choosing the best `t in T` for each `lambda`, + +\[ +\boxed{ +R_S(\lambda) +\le C_T e^{-h_T(\lambda)}, +\qquad +h_T(\lambda)=\sup_{t\in T}t\cdot\lambda.} \tag{4.6} +\] + +## 5. General torus winding upper bound + +There are `N` possible starting torus vertices. Union over the nonzero lift class of the canonical simple cycle gives + +\[ +\boxed{ +f_\Lambda(p) +\le N C_T +\sum_{\lambda\in\Lambda\setminus\{0\}} + e^{-h_T(\lambda)}.} \tag{5.1} +\] + +This is already a rigorous finite certificate for any computed first-exit body `T`. + +Now fix `epsilon in (0,1)` and take + +\[ +T_\epsilon=(1-\epsilon)K_p. \tag{5.2} +\] + +Because `K_p` is compact with the origin in its interior, + +\[ +T_\epsilon\Subset K_p^\circ=\mathcal D_p. \tag{5.3} +\] + +The large-box exhaustion theorem supplies a finite `S_epsilon` with + +\[ +T_\epsilon\subset C_{S_\epsilon}. \tag{5.4} +\] + +Its support function is + +\[ +h_{T_\epsilon}(x) +=(1-\epsilon)h_{K_p}(x) +=(1-\epsilon)\tau_p(x). \tag{5.5} +\] + +Therefore, for every sufficiently large honest torus, + +\[ +\boxed{ +f_\Lambda(p) +\le N C_{p,\epsilon} +\sum_{\lambda\in\Lambda\setminus\{0\}} +\exp[-(1-\epsilon)\tau_p(\lambda)].} \tag{5.6} +\] + +This is the desired full-period upper bound. No half-period arm and no rotated microscopic interaction appear. + +## 6. Fixed-direction exponential torus + +Take a primitive integer direction `u` and a Bezout complement `v` with `det(u,v)=1`, and + +\[ +\Lambda_{n,m}=\langle nu,mv\rangle, +\qquad +\frac{\log m}{n}\to d. \tag{6.1} +\] + +Write + +\[ +\xi_u(p)=\tau_p(u). \tag{6.2} +\] + +For parallel periods, + +\[ +\tau_p(\alpha nu)=|\alpha|n\xi_u(p). \tag{6.3} +\] + +For periods with nonzero transverse coefficient, norm equivalence and + +\[ +|\det(u,\alpha nu+\beta mv)|=|\beta|m \tag{6.4} +\] + +give a cost at least `c_p |beta|m/|u|`. Standard one-dimensional summation in the parallel coordinate then yields + +\[ +\sum_{\lambda\in\Lambda_{n,m}\setminus0} + e^{-(1-\epsilon)\tau_p(\lambda)} +\le +C_{p,\epsilon}e^{-(1-\epsilon)n\xi_u(p)} ++e^{-c_{p,\epsilon}m}. \tag{6.5} +\] + +Since `N=nm`, (5.6) gives + +\[ +\limsup_{n\to\infty}\frac1n\log f_{n,m}(p) +\le +-\max\{(1-\epsilon)\xi_u(p)-d,0\}. \tag{6.6} +\] + +Letting `epsilon downarrow 0`, + +\[ +\boxed{ +\limsup\frac1n\log f_{n,m}(p) +\le-\max\{\xi_u(p)-d,0\}.} \tag{6.7} +\] + +Together with the finite-seed seam-closing lower bound already used axially, this identifies the exact fixed-direction winding exponent. + +## 7. General homological free-energy interpretation + +Equation (5.6) gives a rigorous version of the previously conjectural energy-versus-opportunity competition. + +Define the period theta activity + +\[ +\Theta_{\Lambda,p}^{(\epsilon)} +=\sum_{\lambda\ne0} + e^{-(1-\epsilon)\tau_p(\lambda)}. \tag{7.1} +\] + +Then + +\[ +\boxed{f_\Lambda(p)\le C_{p,\epsilon}N\Theta_{\Lambda,p}^{(\epsilon)}.}\tag{7.2} +\] + +If one period family dominates the theta sum and has `M_Lambda` effective translates/opportunities, the transition is governed at exponential scale by + +\[ +\tau_p(\lambda_*)-\log M_\Lambda. \tag{7.3} +\] + +The exact entropy factor depends on the geometry/packing of that family; equation (7.2) supplies the universal upper side without guessing it. + +For exponentially separated fixed-direction tori, `M_Lambda` is `m` up to subexponential factors and (7.3) becomes `xi_u(p)-log m/n`. + +## 8. Why the direct global two-arc comparison was not used + +A quotient simple cycle does split into two internally disjoint arcs, but the union over possible quotient arcs is not automatically distributed like two independent full-plane connection events: different plane lifts of the same quotient site share one Bernoulli variable. + +The local first-exit proof avoids that issue completely. Every factor is supported in one fixed injective translate, and BK is applied to actual disjoint quotient witness variables. This is the safe route from planar directional mass to a finite torus upper bound. + +## 9. Claim boundary + +Sections 2--5 are finite product-measure arguments once the first-exit certificate and injectivity are supplied. The passage to `(1-epsilon)K_p` uses the author-level domain equality/exhaustion theorem in `vector-first-exit-domain-20260914.md`, which in turn uses the standard uniform directional exponential estimate for the subcritical SITE norm. The fixed-direction lower bound remains the finite-seed/Harris/packing construction, not an OZ prefactor. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/first-noncommon-correction-channels-20260914.md b/docs/manuscripts/geometric-balance/first-noncommon-correction-channels-20260914.md new file mode 100644 index 000000000..ec171c996 --- /dev/null +++ b/docs/manuscripts/geometric-balance/first-noncommon-correction-channels-20260914.md @@ -0,0 +1,434 @@ +> # ⚠️ 勘误与独立核验裁定（2026-09-14，由独立核验代理追加） +> +> **本笔记的 `§0` 第 4 条（「动量 0 排除 spin±4」）已由独立核验降级为【条件性声称】。** +> 核验报告：`docs/manuscripts/geometric-balance/verify-768-momentum-exclusion-20260914.md`； +> 冲突清单（含最小复现）：`docs/manuscripts/geometric-balance/verify-768-issues-20260914.md`。 +> +> **核验结论摘要（请以核验报告为准）** +> 1. 该排除**作为一阶矩阵元陈述成立**（`δ_{h,h̄}` 推导精确；`Θ_w`/`Δ_w` 作为"每行扇区能量差"确是动量 0）。 +> ⚠️ 但「矩阵元 = 0」只**一阶**成立，本笔记未证明 `w^{−17/4}` 是一阶 ⇒ **缺环**。 +> 2. **「区分两个候选」不成立**：动量规则只能杀 `spin ≠ 0`，**永远选不出标量** +> （`k=2..7` 的标量 arm 通道全在 `x ≤ 21/4` 且未被排除）；8-arm 的**非零性**也未被这条规则建立。 +> 3. ⚠️ **`§3.4 判据 D3` 与其自身 `§1.2` 直接矛盾**： +> D3 用 `(ii)/(iii)` 的类计数推出 `x=21/4` 处有 **2 个**独立动量 0 幅度； +> 但那个「多出来的 level-2 类」**正是零范态 `χ` 的方向本身** +> （`verma_mod_Lm1_dims[2]=1` 而 `mod_Lm1_dims[2]=0`、`chi_norm="0"`） +> ⇒ **零范场关联恒为 0 ⇒ 该 `(2,2)` 类贡献 0** ⇒ **D3 对 `(ii)` 是错的**； +> 对 `(iii)` 本笔记自述代数不定。 +> 而 `§1.2` 明说「区分 (i) 与 (ii) 没有物理意义」、(iii) 才是物理模 +> ⇒ **用 `(ii)` 的计数去支持 `(iii)` 的效果，是混用。** +> 4. ⇒ **正确表述**：该排除在 **(i) 与 (ii) 下都成立**，**只有在 (iii) 且该类为真算子时才失效** —— +> 而本笔记自己主张物理模是 (iii)。**它是条件性的，不是无条件的。** +> 5. ⚠️ **本笔记 JSON 的 flag `rank_equals_p_shift2: false` 是脚本 bug**（拿 `rank` 去比 `p(n−2)`）； +> 正确关系是 `gram_rank = p(n) − p(n−2)`，**表与结论本身是对的**。 +> 只读 JSON 会误判该声称失败。详见冲突清单 IF-1。 +> +> **本笔记的其余部分（`§1.1` 的等级计数表、`§1.2` 的图表、`§1.3`–`§1.4`、`§3.5` 的诚实负面结论） +> 经核验成立或未被推翻。** `§3.5` 那句「仅凭 `(T,N)` 分离不了纯 dual-odd 的两个候选」核验认可。 +> +> **正文保持原样，未作任何改动**（以保留原始交付的可追溯性）。 + +# #768 深度理论：首个非公共扇区修正的实际矩阵元 + +**日期**：2026-09-14　**作者**：`theory768`（本机只做读写/上传下载/汇总；**全部计算在云机 `DevEnvC_TV2N0X` 上执行**） +**交付对应**：`#768` 主单（G1 最低性/非零性、G2 模块与总导数、G3 不同预测）。 + +> **诚实性声明**（每节都适用） +> 1. 本文所有数字都由我在云机跑的脚本产出，脚本在 `scripts/`（`g2_virasoro.py`、`g2b_derivatives.py`、 +> `g2c_bilevel.py`、`g1g3_channels_predictions.py`），输出在 `out/*.json`。 +> 2. **引用文献一律标「引用，未独立重算」**，并给可检索标识（arXiv id + 卷页 + 式号）。 +> 凡我没读到的，写「未找到 / 未读取」。 +> 3. 严格区分 **精确恒等式 / 有限枚举事实 / 条件定理 / 假设 / 猜想 / 路线提示**。 +> 4. **没有**触碰 `#802` 的 `(T_g,N_g)` 数值计算（另一代理在做）。本文只做理论侧。 +> 5. `#585` 未读取（GitHub API 限流 403，见 `issues-found.md`）。 + +--- + +## 0. 五句话结论 + +1. **G2 的「`0,0,1,1`」成立**——但**只对"把 level-2 奇异向量商掉的不可约商"这一个模**。 + 我自己用精确有理数复算：`p(n)=1,1,2,3,5,7,11`；`rank G_n = 1,1,1,2,3,4,6 = p(n)−p(n−2)`； + 奇异向量只在 level 2 出现（维数 1）；`χ = −3L_{-2}+2L_{-1}²`（即 `∝ L_{-2}−(2/3)L_{-1}²`）， + **范数恰为 0**；level 1..4 模去 `L_{-1}` 的维数 = **`0,0,1,1`**。 +2. **null 的去向**：代数上 `χ` 是**零范**奇异向量，`V(5/8)` 的极大真子模**只由它生成** + （我算出的 `rank G_n = p(n)−p(n−2)` 就是这个事实），所以不可约商 `L_{2,1}` 定义良好 ⇒ **(i) 与 (ii) 在代数上都可站住**。 + 但 **(i) 与 (ii) 的关联函数完全相同**（零范且被正模湮灭 ⇒ 对任何期待值贡献为零）， + **真正改变物理的只有 (iii)**。而 c=0 渗流的物理模**就是 (iii)**：能量算符是零范态且是**对数多重态的 bottom field** + （引用：He, arXiv:2411.18696, SciPost Phys. 19, 008 (2025)；Vasseur–Jacobsen–Saleur, arXiv:1206.2312, J. Stat. Mech. L07001 (2012)）。 +3. **G1 里被"证明"消去的只有两类**：identity 家族（dual-even，在差里恒等消去）与所有 C4 禁止的通道 + （spin `1,2,3 mod 4`）。**`k=2..7` 的标量 arm 通道（`x=1/4,2/3,5/4,2,35/12,4`）没有被证明消去**—— + 它们的 matching parity 正是未建立的 map/sector dictionary（与 `round45-out/arms-audit.md` §2 的裁定一致）。 +4. **G3 的关键**：在**平移不变**的圆柱/环面上，权重 `(h,h̄)` 的算符一点函数 `∝ δ_{h,h̄}`（动量守恒）。 + 因此**热族 level-4 手征后代（spin ±4, `x=21/4`）在"纵向每行能量差"这类动量 0 观测量里矩阵元为 0**。 + 它与 scalar 8-arm（`(21/8,21/8)`，spin 0，动量 0）**预测像不同**，且这是**可行的排除**（#768 允许的"导致 spin4 方案失败的模块/矩阵元"）。 + 附加结论：只有**保留零范态**时，热模才在 `(a,b)=(2,2)` 多出一个 **spin 0**、`x=21/4` 的非导数类，才能进入动量 0 观测量。 +5. **不能宣称**：不能宣称 level-4 类的**差矩阵元非零**；不能把 `∂⁴ε` 当插入（我显式验证它的类是 0）； + 不能宣称 8-arm 或 spin4 谁胜出；不能宣称 `w^{−17/4}` 已被解释。 + +--- + +## 1. G2（主攻）：`c=0, h=5/8` 的 level-2 奇异向量与等级计数 + +### 1.1 我算出来的东西（精确有理数，脚本 `scripts/g2_virasoro.py`） + +Virasoro 代数 `[L_m,L_n]=(m−n)L_{m+n}+(c/12)(m³−m)δ_{m+n,0}`，在 PBW 基 +`L_{−n_1}…L_{−n_k}|h>`（`n_1≥…≥n_k≥1`）上用交换关系 `L_{−a}L_{−b}=L_{−b}L_{−a}+(b−a)L_{−(a+b)}` +做正规化，构造各级 `Gram` 矩阵（反变量形式，`⟨h|h⟩=1`）。**`c=0, h=5/8`：** + +| `n` | `p(n)` | `rank G_n` | `nullity` | 奇异向量维数 | `p(n)−p(n−2)` | +|---:|---:|---:|---:|---:|---:| +| 0 | 1 | 1 | 0 | 0 | 1 | +| 1 | 1 | 1 | 0 | 0 | 1 | +| 2 | 2 | **1** | **1** | **1** | 1 | +| 3 | 3 | 2 | 1 | 0 | 2 | +| 4 | 5 | 3 | 2 | 0 | 3 | +| 5 | 7 | 4 | 3 | 0 | 4 | +| 6 | 11 | 6 | 5 | 0 | 6 | + +- **`rank G_n = p(n) − p(n−2)`（`p(<0)=0`）在所有 `n≤6` 上成立**（精确有理数）。 + `Gram` 矩阵的根（radical）在 level `n` 的维数 = 极大真子模在该层的维数（Kac 标准事实）， + 所以这条等式就是：**极大真子模恰好是 level-2 奇异向量平移 `2` 层后的 Verma 模** ⇒ + `V(5/8)` **只有一个极大真子模**，不可约商 `L_{2,1}` 定义良好。 +- 奇异向量**只在 level 2 出现**（`n=0..6` 内维数分别为 `0,0,1,0,0,0,0`）。 +- **`χ = −3·L_{−2}|h> + 2·L_{−1}²|h>`**（在 level-2 基 `[(2,),(1,1)]` 上；即 `∝ L_{−2} − (2/3)L_{−1}²`）。 + 验证：`L_1χ = 0`✓、`L_2χ = 0`✓、**`⟨χ|χ⟩ = 0` 精确**（不是"数值小"）。 +- `det G_2(c=0) = 4h²(8h−5)`（我算的因式分解）⇒ `h=0` 与 `h=5/8` 是 c=0 的 level-2 退化权。✓ + +**模去 `L_{-1}` 的维数**（`d_n = rank G_n`，`e_n = dim(L_{−1} 在 level n 商中的像)`）： + +| `n` | 0 | 1 | 2 | 3 | 4 | 5 | 6 | +|---|---:|---:|---:|---:|---:|---:|---:| +| `d_n` | 1 | 1 | 1 | 2 | 3 | 4 | 6 | +| `e_n` | 0 | 1 | 1 | 1 | 2 | 3 | 4 | +| **`d_n − e_n`** | 1 | **0** | **0** | **1** | **1** | 1 | 2 | + +> **裁定：`WORK-NOW` 第 2 条理论推进里"模去 `L_{−1}` 后 level 1..4 的维数为 `0,0,1,1`"——成立。** +> 但它成立的前提是"`χ` 被商掉"，即场景 **(i)**。 + +### 1.2 三种去向的裁定 + +| 场景 | 含义 | 我的计算能否判定 | 结果 | +|---|---|---|---| +| **(i) 被商掉** | 物理态空间 = 不可约商 `L_{2,1}` | ✅ 能 | `0,0,1,1` **成立**（上表） | +| **(ii) 零范但保留** | 物理态空间 = Verma，`χ` 仍是一个态 | ✅ 能 | 我算的 Mod-`L_{−1}` 维数 level 1..4 = **`0,1,1,2`** | +| **(iii) 嵌入 log 模块** | 物理模是不可分解扩张 | ❌ **不能**（代数不定，是物理模的性质） | 文献说**是 (iii)** | + +**关于 (i) 与 (ii) 的关键澄清（我自己的推理，非引用）：** +`χ` 被所有正模湮灭（奇异向量）且与一切态内积为 0（零范）。 +因此无论是否把它从态空间里减掉，**任何关联函数/任何期待值都不受影响**。 +⇒ **区分 (i) 与 (ii) 没有物理意义；(ii) 不是 (i) 的替代解释。** 这不是新定理，只是把"零范态"的定义写清楚。 + +**关于 (iii) 的文献依据（引用，未独立重算）：** +- **Vasseur–Jacobsen–Saleur, arXiv:1206.2312, J. Stat. Mech. L07001 (2012)**： + 在 `Q→1` 的 Potts/渗流里，能量算符 `ε` 与"对数伙伴"（识别为 **4-leg / tensor 算符**，即产生两条传播簇的场） + **有相同的标度维数 `Δ_ε = Δ_ψ̂ = 5/4`**（该文 §4，(7) 之后正文），并构成 **rank-2 Jordan cell**： + `(11) ψ̂_{ab}(Λr) = Λ^{−5/4}[ ψ̂_{ab}(r) + (2√3/π) logΛ · ε(r) ]`； + `(8)` 给出对数两点函数；`(17)` 给出纯对数组合 `F(r) ~ θ + (2√3/π) log r`； + 不可分解参数 = `2√3/π ≈ 1.1026`（`(9)` 的极限），MC 复核 `1.15 ± 0.05`。 + ⇒ 这**就是 (iii)**：`(5/8,5/8)` 的模在物理上是**可分解性的**，零范态是 Jordan cell 的 bottom。 +- **Yifei He, arXiv:2411.18696, SciPost Phys. 19, 008 (2025)**：摘要原文 + "**At c=0, Kac operators become zero-norm states and the bottom fields of logarithmic multiplets**"； + 并讨论"the rank-2 pair of the energy operator mixing with the hull operator"， + 以及"the rank-3 Jordan block associated with the second energy operator"；建议可能存在任意高阶 Jordan 块。 +- **Mathieu–Ridout, arXiv:0708.0802, Phys. Lett. B 657 (2007) 120**： + c=0 扩展 Kac 表 `h_{r,s}=((3r−2s)²−1)/24`，`h_{r,s}=h_{r+2,s+3}`； + **`r` 偶（或 `s` 为 3 的倍数）的 `V_{r,s}` 的极大真子模由 grade `rs` 的单个奇异向量生成**。 + 我们的 `h=5/8` 是 `h_{2,1}`（`r=2` 偶）⇒ grade `2` 的单个奇异向量。 + **这与我的 `rank G_n = p(n)−p(n−2)` 独立一致。** + +### 1.3 为什么 (i)/(iii) 的区分在物理上很重要——非手征 `(a,b)` 表 + +热模 `(h_t,h̄_t)=(5/8,5/8)` 的 `(a,b)` 层：`x = 5/4 + a + b`，`spin = a − b`。 +手征与反手征因子独立，非导数类维数 `= q_a · q_b`（`q_n` = 该模 level `n` 的**非导数手征类数**， +即 `d_n − e_n`）。脚本 `scripts/g2c_bilevel.py` 用上面算出的 `q` 直接展开（`q` 不可超过已算层数，这里 `a,b≤4<6` 全程在已验证范围内）： + +**不可约商 (i)：`q = [1,0,0,1,1,1,2]`** + +| `(a,b)` | spin | `x` | 维数 | 动量 0? | C4 允许? | 可在动量 0 观测量中可见? | +|---|---|---:|---:|---|---|---| +| (0,0) | 0 | 5/4 | 1 | ✅ | ✅ | ✅ | +| (3,0)/(0,3) | ±3 | 17/4 | 1 | ❌ | ❌ | ❌ | +| (4,0)/(0,4) | ±4 | **21/4** | 1 | ❌ | ✅ | **❌** | +| (1,3)/(2,2)/(3,1) | — | 21/4 | **0** | — | — | — | + +**保留零范态 (ii)/(iii)：`q = [1,0,1,1,2,2,4]`** + +| `(a,b)` | spin | `x` | 维数 | +|---|---|---:|---:| +| (0,0) | 0 | 5/4 | 1 | +| (2,0)/(0,2) | ±2 | 13/4 | 1 | +| (3,0)/(0,3) | ±3 | 17/4 | 1 | +| (4,0)/(0,4) | ±4 | 21/4 | 2 | +| **(2,2)** | **0** | **21/4** | **1** | + +**⇒ 这是本次最有用的一条结果：** + +> 在 `x = 21/4` 处，热模的**动量 0（spin 0）内容完全取决于 level-2 null 是否被商掉**： +> - 场景 **(i)**：**没有** spin-0 类（只有 spin±4，动量 ≠ 0）⇒ **热模不能贡献到"纵向每行能量差"这类动量 0 观测量**； +> - 场景 **(ii)/(iii)**：在 `(2,2)` 有一个 spin-0 类 ⇒ 热模**可以**贡献。 +> +> 而 **scalar 8-arm（`(21/8,21/8)`，spin 0）本身是动量 0 的**，在两种场景下都能贡献。 + +### 1.4 `∂⁴ε` 的处置（#768 明确点名） + +- 我显式展开：`L_{−1}^4|h> = (1,1,1,1)`（单一 PBW 词），**它属于 `L_{−1}(level 3)` 的像**。 +- 因此它在"模去 `L_{−1}`"的商里**类为 0**（脚本 `g2b_derivatives.py` 第 1、2 段：level 4 的 + `(1,1,1,1)` 判为"在导数子空间里 = True"，而 `(4,)`、`(2,2)`、`(3,1)` 判为 False）。 +- **所以不能把 `∂⁴ε` 当成非零插入**——它是**总导数**，在平移不变背景上一点函数 `= ∂⁴⟨ε⟩ = 0`。 + #768 的这条限制**成立**，而且我现在给出显式证据。 +- 真正在 level-4 商里的那个**非导数类**是一个具体向量（在基 `[(4,),(3,1),(2,2),(2,1,1),(1,1,1,1)]` 上）： + `(1/2, −1, 2/3, 1, 0)`，即 `½L_{−4} − L_{−3}L_{−1} + ⅔L_{−2}² + L_{−2}L_{−1}²`；它不是 `ker G_4` 的元素。 + （脚本给的是"与 `span(像+ker G_4)` 欧氏正交的唯一方向"——这是等价刻画。） + +### 1.5 G2 **能**与**不能**结论什么 + +**能**： +- `0,0,1,1` 对该不可约商是**精确正确**的（我用精确有理数独立复算，不是抄 `WORK-NOW` 的表）。 +- "首个 **C4 允许** 的非导数手征类在 level 4 ⇒ `x = 5/4+4 = 21/4`" **对 (i)/(ii)/(iii) 都成立**： + level 1、2、3 的 spin 是 ±1、±2、±3，`s≡0 (mod 4)` 都不满足；level 4 才满足。 + **这条结论对 null 去向是稳健的（robust）。** +- `χ` 是**零范**的、且是 level 2 唯一的奇异向量；极大真子模只由它生成。 +- `∂⁴ε` 不是有效插入（显式验证）。 + +**不能**： +- ❌ 不能宣称 level-4 那个非导数类在**实际扇区中的差矩阵元非零**。等级计数只说"它是唯一候选"，不说振幅。 +- ❌ 不能宣称物理模就是不可约商（**(i)**）。c=0 的物理模更可能是 **(iii)**（两条独立文献）。 +- ❌ 不能宣称 `0,0,1,1` 是"物理态空间的计数"——它只是**不可约商**的计数； + 在 (iii) 下物理模还多出 Jordan 伙伴，计数**必然更大**（具体数目**不由 Virasoro 代数决定**，我不给数字）。 +- ❌ 不能排除其他模（8-arm scalar、`k=2..7` 标量、log 伙伴、contact/seam 项）。 +- ❌ 不能用等级计数去谈 `w` 的幂——等级计数是 `h` 的加性结构，与 `w` 的指数只通过 `x=2h` 挂钩。 + +--- + +## 2. G1：低于 `x = 21/4` 的通道（四件事分开） + +### 2.1 先把四件事分开 + +| | 含义 | 判定手段 | 强度 | +|---|---|---|---| +| (a) **C4 允许** | 方格格点 `C4` 下不变：spin `s ≡ 0 (mod 4)` | 精确（群论） | 定理 | +| (b) **dual-odd** | 在 primal/dual（黑↔白、`p↔1−p`）交换下变号 | 需要 map/sector dictionary | **多数通道未建立** | +| (c) **实际可出现** | 真的在 square-site 渗流谱里 | CFT 谱 + 数值 | 部分 | +| (d) **差矩阵元非零** | 在 NN-safe 与 matching-safe 两扇区**不同且非零** | 需要实际插入计算 | **未建立** | + +`Δ_w(p) = I⁰_{4,w}(p) − I⁰_{8,w}(1−p)` 在 dual 交换下**变号**， +所以 **(b) 是必要的**：dual-even 算符对 `Δ_w` 的贡献**恒等消去**（不是小，是零）。 +但 (b) **不充分**——`#776` 的 triangular self-matching 会让所有 dual-odd 贡献同时为零， +所以它只是**回归控制**，不能单独挑出 8-arm 或 spin4。✓（与 `#776` 原文一致） + +### 2.2 通道清单（`c=0` 扩展 Kac 标量，精确有理数，脚本 `g1g3_channels_predictions.py`） + +`h_{r,s}=((3r−2s)²−1)/24` 只依赖 `k:=3r−2s`：`h(k)=(k²−1)/24`，`x=2h=(k²−1)/12`。 +（**注意**：`x(k)` 与 `arm` 指数 `α_j=(j²−1)/12` 是**同一个函数**——`round45-out/arms-audit.md` §1.3 已裁定， +我这次是**独立重算的算术**，不是引用。） + +| `k` | `h` | `x` | `≤ 21/4`? | C4（作为标量） | +|---:|---|---:|---|---| +| 0 | −1/24 | −1/12 | ✅（相关） | ✅ | +| 1 | 0 | 0 | ✅ | ✅ | +| 2 | 1/8 | 1/4 | ✅ | ✅ | +| 3 | 1/3 | 2/3 | ✅ | ✅ | +| 4 | **5/8** | **5/4** | ✅（= thermal） | ✅ | +| 5 | 1 | 2 | ✅ | ✅ | +| 6 | 35/24 | 35/12 | ✅ | ✅ | +| 7 | 2 | 4 | ✅ | ✅ | +| 8 | **21/8** | **21/4** | ✅（目标） | ✅ | + +**thermal 手征塔**（`x = 5/4 + s`，`spin = ±s`）：`s=0→5/4`（C4✅）、`s=1→9/4`（❌）、 +`s=2→13/4`（❌）、`s=3→17/4`（❌）、**`s=4→21/4`（✅）**、`s=5→25/4`（❌）。 + +**`C4` 选择定则（我算的）**：允许 `s = 0,4,8,…`；禁止 `s = 1,2,3,5,6,7`。 + +### 2.3 四分表（`x < 21/4`） + +| 通道 | `x` | spin | (a) C4 允许 | (b) dual-odd | (c) 实际可出现 | (d) 差矩阵元 ≠ 0 | 消去理由 / 证据强度 | +|---|---:|---|---|---|---|---|---| +| identity, `T`, identity 塔 | 0,1,2,… | 0,±1,… | s=0,4,…✅；s=1,2,3❌ | **even** | ✅ | **0（恒等消去）** | `Δ_w` 是 dual-odd；complement 关系给出**精确**消去（本项目已确立的对称性，非本文新证） | +| stress tensor | 2 | ±2 | ❌ | even | ✅ | 0 | C4 禁止 + even | +| thermal primary `ε` | 5/4 | 0 | ✅ | **odd**（与 4-leg 构成 Jordan pair） | ✅ | 不是 `p_c` 处的无关修正；它给 `Θ'_w ~ w^{−1/4}` | 相关场 ⇒ 只进分母（根位移），不进 `Θ_w(p_c)` 的首项 | +| thermal `s=1,2,3` | 9/4,13/4,17/4 | ±1,±2,±3 | ❌ | odd | (i) 下 level1,2 类为 0、level3 类为 1；(ii) 下都在 | ❌（C4 禁止 + 动量≠0） | 见 §3 的动量选择定则 | +| **标量 arm 通道 `k=2..7`** | 1/4,2/3,5/4,2,35/12,4 | 0 | ✅（全是标量） | **未建立** | **未建立**（Kac 表 ≠ 谱） | **未建立** | ⚠️ **这是本轮最大的空白**：#768 的 kill test 从未执行（`round45-out/arms-audit.md` §2 已裁定"是要求不是证明"） | +| log 通道：`h=0` 伙伴 | 0 | 0 | ✅ | 未建立 | 文献支持 | 未建立 | 引用：He arXiv:2411.18696（c=0 的 Kac 算符都是零范/log bottom） | +| log 通道：`(ε, ψ̂)` 对 | 5/4（两者同权） | 0 | ✅ | `ε` odd；`ψ̂`=? | ✅（VJS 显式构造 + MC） | 不可分解参数 `2√3/π ≈ 1.1026 ≠ 0`（VJS (9)(11)(17)） | 引用，未独立重算 | + +### 2.4 G1 的**动量选择定则**（本轮新增，我认为是关键） + +在**平移不变**的圆柱（横向周期）或环面（双周期）上， +一个权重 `(h,h̄)` 的原位算符的一点函数 +`⟨φ⟩ ∝ δ_{h,h̄}`， +因为横向平移生成元 `P = L_0 − L̄_0` 湮灭基态（真空），而 `[P, φ] = (h−h̄)φ`。 +（这是教科书级别的 CFT 事实，我在此**显式写出**，因为它是本轮最有力的筛选。） + +**直接后果**：任何 `spin = h − h̄ ≠ 0` 的算符，**不能移动动量 0 能级的有限尺寸修正**。 +"纵向每行的扇区能量差 `Θ_w`"（动量 0 的谱量）只接收 `h = h̄` 的算符。 + +**文献侧的一致证据（引用，未独立重算）**： +- **Javerzat–Grijalva–Rosso–Santachiara, arXiv:2005.11830, SciPost Phys. 9, 050 (2020)**： + 摘要明确"exploiting the anisotropy of the rectangular torus (M≠N), we directly test the presence of the + two components of the traceless stress-energy tensor"；正文用"沿两条正交轴的两点连通性之差 + `p12(x_h) − p12(x_v)`"做探针，并在方形环面（`q=e^{−2π}`）上得 `c_T(e^{−2π})=0`。 + ⚠️ **我只读了摘要页 + 部分 HTML 正文（提取出的公式编号有 OCR 混杂），等式编号与推导未逐条核实**。 + ⇒ 我把这条只当作**方向一致的外部佐证**，不作为我的论证依赖。 + +**对 #768 的直接含义**（这是我要交付的"硬结论"）： +> **热族 spin±4 那个候选（`(h_t+4, h̄_t)=(37/8,5/8)`）在平移不变的方形圆柱/环面上， +> 对 `Θ_w` 这种动量 0 量的矩阵元为 0。** 因此： +> - 要么 **spin4 作为"`w^{−17/4}` 的载体"这一读法要撤回**（`#768` 明确允许交付"一个导致 spin4 方案失败的模块/矩阵元"）； +> - 要么 **`Θ_w` 必须定义成方向分辨/各向异性（`M≠N`、缝方向、各向异性格点）的量**，spin±4 才可见。 +> 两者都是可判定的，且**不改动任何已有数据**。 +> +> **注意**：`WORK-NOW`/`#768` 里"`C4` 允许 spin `s` 仅当 `s≡0 mod 4`，故首个是 `s=4` ⇒ `x=21/4`" +> 这条推理**不完整**：对**动量 0** 的观测量，先需要 `h=h̄`；`C4` 只是在动量 0 的类里再筛。 +> 加上动量条件后，`x=21/4` 处动量 0 的候选在**热模里**要么不存在（场景 (i)）， +> 要么是 `(2,2)` 那个 **spin 0** 类（场景 (ii)/(iii)），**不是** spin±4 那个。 + +### 2.5 必须一起写进去的已知限制（沿用，不重复发现） + +- 有限 `T3`/`jump2` 的 pair-even 几何**既不证明也不否定** odd 六臂幅度。 +- triangular self-matching 使所有 dual-odd 贡献同时为零 ⇒ **只是回归控制**，不能单独挑出 8-arm 或 spin4。 +- `x_ℓ^P = (ℓ²−1)/12` 是**已发表**公式（**Aizenman–Duplantier–Aharony 1999, PRL 83, 1359, arXiv:cond-mat/9901018** 式 (1)(2)；引用，未独立复核）， + 所以"用 8-arm 反推 21/8"是**重述一个已发表公式**，不携带独立证据。 +- 关于 square-site matching function 的有限尺寸指数，**两份互相冲突的测量**： + Jacobsen（`1507.03027` §7.1 式 (36)：`Δ₁ = 4.000 1(2)`，并直述 "It appears inevitable to admit that `Δ₁ = 4` exactly"； + §7.1 式 (40)：`p_c(n)=p_c+Σ A_k/n^{2(k+1)}`；§7.2：kagome 令 `A₁=0`、`Δ₂=6.00(5)`， + 且原话把差别归因于 "**the three-fold rotational symmetry of the kagome lattice (which replaces the four-fold + symmetry of the square lattice)** has the effect of setting `A₁=0`"）； + Mertens–Ziff（`1603.07289` §IV：`2−x=−3.42`、`2−y=0.705`、`w=2−x−1/ν=−4.17`，直接测 `p_L*−p_c` 斜率 `−4.07`， + 并自认 "**Presumably, larger systems are needed to find the true behavior**"）。 + **`21/8` 的生死尚未判定**（沿用 `ROUND-STATE` §1.12）。 +- `M_L` 的指数 `2−x` **只有 MZ16 一份测量**。 + +### 2.6 G1 **能**与**不能**结论什么 + +**能**： +- 严格消去 identity 家族（dual-even）与所有 C4 禁止的自旋通道（精确群论 + 精确奇偶）。 +- 严格指出：`x=21/4` 处**动量 0** 的候选在热模里是 `(2,2)`（条件性）而不是 spin±4。 +- 严格指出：`k=2..7` 的标量 arm 通道**没有被证明消去**——消除它们是 requirement，不是 theorem。 + +**不能**： +- ❌ 不能宣称 `k=2..7` 已被 matching balance 消去。**这是 #768 自己点出的核心定理义务，仍然开着。** +- ❌ 不能宣称 8-arm 是 dual-odd。 +- ❌ 不能宣称哪个候选"实际出现"（Kac 表 ≠ 谱）。 +- ❌ 不能宣称 spin4 被"排除"——只能说它**在平移不变观测量的矩阵元为 0**（选定几何下的排除）， + 且**这条依赖"观测量是动量 0"这一前提**，需要与 `#802` 的 `Θ_w` 定义对齐后才算定案。 + +--- + +## 3. G3：两种仍存活候选的 `(T,N)` 预测 + +### 3.1 恒等式（我自己符号验证，脚本 `g1g3_channels_predictions.py`） + +由 `b(p*(g),g)=0` 微分： `b_p·dp*/dg + b_g = 0` ⇒ **`T_g := dp*/dg = −b_g/b_p`** ✓ +**`N_g := e_g − (e_p/b_p)·b_g`** ✓ +**若源纯 dual-odd（`e_g = 0`）：`N_g = e_p · T_g`** ✓（`sympy` 判定 `True`） + +### 3.2 反例族（我独立复核 `#802` 给的族） + +`b_L(p) = −L^y (p − p_c − aL^{−4})`，`e_L(p) = C_0(b_L(p))`，`C_0` 任意正偶函数： +- `dp*/da = L^{−4}` **精确**（与 `y`、`C_0` 无关）⇒ `p* − p_c = aL^{−4}` 对任意 `a` 成立； +- `N_a = e_a − (e_p/b_p)b_a = 0` **精确**（与 `C_0` 无关）。 + +⇒ **"单一 odd 修正导致根按 `L^{−4}` 移动"不推出"内禀曲线 `C_L(b)` 必有非零 odd 首项"。** +`O_L` 为零/小**不能**单独排除非零根位移机制。这条**成立**，且我现在有独立的符号证据。 + +### 3.3 两个候选的预测表（**同一源、同一几何、同一热中心约定**） + +约定：`w` = 圆柱周长；`b = ½log(P0/P2)`（matching-odd）；`e = log[P1/(2√(P0P2))]`（matching-even）； +`Θ_w(p) = I⁰_{4,w}(p) − I⁰_{8,w}(1−p)`；`Δ_w = Θ_w(p_c)`，`B := Θ'_w(p_c)`。 + +| | **候选 S：scalar 8-arm** | **候选 T4：thermal spin±4** | +|---|---|---| +| 算符 | `(h,h̄) = (21/8, 21/8)`；**独立 primary**（Kac `h_{4,2}=h_{2,7}`） | `(h,h̄) = (5/8+4, 5/8)` 与其反手征共轭；**thermal 模 level-4 手征后代** | +| `x` | 21/4 | 21/4（**同幂**） | +| spin | **0（动量 0）** | **±4（动量 ≠ 0）** | +| C4 | ✅ | ✅ | +| 主导 `Δ_w` | `A_S w^{−17/4}` | `A_4 w^{−17/4}` —— **但见下表判据 D1** | +| `Θ'_w` | `B w^{−1/4}` | `B w^{−1/4}` | +| **`T`** | `−(A_S/B) w^{−4}` | `−(A_4/B) w^{−4}·(1+λ log w)`（λ 由 (iii) 的不可分解性给出，**λ 的值我未算**） | +| **`N`** | 纯 odd ⇒ `N = e_p T = −(e_p A_S/B) w^{−4}` | 同上（纯 odd 初阶）；若 log 伙伴带来 even 响应则偏离 | +| log 项 | 首阶无 | (iii) 下有 `log w` | +| 自由幅度数 | **1** | **(i)：1；(ii)/(iii)：≥2** | +| `x<21/4` 的伴随 | **无**（其手征后代全 `≥ 25/4`） | **有**：`(a,b)=(2,0),(0,2)→13/4`；`(3,0),(0,3)→17/4`（(ii)/(iii) 下，见 §1.3 表） | +| `(T,N)` 可达像 | 射线 `N = e_p T` | 同一射线（纯 odd 时），但 `T(w)` 带 `log`；幅度维数与伴随不同 | + +### 3.4 **预测像不同**的判据（这是 G3 的交付） + +**判据 D1（最强，几何 + 动量选择定则）** + +| | 候选 S（spin 0） | 候选 T4（spin ±4） | +|---|---|---| +| 平移不变的方形圆柱/环面（`M=N`，无缝）上的 `Θ_w` | `A_S w^{−17/4} ≠ 0` **允许** | **矩阵元为 0**（动量守恒，§2.4） | +| 各向异性/方向分辨探针（矩形环面 `M≠N`，或 ⟨10⟩ 缝 vs ⟨11⟩ 缝，或沿两条正交轴的连通性之差） | 无新结构（标量） | **必须出现**，且是 spin±4 的指纹 | + +⇒ **这是"预测像不同"**：一个在**各向同性**观测量里存活，另一个只在**方向分辨**观测量里存活。 +形式上可测：把同一 `Θ` 在"方形"与"矩形/方向分辨"两种几何下各测一次。 +**引用侧支撑（未独立重算）**：Javerzat et al. `2005.11830`（SciPost Phys. 9, 050）用 `M≠N` 的环面各向异性 +去测 `Δ≠Δ̄` 分量；在这个框架里 `Δ=Δ̄` 的算符在方形环面上没有同类的压制。 + +**判据 D2（伴随通道，条件性）** +在 (ii)/(iii) 情形，热模**必须**在 `x = 13/4`、`17/4` 处有 spin ±2、±3 的伴随类（§1.3 表）。 +它们同样是动量 ≠ 0 ⇒ 同样只在**方向分辨**观测量里可见。 +候选 S **不预测**任何 `x < 21/4` 的伴随。 +⇒ 观测量：方向分辨的 `x=13/4, 17/4` 修正。**这是我算出来的模结构 + 精确 C4 定则；观测量是否可测未验证。** + +**判据 D3（幅度计数）** +- 场景 (i)：热模在 `x=21/4` 的**动量 0** 内容 = **0**；能贡献的只有 scalar primary（1 个幅度）。 +- 场景 (ii)/(iii)：热模多出 `(2,2)` 的 spin-0 类 ⇒ 若 `(21/8,21/8)` 的 Kac primary 也在谱里， + `x=21/4` 处有 **2 个**独立的动量 0 幅度（彼此因同权而混合）。 +⇒ 在 `(T,N)` 平面上，可达像的**维数**不同（1 vs 2）。 + +### 3.5 G3 的诚实负面结论（同样要保存） + +> **仅凭 `(T,N)` 分离不了"纯 dual-odd"的两个候选。** +> 只要源是纯 dual-odd（`e_g = 0`），**任何**候选都给同一条射线 `N = e_p T`（§3.1 的恒等式）。 +> 所以 `(T,N)` 的"预测像"对候选 S 与候选 T4 **在初阶相同**，差异只在 +> **① `T(w)` 是否带 `log`**、**② 幅度个数**、**③ 是否在动量 0 观测量里存活（D1）**、**④ `x<21/4` 的伴随（D2）**。 +> ⇒ 与 `#802` 的提醒一致："内禀 `C_L(b)` 会消掉纯热切向修正"； +> 本轮补充的是：**`(T,N)` 也消掉两个候选之间的差别，除非把方向/几何自由度打开。** + +### 3.6 G3 **能**与**不能**结论什么 + +**能**： +- 给出两个候选在**同一约定**下的 `(T,N)` 显式形式（含 log 与幅度计数）。 +- 给出**至少一个不共享预测像**的对比：**D1（各向同性 vs 方向分辨）**，且有动量守恒这一硬基础。 +- 独立复核 `#802` 的反例族（`T=aL^{−4}`、`N=0` 精确）。 + +**不能**： +- ❌ 不能宣称 lambda（log 系数）的具体值——我没算； VJS 的 `2√3/π` 是 **(ε,ψ̂) 那一对**的参数， + **不能**直接当作 level-4 descendant 的 `log` 系数。 +- ❌ 不能宣称候选 S 是 dual-odd（未建立）。 +- ❌ 不能宣称 D1 已经在实际数据上被验证——它是**选择定则层面的排除 + 一个可做的实验设计**， + 需要与 `#802` 的 `Θ_w` 定义对齐（`Θ_w` 到底是动量 0 的谱量，还是含缝/方向分辨的量）。 +- ❌ 不给出任何新的 `(T_g,N_g)` 数值（那是 `#802` 的工作）。 + +--- + +## 4. 与 `#802` 的接口（不重复它的计算） + +1. **必须确认 `Θ_w` 的定义**：它是**动量 0 的谱量**（那么 §2.4 的排除直接适用）， + 还是含**缝/方向分辨**的量（那么 spin±4 才能进入）。这一点会**改变结论**，我把它标为**未决**。 +2. `(T,N)` 的联合像在纯 odd 源下是射线（§3.1）；请把**方向/几何**自由度（`M/N` 比、缝方向） + 纳入同一批产出，否则候选分离不出来。 +3. 若要做 `log` 检验：只需对**同一份** `Δ_w` 数据做"带 `log w` 与不带"的两参数拟合， + 比较残差；不需要新宽度。 +4. 我认为**最便宜的判别实验**是：把 `Δ_w`（或 `Θ_w`）在**方形环面**与**矩形环面（`M≠N`）** + 两种几何下各测一次（复用已有 transfer-matrix，不需要新宽度阶梯）。 + - 候选 S：两者都非零、且比值接近 1（标量不敏感）； + - 候选 T4：方形上 ≈ 0，矩形上非零。 + **这是唯一一个我认为值得占 `#802` 机器槽位的、能改变判断的观测量。** + +--- + +## 5. 交付清单 + +| 文件 | 内容 | +|---|---| +| `out/note-first-noncommon-correction.md` | 本文件（G1 四分表 + G2 裁定 + G3 预测 + 不能宣称什么） | +| `out/theory768.json` | 机器可读：等级维数、奇异向量、`(a,b)` 表、通道清单、预测表 | +| `out/lit-directed-768.md` | §3 定向检索（可检索标识 + 式号；找不到的写"未找到 + 已检索关键词"） | +| `out/issues-found.md` | 与包内/文献口径冲突处（含最小复现） | +| `scripts/g2_virasoro.py` | G2 主计算（Verma 等级计数、Gram rank、奇异向量、模 `L_{−1}`） | +| `scripts/g2b_derivatives.py` | `L_{−1}^4|h>` 的展开与导数子空间判定 | +| `scripts/g2c_bilevel.py` | 非手征 `(a,b)` 非导数类表（含动量 0 标记） | +| `scripts/g1g3_channels_predictions.py` | G1 通道清单 + G3 恒等式与预测表 | +| `out/g2_result.json`、`out/g2b_result.json`、`out/g2c_result.json`、`out/g1g3_result.json` | 原始输出 | diff --git a/docs/manuscripts/geometric-balance/fixed-direction-exponential-centres-20260914.md b/docs/manuscripts/geometric-balance/fixed-direction-exponential-centres-20260914.md new file mode 100644 index 000000000..001682620 --- /dev/null +++ b/docs/manuscripts/geometric-balance/fixed-direction-exponential-centres-20260914.md @@ -0,0 +1,373 @@ +# Fixed integer directions: exact exponential-aspect birth centres + +2026-09-14. Author-level continuation of #739/#765. This closes the fixed-primitive-integer-direction case without rotating the microscopic interaction and without importing a bond OZ theorem. + +The proof uses two ingredients already established on this continuation branch: + +1. `first-exit-torus-winding-upper-20260914.md` for the full-period upper bound; +2. the finite directional seed + Harris + translation-packing lower bound, which is the directional analogue of `exponential-birth-centres.md`. + +A previously tempting direct comparison between quotient simple-cycle arcs and full-plane two-point events is **not** used: periodic reuse of Bernoulli variables makes that comparison unsafe. The first-exit skeleton is the rigorous replacement. + +## 1. Directional mass along a fixed integer vector + +Fix a primitive integer vector + +\[ +u=(a,b)\in\mathbb Z^2, +\] + +and choose `v in Z^2` with + +\[ +\det(u,v)=1. \tag{1.1} +\] + +For `G=G4` (NN) or `G8` (matching), let + +\[ +\pi^G_p(x)=P_p^G(0\leftrightarrow x). \tag{1.2} +\] + +The subcritical inverse-correlation norm is + +\[ +\tau_{G,p}(x) +=\lim_{k\to\infty}-\frac1k\log \pi^G_p(\lfloor kx\rfloor). \tag{1.3} +\] + +For the fixed integer direction put + +\[ +\xi_{G,u}(p)=\tau_{G,p}(u). \tag{1.4} +\] + +Equivalently, after normalizing the common occupied endpoint and applying Harris, + +\[ +\xi_{G,u}(p) +=\lim_{k\to\infty}-\frac1k\log s_p^G(ku), +\qquad +s_p(x)=\pi_p(x)/p. \tag{1.5} +\] + +For Euclidean unit direction `e=u/|u|`, + +\[ +\tau_{G,p}(e)=\xi_{G,u}(p)/|u|. \tag{1.6} +\] + +## 2. The exponential torus adapted to u + +Let + +\[ +\Lambda_{n,m}=\langle nu,mv\rangle. \tag{2.1} +\] + +Its index is + +\[ +N=nm. \tag{2.2} +\] + +Assume + +\[ +n\to\infty, +\qquad +\frac{\log m}{n}\to d\in(0,\infty). \tag{2.3} +\] + +Because + +\[ +|\det(u,\alpha nu+\beta mv)|=|\beta|m, \tag{2.4} +\] + +every period with `beta!=0` has Euclidean length at least `|beta|m/|u|`. Hence the only periods on the `O(n)` scale are the multiples of `nu`. + +The physical short length and transverse height are + +\[ +\ell_n=n|u|, +\qquad +h_n=N/\ell_n=m/|u|. \tag{2.5} +\] + +Thus if #765 uses + +\[ +\frac{\log h_n}{\ell_n}\to\delta, \tag{2.6} +\] + +then + +\[ +d=\delta|u|. \tag{2.7} +\] + +## 3. Rigorous full-period upper bound from first-exit certificates + +Let + +\[ +f_G(n,m;p)=P_p^G(r_G>0). \tag{3.1} +\] + +`first-exit-torus-winding-upper-20260914.md` proves that for every fixed subcritical `p` and every `epsilon in (0,1)`, once the torus is large enough, + +\[ +\boxed{ +f_G(n,m;p) +\le N C_{p,\epsilon} +\sum_{\lambda\in\Lambda_{n,m}\setminus0} +\exp[-(1-\epsilon)\tau_{G,p}(\lambda)].} \tag{3.2} +\] + +The proof is local and quotient-safe: + +- choose a shortest nonzero-homology occupied closed walk; it is a torus-vertex-simple cycle; +- apply the finite first-exit skeleton to its lift; +- successive witness interiors are disjoint actual quotient SITE variables; +- each local translate injects, so its coefficient is the genuine planar first-exit coefficient; +- site BK bounds a skeleton by the product of these local coefficients; +- a compact certified Wulff body gives a pointwise factor `exp[-h_T(lambda)]`; +- large first-exit boxes exhaust `(1-epsilon)K_p`, whose support function is `(1-epsilon)tau_p`. + +For the period lattice (2.1), parallel periods have + +\[ +\tau_{G,p}(\alpha nu)=|\alpha|n\xi_{G,u}(p), \tag{3.3} +\] + +while nonparallel periods cost at least `c_p|beta|m/|u|`. Therefore the theta sum in (3.2) satisfies + +\[ +\sum_{\lambda\ne0}e^{-(1-\epsilon)\tau_{G,p}(\lambda)} +\le +C_{p,\epsilon}e^{-(1-\epsilon)n\xi_{G,u}(p)} ++e^{-c_{p,\epsilon}m}. \tag{3.4} +\] + +Since `N=nm`, + +\[ +\limsup_{n\to\infty}\frac1n\log f_G(n,m;p) +\le +-\max\{(1-\epsilon)\xi_{G,u}(p)-d,0\}. \tag{3.5} +\] + +Let `epsilon downarrow 0`: + +\[ +\boxed{ +\limsup\frac1n\log f_G(n,m;p) +\le +-\max\{\xi_{G,u}(p)-d,0\}.} \tag{3.6} +\] + +The complete directional mass appears; no half-period loss remains. + +## 4. Lower bound: repeat one finite directional seed and close the seam + +Fix `p0`. Choose a fixed integer `L` and a finite box around the segment from `0` to `Lu` such that the conditional finite-box connection probability `q` obeys + +\[ +q> +\exp[-(\xi_{G,u}(p)+\epsilon/2)L]. \tag{4.1} +\] + +This is possible by the definition of the directional mass and finite-box exhaustion. + +Write + +\[ +n=kL+r, +\qquad0\le r0. \tag{6.1} +\] + +Choose `d0` so that the Friedgut--Kalai increment satisfies + +\[ +\rho\frac{A-d+\zeta}{d}\epsilon_n=e^{-(A-d+\zeta)n} \tag{6.3} +\] + +for large `n`. The event `r_G>0` is increasing and invariant under the transitive torus translation group on `N=nm` sites, so Friedgut--Kalai forces + +\[ +f_G(n,m;q)>1-\epsilon_n\to1. \tag{6.4} +\] + +But (3.6) with `xi(q)=A>d` gives `f_G(n,m;q)->0`, a contradiction. Hence + +\[ +\boxed{p\mapsto\xi_{G,u}(p)\text{ is strictly decreasing on }(0,pc(G)).}\tag{6.5} +\] + +The small-`p` path count gives `xi->infinity` as `p->0`. As `p->pc(G)`, square symmetry and the norm triangle inequality give + +\[ +\xi_{G,u}(p) +\le(|a|+|b|)\kappa_G(p)\to0. \tag{6.6} +\] + +Thus `xi_{G,u}` continuously and strictly maps `(0,pc(G))` onto `(infinity,0)`. + +## 7. Fixed-direction birth centres + +Let `T_1,T_2` be the two NN rank births on `Lambda_{n,m}`. Since + +\[ +P(T_1\le p)=f_{G4}(n,m;p), \tag{7.1} +\] + +(5.1) and strict invertibility imply + +\[ +\boxed{ +T_1\xrightarrow{P}a_u(d), +\qquad +\xi_{4,u}(a_u(d))=d.} \tag{7.2} +\] + +Digital Alexander duality gives + +\[ +P_p^{G4}(r=2)=P_{1-p}^{G8}(r=0), \tag{7.3} +\] + +so + +\[ +\boxed{ +T_2\xrightarrow{P}b_u(d), +\qquad +b_u(d)=1-c_u(d), +\qquad +\xi_{8,u}(c_u(d))=d.} \tag{7.4} +\] + +Equivalently, with `e=u/|u|` and `delta=d/|u|`, + +\[ +\boxed{ +\tau_{4,a(e,\delta)}(e)=\delta, +\qquad +\tau_{8,1-b(e,\delta)}(e)=\delta.} \tag{7.5} +\] + +## 8. Strict matching mass gap in every fixed direction + +The endpoint-direction-independent enhancement argument gives, on every compact parameter interval inside `(0,pc(G8))`, a `delta_I>0` such that + +\[ +\boxed{ +P_p^{G8}(0\leftrightarrow x) +\ge +P_{p+\delta_I}^{G4}(0\leftrightarrow x)} \tag{8.1} +\] + +for distant endpoints `x`, uniformly in direction. Taking the logarithmic rate along `ku` gives + +\[ +\xi_{8,u}(p) +\le\xi_{4,u}(p+\delta_I). \tag{8.2} +\] + +By (6.5), + +\[ +\xi_{4,u}(p+\delta_I)<\xi_{4,u}(p). \tag{8.3} +\] + +Therefore + +\[ +\boxed{ +\xi_{8,u}(p)<\xi_{4,u}(p), +\qquad0e`, where the arithmetic of `u_n` changes and one needs uniformity of: + +- the directional norm convergence; +- finite first-exit certificates near the moving support point; +- finite-seed lower bounds; +- separation from other comparable period classes. + +The projective Poisson hard-core note on this branch gives the correct topology if several direction classes remain competitive. No new angle scan is required to finish the fixed-direction theorem. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/fixed-direction-poisson-gumbel-20260914.md b/docs/manuscripts/geometric-balance/fixed-direction-poisson-gumbel-20260914.md new file mode 100644 index 000000000..ee692b303 --- /dev/null +++ b/docs/manuscripts/geometric-balance/fixed-direction-poisson-gumbel-20260914.md @@ -0,0 +1,384 @@ +# Fixed primitive directions: no-prefactor Poisson--Gumbel windows + +2026-09-14. Directional extension of `poisson-birth-windows.md` for a **fixed primitive ambient integer direction**. The extension is exact at the lattice level through an `SL_2(Z)` coordinate change; it does not rotate the physical interaction or import a directional OZ prefactor. + +The conclusion is intentionally narrower than `varying-direction-exponential-centres-20260914.md`: centres are now known for varying directions, but the full component-Poisson/Gumbel machinery is proved here only when the integer direction is fixed, so that the transformed finite-range interaction has fixed range. + +## 1. Straightening a fixed integer direction without rotating the model + +Fix a primitive ambient integer vector + +\[ +u=(a,b)\in\mathbb Z^2, +\qquad \gcd(a,b)=1. \tag{1.1} +\] + +Choose `v in Z^2` such that + +\[ +\det(u,v)=1, \tag{1.2} +\] + +and let + +\[ +A=[u\ v]\in SL_2(\mathbb Z). \tag{1.3} +\] + +Use integer coordinates + +\[ +x=Ay. \tag{1.4} +\] + +Then the original torus with periods + +\[ +nu,\qquad mv \tag{1.5} +\] + +is graph-isomorphic to a rectangular `n x m` torus in `y` coordinates. + +The price is not a rotation of NN edges. The original NN step set becomes the fixed finite-range set + +\[ +\mathcal S_{4,u}=\{\pm A^{-1}e_1,\pm A^{-1}e_2\}, \tag{1.6} +\] + +and the matching graph becomes the corresponding fixed transform of NN+NNN, + +\[ +\mathcal S_{8,u}=A^{-1}\mathcal S_8. \tag{1.7} +\] + +All coordinates remain integer. Since `u,v` are fixed, the horizontal and vertical jump ranges + +\[ +R_x(u),R_y(u)<\infty \tag{1.8} +\] + +are fixed constants independent of `n,m`. + +Thus a fixed oblique direction is exactly an axial problem for one fixed anisotropic finite-range SITE graph. + +## 2. Directional mass and centre + +Let + +\[ +\xi_{G,u}(p)=\tau_{G,p}(u) \tag{2.1} +\] + +be the inverse-correlation cost per period copy. On tori with + +\[ +n\to\infty, +\qquad +\frac{\log m}{n}\to d>0, \tag{2.2} +\] + +the directional centre theorem gives unique + +\[ +\xi_{4,u}(a_u(d))=d, \tag{2.3} +\] + +\[ +\xi_{8,u}(c_u(d))=d, +\qquad +b_u(d)=1-c_u(d). \tag{2.4} +\] + +The two births satisfy + +\[ +T_1\to a_u(d), +\qquad +T_2\to b_u(d) \tag{2.5} +\] + +in probability. + +## 3. Complete winding components on the transformed cylinder + +Consider the infinite cylinder + +\[ +C_n\times\mathbb Z \tag{3.1} +\] + +for the transformed graph `G_u`. + +A whole empty **slab of thickness `2R_y+1`** disconnects the graph vertically. Such slabs occur independently at well-separated heights with positive probability. Hence every occupied component is vertically finite almost surely at each fixed `n` and `p<1`. + +For a component with nonzero horizontal homology, anchor it at its lowest `y_2` coordinate, with a deterministic horizontal tie-break. Define + +\[ +\nu^{G,u}_n(p) +=E[\text{number of complete winding-component anchors at level }0].\tag{3.2} +\] + +This is the fixed-direction analogue of the axial component density. + +To define a local truncated anchor with height `H`, use guard **slabs** of thickness at least `R_y` below and above the candidate window. If the candidate component misses both guard slabs, finite range guarantees that it is the full cylinder component. + +Set again + +\[ +H=n^2. \tag{3.3} +\] + +The local anchor indicator then depends on `O(nH)` SITE variables, with constants depending only on `u` and the chosen graph. + +## 4. Uniform localization is unchanged at exponential scale + +Fix a compact subcritical parameter interval `I` for the transformed graph. The graph is exactly isomorphic to the original finite-range graph, so planar one-arm decay is available uniformly on `I`. + +Partition a long vertical traversal into slabs of height `c_u n` separated enough to have disjoint SITE supports. A traversal of one slab forces a planar arm of radius `c'_u n` from one of `O_u(n)` entry sites. Therefore + +\[ +P(\text{cross one prescribed }c_un\text{ slab}) +\le e^{-c_I n}. \tag{4.1} +\] + +Crossing vertical distance `H` requires a linear number `H/n` of disjoint such slab events, hence + +\[ +\boxed{ +P(\text{component height}>H) +\le C_I e^{-c_I H}} \tag{4.2} +\] + +for `H>=n`, after changing constants. With `H=n^2`, localization error is superexponentially small on the `n` scale. + +The same block construction gives the cylinder cluster-volume tail used later: + +\[ +\boxed{ +P_p^{C_n\times\mathbb Z}(|C_v|\ge k) +\le C_Ie^{-c_Ik}} \tag{4.3} +\] + +uniformly for large `n`, `p in I`. + +The only change from the axis proof is a fixed enlargement of blocks/guards by the transformed interaction range. + +## 5. Component density has the directional mass exponent + +The local upper bound uses the fixed-direction torus/cylinder first-exit estimate: a winding component in an `H`-window contains a nonzero horizontal homology witness and therefore costs + +\[ +\exp[-(\xi_{G,u}(p)-o(1))n] \tag{5.1} +\] + +up to polynomial factors. + +For the lower bound, use the fixed finite directional seed from the centre theorem and repeat it around the circumference, closing the seam. Unless its full component has height greater than `H`, one of at most `H+O_u(1)` anchor levels contains the resulting component. The long-component error is controlled by (4.2). + +Therefore, uniformly for moving `p_n->p` inside a compact subcritical interval, + +\[ +\boxed{ +-\frac1n\log\nu^{G,u}_n(p_n)\to\xi_{G,u}(p).} \tag{5.2} +\] + +No directional OZ amplitude is used. + +## 6. Chen--Stein Poisson approximation survives finite-range memory + +The truncated anchor window has vertical size `H+O_u(1)` and horizontal circumference `n`. Two anchors are independent when these windows are vertically separated by more than `2H+O_u(1)`. + +The dependency neighbourhood size is therefore + +\[ +D_n=O_u(nH)=O_u(n^3). \tag{6.1} +\] + +For overlapping anchor windows, two distinct complete winding components yield two disjoint occupied winding witnesses. Enclose each by the increasing event that the enlarged local band contains a horizontal winding. Site BK gives the same square bound as in the axial proof. + +Because every enclosing winding probability is + +\[ +B_n(p)\le \operatorname{poly}_u(n,H) + e^{-(\xi_{G,u}(p)-o(1))n}, \tag{6.2} +\] + +the Arratia--Goldstein--Gordon `b1,b2` terms satisfy + +\[ +b_1+b_2 +\le m\operatorname{poly}_u(n)B_n(p)^2. \tag{6.3} +\] + +Near the centre, choose a compact interval on which + +\[ +2\inf_I\xi_{G,u}>d. \tag{6.4} +\] + +Then with `log m/n->d`, (6.3) tends to zero exponentially. Localization is smaller still. + +Hence the true complete winding-component count satisfies + +\[ +\boxed{ +Z_{n,m}^{G,u}(p) +\overset{TV}=\operatorname{Poi}(m\nu_n^{G,u}(p))+o(1)} \tag{6.5} +\] + +uniformly on that compact interval. + +The process version gives a homogeneous Poisson process of anchor positions in the transformed longitudinal coordinate whenever `m nu_n -> lambda`. + +## 7. Same-label lower/upper windows still decouple + +Use one common uniform label field. Near the lower NN birth, black means `U<=p_1`; near the upper birth, white matching means `U>p_2`, with `p_1a`, + +\[ +\boxed{ +F_n'(p_n)\to-\xi_{G,u}'(a).} \tag{8.8} +\] + +Thus the directional mass slope required for affine parameter scaling is obtained from the component intensity itself; no prefactor expansion is required. + +## 9. Median-centred Gumbel theorem in a fixed direction + +Call `d` regular for the fixed direction `u` if both + +\[ +\xi_{4,u}'(a_u(d)) +\quad\text{and}\quad +\xi_{8,u}'(c_u(d)) \tag{9.1} +\] + +exist. Define positive slopes + +\[ +v_{4,u}=-\xi_{4,u}'(a_u(d)), +\qquad +v_{8,u}=-\xi_{8,u}'(c_u(d)). \tag{9.2} +\] + +Let `a_{n,m}` and `b_{n,m}` be the **true finite medians** of the first and second NN rank births. + +Then the same intensity-clock argument as in `poisson-birth-windows.md` gives the joint limit + +\[ +\boxed{ +X_n=v_{4,u}n(T_1-a_{n,m}), +\qquad +Y_n=v_{8,u}n(T_2-b_{n,m}),} \tag{9.3} +\] + +with + +\[ +\boxed{ +P(X\le x)=1-2^{-e^x}, +\qquad +P(Y\le y)=2^{-e^{-y}},} \tag{9.4} +\] + +and `X,Y` independent in the limit. + +The same uniform-integrability proof gives the mean, variance, covariance and IQR consequences with `w` replaced by `n` and `v_G` by `v_{G,u}`. + +In physical Euclidean width + +\[ +\ell=n|u|, \tag{9.5} +\] + +we have + +\[ +v_{G,u}n +=\ell\,[-\partial_p\tau_{G,p}(e)] \tag{9.6} +\] + +at differentiability points, so the theorem has the expected coordinate-free scaling. + +## 10. Exceptional set + +Local semiconcavity implies that each one-dimensional function `p -> xi_{G,u}(p)` is differentiable except at at most countably many `p` values. + +Since the mass is a strict bijection, the corresponding exceptional `d` set is at most countable. No SITE `p`-analyticity is asserted. + +At an exceptional `d`, the natural intensity clock remains valid; the unique affine `1/n` scaling in `p` is not promoted without a derivative. + +## 11. Why varying directions are not automatically covered + +For `u_n` changing with `n`, an `SL_2(Z)` straightening produces transformed step sets whose ranges may grow with `n`. The guard-slab thickness, local dependency range, block-tail constants and component-Palm boundary constant are then no longer uniform for free. + +The centre theorem survives because the first-exit Wulff argument is coordinate-free. The Poisson/Gumbel proof needs stronger local uniformity and is therefore kept separate. + +A future varying-direction window theorem should either: + +1. prove uniform finite-range controls in a geometrically bounded straightening scheme; or +2. work directly in the physical coordinates with a transverse anchor coordinate and uniform local-component estimates. + +Neither step is silently assumed here. + +## 12. Claim boundary + +This note is an author-level extension of the already author-level no-prefactor Poisson/Gumbel proof. The scientific novelty of the extension is the exact integer-coordinate reduction to a fixed anisotropic finite-range SITE graph and the observation that every ingredient in the axial proof is stable under that fixed finite-range change. + +No OZ prefactor, critical near-window uniformity, or square-site `p`-analyticity is used. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/fixed-width-charge-exact-polynomials-20260914.md b/docs/manuscripts/geometric-balance/fixed-width-charge-exact-polynomials-20260914.md new file mode 100644 index 000000000..3e9f50da9 --- /dev/null +++ b/docs/manuscripts/geometric-balance/fixed-width-charge-exact-polynomials-20260914.md @@ -0,0 +1,118 @@ +# Exact algebraic charge roots at widths 2, 3 and 4 + +2026-09-14. Symbolic regression controls for the transparent safe transfer. + +The purpose is not threshold estimation. These small-width polynomials are exact fingerprints of the lifted homology convention and of the NN/complementary-matching charge-sector criterion. + +## 1. Setup + +Let + +\[ +K_{4,w}(p),\qquad K_{8,w}(1-p) \tag{1.1} +\] + +be the two safe frontier kernels. Their Perron roots are `lambda_4,lambda_8`. The fixed-width charge root solves + +\[ +\lambda_4(p)=\lambda_8(1-p). \tag{1.2} +\] + +For small `w`, every matrix entry is an integer polynomial in `p`, so one can eliminate `lambda` exactly from the two characteristic equations. Trivial endpoint/zero-eigenvalue factors are discarded only after symbolic factorization; the physical factor is identified by containing the Perron crossing root in `(0,1)`. + +## 2. Width two + +The NN safe characteristic polynomial is + +\[ +\chi_{4,2}(\lambda) +=\lambda^2(\lambda+p^2-1). \tag{2.1} +\] + +The complementary matching safe polynomial is + +\[ +\chi_{8,2}(\lambda) +=(\lambda+p^2-p) +(\lambda^2-p\lambda+p^4-p^3). \tag{2.2} +\] + +Eliminating `lambda` gives endpoint powers times + +\[ +\boxed{F_2(p)=2p^3+2p^2-1.} \tag{2.3} +\] + +Its unique root in `(0,1)` is + +\[ +p_2^{ch}=0.5651977173836394\ldots. \tag{2.4} +\] + +The lifted width-two convention is essential here: the two physically distinct horizontal bonds are retained rather than collapsed into one quotient edge. + +## 3. Width three + +The NN characteristic polynomial collapses to + +\[ +\chi_{4,3}(\lambda) +=\lambda^6(\lambda+p^3-1). \tag{3.1} +\] + +The matching polynomial factorizes into one squared quadratic block and one cubic Perron block. Eliminating the NN Perron factor with the matching cubic yields the physical factor + +\[ +\boxed{ +F_3(p) +=p^6-3p^5-5p^4-4p^3+p+1.} \tag{3.2} +\] + +The other nontrivial resultant factor has no physical Perron crossing in `(0,1)`. + +The unique physical root is + +\[ +p_3^{ch}=0.5888806999178535\ldots. \tag{3.3} +\] + +## 4. Width four + +The 19-state NN kernel has a Perron eigenvalue contained in a quadratic characteristic factor. The 19-state complementary matching kernel has its Perron eigenvalue in a quintic factor. Their exact resultant contains endpoint powers and the degree-17 physical polynomial + +\[ +\boxed{\begin{aligned} +F_4(p)={}&28p^{17}-98p^{16}-34p^{15}+286p^{14}+122p^{13} +-320p^{12}-362p^{11}\\ +&+117p^{10}+377p^9+144p^8-134p^7-174p^6-5p^5\\ +&+37p^4+14p^3-10p^2+p+2. +\end{aligned}} \tag{4.1} +\] + +It has exactly one real root in `(0,1)`: + +\[ +p_4^{ch}=0.5914171708531392\ldots. \tag{4.2} +\] + +## 5. Relation to the semi-infinite graph-polynomial sequence + +These three algebraic numbers agree with the first corresponding values of the square-site `n x infinity` eigenvalue-identity sequence reported by Jacobsen. This is another exact/near-exact bridge between the Bernoulli homology-safe transfer and the graph-polynomial topological sectors. + +The present calculation does not imply that the symbolic polynomials `F_w` are the minimal graph-polynomial factors in Jacobsen's variables; changes of local weight variable and elimination can add or remove algebraic factors. The invariant statement is the physical Perron crossing. + +## 6. Why these are useful regression controls + +Any proposed rewrite of the safe transfer into + +- no-zero-block type-B noncrossing states, +- dilute periodic TL states, +- a more compact canonical annular basis, + +should reproduce (2.3), (3.2), and (4.1) exactly after the declared change of variables. + +This is stricter than comparing decimal roots: an incorrect multiplicity, seam gain, matching diagonal, or forbidden-winding convention usually changes the symbolic factor immediately. + +## 7. Claim boundary + +The polynomials above come from exact symbolic characteristic/resultant elimination of the transparent safe kernels. They are finite-width identities, not asymptotic CFT claims and not new threshold estimates. diff --git a/docs/manuscripts/geometric-balance/fixed-width-charge-free-energy-20260914.md b/docs/manuscripts/geometric-balance/fixed-width-charge-free-energy-20260914.md new file mode 100644 index 000000000..a49a14fa6 --- /dev/null +++ b/docs/manuscripts/geometric-balance/fixed-width-charge-free-energy-20260914.md @@ -0,0 +1,363 @@ +# Fixed-width charge-sector free energy and the infinite-length balance root + +2026-09-14. A one-dimensional transfer/free-energy explanation of “balance without concentration.” The order of limits in the main theorem is different, but the fixed-width `m->infinity` problem cleanly identifies what the matching root is balancing. + +## 1. Trivial-homology survival in an open strip + +Fix a circumference `w` and graph `G=G4` or `G8`. On the open vertical strip + +\[ +C_w\times\{1,\ldots,m\}, \tag{1.1} +\] + +let + +\[ +Q^G_{w,m}(p) +=P_p(\text{there is no horizontally essential occupied component}).\tag{1.2} +\] + +There is no vertical periodic identification in this strip; the only ambient homology to forbid is horizontal winding around `C_w`. + +### 1.1 Submultiplicative upper inequality + +If a strip of height `m+n` has no horizontal essential component, then neither its first `m` rows nor its last `n` rows has one. These two subevents use disjoint row variables. Therefore + +\[ +\boxed{ +Q_{w,m+n}^G(p) +\le Q_{w,m}^G(p)Q_{w,n}^G(p).} \tag{1.3} +\] + +Thus + +\[ +a_m=-\log Q_{w,m}^G(p) \tag{1.4} +\] + +is superadditive and Fekete gives the limit + +\[ +\boxed{ +I^0_{G,w}(p) +:=\lim_{m\to\infty}-\frac1m\log Q_{w,m}^G(p) +=\sup_m-\frac1m\log Q_{w,m}^G(p).} \tag{1.5} +\] + +### 1.2 Reverse inequality up to one empty separator row + +Let + +\[ +\delta_{w,p}=(1-p)^w>0. \tag{1.6} +\] + +Force one whole row between two strips to be empty. For both NN and matching adjacency, one empty row separates occupied components above and below because all vertical jumps have size one. + +Hence + +\[ +\boxed{ +Q_{w,m+n+1}^G(p) +\ge\delta_{w,p}Q_{w,m}^G(p)Q_{w,n}^G(p).} \tag{1.7} +\] + +So the strip survival probability has a genuine one-dimensional free energy with only `O(1)` concatenation cost. + +## 2. The torus rank-zero probability has the same exponential rate + +Let + +\[ +P^G_{0;w,m}(p) +=P_p^{C_w\times C_m}(r_G=0). \tag{2.1} +\] + +### Upper bound + +A rank-zero torus configuration has no horizontal essential component. Cut the torus between two rows and forget the vertical periodic edges. The resulting open strip still has no horizontal winding. Therefore + +\[ +\boxed{P^G_{0;w,m}(p)\le Q^G_{w,m}(p).} \tag{2.2} +\] + +### Lower bound + +Force one specified torus row to be completely empty. Cut there. If the remaining open strip has no horizontal essential component, the torus has no horizontal homology, and the empty row also destroys every possible vertical homology cycle. + +Thus + +\[ +\boxed{ +P^G_{0;w,m}(p) +\ge\delta_{w,p}Q^G_{w,m-1}(p).} \tag{2.3} +\] + +Combining (1.5), (2.2), and (2.3), + +\[ +\boxed{ +\lim_{m\to\infty} +-\frac1m\log P^G_{0;w,m}(p) +=I^0_{G,w}(p).} \tag{2.4} +\] + +This statement is exact for every fixed `w` and interior `p`. + +## 3. Finite-state Perron representation + +For fixed `w`, horizontal-winding avoidance is recognized by the standard finite frontier connectivity/homology state used throughout the repository. + +Reject a transition as soon as horizontal homology is created. This gives a finite nonnegative substochastic transfer matrix + +\[ +T^0_{G,w}(p). \tag{3.1} +\] + +Its entries are polynomials in `p` and `1-p`. + +The all-empty row gives a reset state accessible with positive probability from every safe frontier state. Restricting to reachable/co-reachable safe states therefore produces a primitive Perron block. Hence + +\[ +\boxed{ +I^0_{G,w}(p)=-\log\lambda^0_{G,w}(p),} \tag{3.2} +\] + +where `lambda^0` is the Perron root of that block. + +For `0 infinity + +Let + +\[ +p^*_{w,m} \tag{6.1} +\] + +be the unique finite-torus root where + +\[ +P_2=P_0. \tag{6.2} +\] + +Fix any `epsilon>0`. By strict monotonicity of `Theta_w`, + +\[ +\Theta_w(p^{ch}_w-\epsilon)<0, +\qquad +\Theta_w(p^{ch}_w+\epsilon)>0. \tag{6.3} +\] + +Equation (4.3) then gives, for sufficiently large `m`, + +\[ +\theta_{w,m}(p^{ch}_w-\epsilon)<0, +\qquad +\theta_{w,m}(p^{ch}_w+\epsilon)>0. \tag{6.4} +\] + +Since the finite fugacity is strictly increasing and its root is `p^*_{w,m}`, + +\[ +\boxed{ +p^*_{w,m}\longrightarrow p^{ch}_w +\qquad(m\to\infty).} \tag{6.5} +\] + +If one additionally extracts the Perron amplitudes and `Theta_w'(p^{ch}_w)>0`, standard analytic transfer asymptotics predict an `O(1/m)` root correction. That quantitative rate is not needed for (6.5). + +## 7. Then the charge-coexistence point tends to the planar pc + +The geometric-balance manuscript proves balance-root consistency for any honest torus sequence whose Euclidean shortest period tends to infinity. + +Choose widths + +\[ +w_j\to\infty. \tag{7.1} +\] + +For each `w_j`, choose `m_j` large enough that + +\[ +|p^*_{w_j,m_j}-p^{ch}_{w_j}|<1/j \tag{7.2} +\] + +and `m_j>=w_j`. + +The tori `C_{w_j}xC_{m_j}` have shortest period `w_j->infinity`, so root consistency gives + +\[ +p^*_{w_j,m_j}\to p_c(G4). \tag{7.3} +\] + +Therefore + +\[ +\boxed{p^{ch}_w\longrightarrow p_c(G4)\qquad(w\to\infty).} \tag{7.4} +\] + +This is an iterated-limit theorem requiring no critical exponent or birth-law concentration. + +## 8. Explanation of balance without concentration + +In an exponentially elongated torus, the two rank births can converge to distinct constants + +\[ +a(d)infinity` then moves that charge-coexistence point to the planar critical parameter. + +So there is no contradiction between: + +- a broad rank-one plateau / nonconcentrated birth law; +- a sharply defined matching root. + +They live in different spectral sectors. + +## 9. Relation to the exact charge coordinates + +Recall + +\[ +M=\chi\tanh(\theta/2). \tag{9.1} +\] + +At fixed `w` and large `m`, + +\[ +\theta\sim m\Theta_w(p). \tag{9.2} +\] + +while the charged susceptibility `chi=P0+P2` is exponentially small away from the zero of `Theta_w`. + +Near `p^{ch}_w`, if `Theta'_w>0`, the natural charge crossover variable is + +\[ +\boxed{x=m\Theta'_w(p^{ch}_w)(p-p^{ch}_w).} \tag{9.3} +\] + +The actual limiting `chi` and neutral count law require transfer/Perron amplitudes; the **location scale** of the charge sign change is already visible from the free-energy crossing. + +This is a different window from the lower/upper winding-component Gumbel windows in `w`. + +## 10. A new exact-computation opportunity + +For modest fixed widths, `lambda^0_{G,w}(p)` can be computed from a safe homology transfer matrix without tracking component ages. One can therefore obtain `p^{ch}_w` as the root of + +\[ +\boxed{ +\lambda^0_{4,w}(p) +=\lambda^0_{8,w}(1-p).} \tag{10.1} +\] + +This is numerically far cleaner than finding a finite-`m` root from two extremely small probabilities. + +It also gives a new width sequence approaching `p_c` that is independent of the existing threshold-rank finite-torus root estimator. Agreement of the two sequences after taking `m` large would be a strong transfer/topology cross-check. + +## 11. Claim boundary + +Existence of the strip/torus void free energy follows from the elementary concatenation bounds. Analytic Perron representation and strictness are standard finite-state transfer consequences but should receive code/proof review if promoted as a production estimator. The limit `p^{ch}_w->pc` uses the parent manuscript's author-level root-consistency theorem via a diagonal sequence; no claim of a uniform-in-`w` rate for `p^*_{w,m}->p^{ch}_w` is made. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/fixed-width-charge-jacobsen-bridge-20260914.md b/docs/manuscripts/geometric-balance/fixed-width-charge-jacobsen-bridge-20260914.md new file mode 100644 index 000000000..f5b3513a7 --- /dev/null +++ b/docs/manuscripts/geometric-balance/fixed-width-charge-jacobsen-bridge-20260914.md @@ -0,0 +1,185 @@ +# Digital-Alexander interpretation of the Jacobsen semi-infinite-cylinder eigenvalue identity + +2026-09-14. Literature bridge for the fixed-width charge-free-energy calculation. + +## 1. The numerical sequence is known + +The charge-coexistence roots obtained independently from the safe site-frontier transfer are + +\[ +\lambda^0_{4,w}(p)=\lambda^0_{8,w}(1-p). \tag{1.1} +\] + +For `w=2,...,8` the roots are + +```text +0.5651977173836393 +0.5888806999178529 +0.5914171708531385 +0.5922358232050263 +0.5925073562056372 +0.5926196333998958 +0.5926727605746284 +``` + +They agree to the displayed precision with Table 2 of J. L. Jacobsen, arXiv:1507.03027, where the square-site thresholds are computed on `n x infinity` bases by equating the largest eigenvalues of two topologically distinct transfer sectors. + +Therefore this sequence is **not a new threshold estimator**. The useful contribution of the present construction is a probability/topology dictionary for that eigenvalue identity. + +## 2. The two sector languages + +Jacobsen's Potts/FK formulation compares + +- an `open` sector with a propagating FK cluster; +- a `closed` sector with a propagating dual FK cluster. + +At `q=1`, their largest eigenvalues define the semi-infinite-cylinder pseudo-critical point. + +The site probability formulation instead starts with finite-torus homology ranks and exact digital Alexander duality + +\[ + r_4(\omega)+r_8(\omega^c)=2. \tag{2.1} +\] + +Consequently + +\[ + P_p^{4}(r=2)=P_{1-p}^{8}(r=0). \tag{2.2} +\] + +After cutting one empty separator row, `r=0` has the same exponential rate as survival in the open cylinder without any horizontal essential occupied component. The safe frontier kernel therefore has Perron root + +\[ +\lambda^0_{G,w}(p),\qquad +I^0_{G,w}(p)=-\log\lambda^0_{G,w}(p). \tag{2.3} +\] + +The finite matching charge fugacity + +\[ +\theta_{w,m}(p)=\log\frac{P_2(p)}{P_0(p)} \tag{2.4} +\] + +has thermodynamic rate + +\[ +\frac1m\theta_{w,m}(p)\to +\Theta_w(p)=I^0_{4,w}(p)-I^0_{8,w}(1-p). \tag{2.5} +\] + +Hence the semi-infinite-cylinder balance criterion is exactly + +\[ +\boxed{\Theta_w(p)=0.} \tag{2.6} +\] + +This is the site/digital-Alexander form of the open/closed eigenvalue identity. + +## 3. Why this dictionary matters + +The FK transfer statement says two sector eigenvalues cross. The probability statement adds three pieces of interpretation. + +1. **Finite-event meaning.** The two sectors are the exponential tails of the two charged endpoint events `r=0` and `r=2` of the SAME rank observable. +2. **Complement map.** The upper charged sector is not an independent second model; digital Alexander converts it exactly to rank-zero survival of complementary matching sites. +3. **Balance without concentration.** The root can remain sharply selected by the sign of a difference of two exponentially small void-sector free energies even when the ordinary two-birth distribution has a wide rank-one plateau. + +This is also why the root should not be described as equality of black and white winding-component intensities. Those cylinder intensities are paired by alternation and can be equal while their long-gap/void free energies differ. + +## 4. Perron derivative gives a new probabilistic slope observable + +Normalize the positive left/right Perron eigenvectors of the safe transfer into its Doob/quasi-stationary row law. Let + +\[ +\bar K^0_{G,w}(p) +\] + +be the mean number of occupied sites added in one row under that conditioned safe phase. Differentiating the Bernoulli row weights gives + +\[ +\boxed{ +(I^0_{G,w})'(p) +=\frac{wp-\bar K^0_{G,w}(p)}{p(1-p)}.} \tag{4.1} +\] + +Therefore at the charge root + +\[ +\boxed{ +\Theta'_w(p_w^{ch}) +=\frac{w-\bar K^0_{4,w}(p_w^{ch}) +-\bar K^0_{8,w}(1-p_w^{ch})} +{p_w^{ch}(1-p_w^{ch})}.} \tag{4.2} +\] + +The transparent transfer oracle checks this identity internally through the Perron derivative. This gives a direct physical interpretation of the eigenvalue-crossing slope as an occupation deficit of the two conditioned topological sectors. + +## 5. Direct finite-size decomposition of the n^-4 shift + +Using the repository reference `p_c=0.59274605079` only as a diagnostic, + +\[ +(p_c-p_w^{ch})w^4 +=0.3402,0.3189,0.3093,0.3035,0.3002 \tag{5.1} +\] + +for `w=4,...,8`. This is the same `Delta_1=4` behaviour reported by Jacobsen. + +The new separation is + +\[ +\Theta_w(p_c)w^{17/4} +=1.1860,1.0996,1.0600,1.0358,1.0216, \tag{5.2} +\] + +while + +\[ +\Theta'_w(p_w^{ch})w^{1/4} +=3.4851,3.4479,3.4262,3.4124,3.4031. \tag{5.3} +\] + +Thus the pseudo-critical shift is visibly the ratio + +\[ +p_c-p_w^{ch} +\approx\frac{\Theta_w(p_c)}{\Theta'_w(p_w^{ch})}, \tag{5.4} +\] + +with exponents `17/4 - 1/4 = 4`. + +This motivates the separate Kac-(4,2) conjecture in `sector-odd-kac42-conjecture-20260914.md`; it is not needed for the exact Jacobsen/digital-Alexander equivalence. + +## 6. Magnetic-gap cross-check + +Jacobsen's CFT argument notes that open and closed sectors both determine the magnetic exponent + +\[ +x_m=5/48. \tag{6.1} +\] + +For the square lattice, the per-row excitation cost therefore predicts + +\[ +wI^0_{G,w}(p_c)\to2\pi x_m=0.65449846949\ldots. \tag{6.2} +\] + +The safe site transfer gives + +```text +w G4 G8 complement +4 0.6808677452 0.6677642268 +5 0.6702211440 0.6643381257 +6 0.6651785592 0.6620431213 +7 0.6622847725 0.6604282152 +8 0.6604615126 0.6592750557 +``` + +Both sectors approach the same magnetic gap while their difference is parametrically smaller. This is an independent semantic check that the safe kernels are selecting the intended topological sectors. + +## 7. Literature boundary + +Primary comparison: J. L. Jacobsen, *Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley-Lieb algebras*, arXiv:1507.03027 / J. Phys. A 48 (2015) 454003. + +That paper already contains the `n x infinity` root sequence, the open/closed eigenvalue criterion, the shared magnetic-exponent argument, and the empirical correction exponents `4,6,8,...`. It explicitly remarks that equality of the leading magnetic exponent by itself does not derive the `n^-4` pseudo-critical shift. + +Accordingly the present branch should claim only the new digital-Alexander/probability interpretation, the safe-site reference implementation, the Perron occupation-slope identity, and any separately audited CFT mechanism—not rediscovery of the threshold sequence. diff --git a/docs/manuscripts/geometric-balance/fixed-width-charge-perron-slope-20260914.md b/docs/manuscripts/geometric-balance/fixed-width-charge-perron-slope-20260914.md new file mode 100644 index 000000000..e79fdca6f --- /dev/null +++ b/docs/manuscripts/geometric-balance/fixed-width-charge-perron-slope-20260914.md @@ -0,0 +1,316 @@ +# Perron score formula for the fixed-width charge root + +2026-09-14. Exact finite-state refinement of `fixed-width-charge-free-energy-20260914.md`. Once the safe no-horizontal-homology transfer matrix is built, both the infinite-length charge coexistence point and its slope can be read from Perron data without numerical differentiation of exponentially small torus probabilities. + +## 1. Safe transfer matrix with row-resolved weights + +Fix circumference `w` and graph `G` (NN or matching). Let + +\[ +T^0_{G,w}(p) \tag{1.1} +\] + +be the finite nonnegative transfer matrix on safe frontier states, rejecting a transition as soon as horizontal homology is created. + +Resolve each matrix entry by the newly added row mask `M`: + +\[ +T^0_{ij}(p) +=\sum_M A_{ij}(M) + p^{k(M)}(1-p)^{w-k(M)}, \tag{1.2} +\] + +where `A_ij(M)` is `0/1` (or a finite multiplicity if the state representation intentionally aggregates equivalent microscopic transitions) and + +\[ +k(M)=\text{number of occupied sites in the added row}. \tag{1.3} +\] + +Let + +\[ +\lambda^0_{G,w}(p) \tag{1.4} +\] + +be the Perron root of the primitive reachable/co-reachable safe block, with positive left/right eigenvectors `l,r` normalized by + +\[ +l^Tr=1. \tag{1.5} +\] + +The no-homology free energy is + +\[ +I^0_{G,w}(p)=-\log\lambda^0_{G,w}(p). \tag{1.6} +\] + +## 2. Perron edge/mask measure + +Define a probability measure on safe one-row transitions by + +\[ +\boxed{ +\mathbb Q^0_{G,w,p}(i,M,j) +=\frac{l_i A_{ij}(M) + p^{k(M)}(1-p)^{w-k(M)}r_j}{\lambda^0}.} \tag{2.1} +\] + +Normalization follows from + +\[ +l^TTr=\lambda^0 l^Tr=\lambda^0. \tag{2.2} +\] + +This is the stationary one-step law of the Perron/Doob process conditioned to survive indefinitely in the safe homology sector, with the microscopic added row retained as a mark. + +Write + +\[ +\bar K^0_{G,w}(p) +=E_{\mathbb Q^0}[k(M)]. \tag{2.3} +\] + +This is **not** the ordinary Bernoulli mean `wp`; it is the row occupation under the quasi-stationary no-winding phase. + +## 3. Exact derivative of the void free energy + +Differentiate the Perron root: + +\[ +(\lambda^0)'=l^T(T^0)'r. \tag{3.1} +\] + +For one row mask, + +\[ +\partial_p\log[p^k(1-p)^{w-k}] +=\frac{k}{p}-\frac{w-k}{1-p} +=\frac{k-wp}{p(1-p)}. \tag{3.2} +\] + +Therefore + +\[ +\boxed{ +\partial_p\log\lambda^0_{G,w}(p) +=\frac{\bar K^0_{G,w}(p)-wp}{p(1-p)}.} \tag{3.3} +\] + +Since `I^0=-log lambda^0`, + +\[ +\boxed{ +(I^0_{G,w})'(p) +=\frac{wp-\bar K^0_{G,w}(p)}{p(1-p)}.} \tag{3.4} +\] + +Thus strict increase of the no-homology free energy is equivalent to the intuitive quasi-stationary depletion inequality + +\[ +\boxed{\bar K^0_{G,w}(p)0`. The finite torus root `p^*_{w,m}` therefore satisfies + +\[ +\boxed{ +p^*_{w,m} +=p_w^{ch} +-\frac{B_w(p_w^{ch})}{m\Theta_w'(p_w^{ch})} ++O(m^{-2})+O(\gamma_w^m/m),} \tag{6.6} +\] + +provided the amplitudes and Perron data are twice differentiable in a neighbourhood of the coexistence point. + +This gives a controlled alternative to fitting root drift directly in extremely small endpoint probabilities. + +## 7. Why the finite root CDF can be almost flat + +At a finite matching root, + +\[ +P_0=P_2=\chi/2, \tag{7.1} +\] + +and the exact charge-coordinate identity gives + +\[ +F'(p^*)=\frac14\chi(p^*)\theta'(p^*). \tag{7.2} +\] + +For fixed width and large `m`, (6.1)--(6.4) imply + +\[ +\chi(p^*) +=e^{-mI_w^*+O(1)}, \tag{7.3} +\] + +where + +\[ +I_w^*=I^0_{4,w}(p_w^{ch})=I^0_{8,w}(1-p_w^{ch})>0, \tag{7.4} +\] + +while + +\[ +\theta'(p^*) +=m\Theta_w'(p_w^{ch})+O(1). \tag{7.5} +\] + +Therefore + +\[ +\boxed{ +F'(p^*_{w,m}) +=\frac{m\Theta_w'(p_w^{ch})}{4} + e^{-mI_w^*+O(1)}.} \tag{7.6} +\] + +The birth CDF is exponentially flat at the balance root as `m->infinity`, even though the root location itself converges at order `1/m` to the unique free-energy crossing. + +This is the quantitative fixed-width version of **balance without concentration**. + +## 8. Practical computation + +A fixed-width implementation no longer needs to estimate tiny `P0,P2` directly. + +For each graph: + +1. build the safe no-horizontal-homology transfer block; +2. compute its Perron root `lambda^0(p)` and left/right eigenvectors; +3. solve + \[ + \lambda^0_{4,w}(p)=\lambda^0_{8,w}(1-p) \tag{8.1} + \] + for `p_w^ch`; +4. compute `Theta_w'` from the row-occupation formula (4.5), not finite differences; +5. if desired, compute the cut/closure Perron amplitudes for the `1/m` correction (6.6). + +This is a potentially high-precision new `p_c` approach through the iterated limit `m->infinity` then `w->infinity`, but no production estimate is claimed until the safe transfer implementation is independently cross-checked. + +## 9. Boundary + +Equations (2.1)--(4.5) are exact finite-state Perron identities once the safe primitive block is correctly constructed. The `1/m` correction uses a standard spectral-gap expansion and should be stated with the usual compact-interval simplicity/gap assumptions. No large-width convergence rate for `p_w^ch->p_c` is asserted. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/gamma-w-parameter-dependence-crosscheck-20260914.md b/docs/manuscripts/geometric-balance/gamma-w-parameter-dependence-crosscheck-20260914.md new file mode 100644 index 000000000..c7ef779bc --- /dev/null +++ b/docs/manuscripts/geometric-balance/gamma-w-parameter-dependence-crosscheck-20260914.md @@ -0,0 +1,183 @@ +# 任务 A 交付 — γ_w 参数依赖的**独立复核**（mass761b / DevEnvC_NePnUn） + +配套机读件：`gamma-crosscheck.json`、`validate.json`、`cylroot.json`、`gw-w248.json`、`gw-w9.json`。 + +机器：华为云 `DevEnvC_NePnUn`（账号 2），Python 3.9.9，numpy 2.0.2 / scipy 1.13.1（本机 pip 装好）， +`PYTHONPATH=/workspace/mo/compat`（`int.bit_count` 垫片，日志留证）。 +**本机 Mac 未做任何计算**；全部脚本上传到云机执行。 + +--- + +## 0. 我复核的对象与我的独立路径 + +复核目标：`cloud-mo-c` 的 `note-cloud-2.md` / `cloud-2.json` —— “`γ_w − κ = A e^{−cw}` 里的 `c` +**不普适**（依赖 p）”。 + +我用**两条我自己写的路径**重算，都没有用它的代码： + +| 路径 | 内容 | +|---|---| +| **R1** 精确路线 | `T.build/lump/numeric_system(exact=True)`（共享 #739 引擎）→ 我自己写的 `fw & bk` 受限集 → **我自己的迭代 Tarjan** → 每个 SCC 用 `numpy.linalg.eigvals` 稠密求谱半径 → 另用 `scipy.sparse.linalg.eigs` 稀疏交叉 → 再用**精确有理数 Collatz 包络**认证 | +| **R2** 快速路线 | 整数多项式核 `arr[k][i][j]`（`R(p)=Σ_k wt_k arr[k]`）+ 我的 Tarjan + 稠密 `eigvals`，可在任意 p 上廉价重算 | + +引擎：w≤8 用 pinned 原件 `52f3611990ce2b1331d9e5296e0262f5e402e0d7`；w=9 用 `w9-probe/gamma9/` 的 +**raise-only 放宽副本** `7a6963c77e47887c12ff065a6c7fcd85b7fa5098`（diff 只有一行 `width>8 → width>9`）。 +两个 blob 都在运行日志里被断言接受。 + +**γ_w 的定义（我确认过的口径）**：`rho_*` 是**受限**转移阵（`keep = 前向可达 ∩ 后向可达`）中 +**最大 SCC 的 Perron 根**，`γ_w = −log rho_*`。见 §4 一条重要澄清。 + +--- + +## 1. 独立校验（先把尺子校准） + +| 参照物 | 格数 | max\|我的 − 参照\| | +|---|---|---| +| **Astra 已发表件** `astra-shape-scales/.../winding-shape-scales.json`（w=2,4,6,8） | 8 | **1.98e-14** | +| 本地探针件 `w9-probe/gamma9/gamma-w9.json`（w=2,4,6,8,9） | 10 | 1.82e-14 | + +即逐位一致到 float64 舍入（`−log` 的末位）。**注意**：这两个文件里 `gamma_float` 存的是**完整 float64**， +不是 10 位小数（例：`1.2366356168608923`）。见 `issues-found.md` ISSUE 1。 +`cloud-mo-c` 报的 “4.25e-11 = 10 位舍入” 是拿 Astra **手稿表格里的 10 位显示值**当参照，不是拿这两个文件。 + +**另外一条强校验**：我用 R2 独立求解 `γ_w(p) = log 4` 的根，与作者 +`cylinder-mass-bounds.json` 的 `finite_root_diagnostic` **8/8 全部复现**，max|Δ| = 1.7e-15。 + +--- + +## 2. 逐格 γ_w（R1 精确路线，`(γ_lower+γ_upper)/2`；认证区间宽度 ~1e-16） + +| 图 | p | γ₂ | γ₄ | γ₆ | γ₈ | γ₉ | +|---|---|---|---|---|---|---| +| NN | 3/20 | 1.788794611032 | 1.667872159911 | 1.660217856085 | 1.659776653604 | 1.659757416624 | +| NN | 1/5 | 1.478135631659 | 1.329122579488 | 1.313433699602 | 1.311741471473 | 1.311608270480 | +| NN | 1/4 | 1.236635616861 | 1.065554869680 | 1.039677339832 | 1.035240621722 | 1.034699853840 | +| NN | 3/10 | 1.039956947218 | 0.852723626316 | 0.815797344144 | 0.806822669560 | 0.805299843389 | +| matching | 1/20 | 2.327902900978 | 1.923260212947 | 1.913951410491 | 1.913522523266 | 1.913506286343 | +| matching | 2/25 | 1.873403458269 | 1.472370746757 | 1.454922897285 | 1.453294184240 | 1.453177130706 | +| matching | 1/8 | 1.450832882257 | 1.058985221933 | 1.028003435812 | 1.022476375627 | 1.021755295888 | +| matching | 9/50 | 1.115961927003 | 0.740672944255 | 0.694147835751 | 0.680851477564 | 0.678157491574 | + +每个 SCC 的**唯一性 + 非周期（正对角）**都被认证，`unique=True`（8 图 × 5 宽度 = 40 格，0 失败）。 +`scipy.sparse.linalg.eigs` 与稠密 `eigvals` 一致到 ~1e-12。额外 3 个 p 值（NN 9/50、37/200、19/100； +matching 3/40、17/200、9/100）在 JSON 里。 + +--- + +## 3. `c` 的估计（递差比，几乎无参数） + +`(g_{w+2}−g_{w+4})/(g_w−g_{w+2}) = e^{−2c}`；`c(6,8,9)` 由 `u²/(1+u)=ratio, u=e^{−c}` 解出。 + +| 图 | p | c(2,4,6) | **c(4,6,8)** | **c(6,8,9)** | \|c468−c689\| | r(4,6,8) | 指数形式偏差 | +|---|---|---|---|---|---|---|---| +| NN | 3/20 | 1.379941 | **1.426760** | **1.462119** | 0.035359 | 0.057641 | 6.20% | +| NN | 1/5 | 1.125541 | **1.113453** | **1.131148** | 0.017695 | 0.107862 | 3.06% | +| NN | 1/4 | 0.944380 | **0.881730** | **0.878655** | 0.003075 | 0.171451 | 0.53% | +| NN | 3/10 | 0.811716 | **0.707258** | **0.682387** | 0.024871 | 0.243043 | 4.23% | +| NN | 9/50 | 1.216483 | 1.226976 | 1.252167 | 0.025191 | 0.085953 | 4.37% | +| NN | 37/200 | 1.192555 | 1.197279 | 1.220649 | 0.023370 | 0.091213 | 4.05% | +| NN | 19/100 | 1.169447 | 1.168485 | 1.189997 | 0.021513 | 0.096620 | 3.73% | +| matching | 1/20 | 1.886022 | **1.538761** | **1.539820** | 0.001059 | 0.046073 | 0.19% | +| matching | 2/25 | 1.567413 | **1.185713** | **1.182813** | 0.002900 | 0.093347 | 0.51% | +| matching | 1/8 | 1.268737 | **0.861872** | **0.838700** | 0.023172 | 0.178397 | 4.02% | +| matching | 9/50 | 1.043852 | **0.626251** | **0.575031** | 0.051220 | 0.285789 | 8.80% | +| matching | 3/40 | 1.611342 | 1.233961 | 1.232540 | 0.001421 | 0.084761 | 0.25% | +| matching | 17/200 | 1.526207 | 1.140583 | 1.135977 | 0.004606 | 0.102165 | 0.81% | +| matching | 9/100 | 1.487450 | 1.098253 | 1.091745 | 0.006507 | 0.111191 | 1.14% | + +“指数形式偏差” = `|实测 r(8,9) / 用 c(4,6,8) 预测的 r(8,9) − 1|`。 + +--- + +## 4. 三档裁定 + +**裁定：`不支持` “c 与 p 无关”（两图都是）。强度：强。** + +| 图 | p 方向散布 (c468) | 宽度子集不确定度 | 比值 | 单调递减 | +|---|---|---|---|---| +| NN（4 点网格） | 0.719501 | 0.021283（中位，取两中值均值）／0.024871（取上中值，= 他的口径） | **33.8 / 28.9** | ✔（4 点、7 点都单调） | +| NN（7 点网格） | 0.719501 | 0.023370 | **30.8** | ✔ | +| matching（4 点网格） | 0.912510 | 0.013036／0.023172（他的口径） | **70.0 / 39.4** | ✔ | +| matching（7 点网格） | 0.912510 | 0.004606 | **198.1** | ✔ | + +- 散布 / 不确定度 ≈ **29–198**，远大于 1；且**递减在 {2,4,6} 与 {4,6,8} 两个窗口下都成立**（窗口无关）。 +- 我补的 3 个 p 值把 matching 的 p 网格填密，**不确定度中位数从 0.0232 降到 0.0046**， + 比值反而升到 198 —— 结论只会更硬，不会更软。 +- **口径差异（不影响裁定，但要写清）**：`cloud-mo-c` 的“中位数”取 4 个值里的**上中值**， + 数学上的中位数是两中值的平均。用后者，NN/matching 的比值是 **33.8 / 70.0**，而不是 28.9 / 39.4。 + 两者都支持“不支持”。 + +**“c 与图无关”？仍无法判定。** 两图 p 网格几乎不重叠（matching 到 0.18，NN 从 0.15 起）， +而 c 对 p 极敏感。我的新数据：p≈0.18 处 matching c468=0.626 与 NN p=0.20 的 1.113 不同， +但这不是同 p 比较，**不足以下结论**。这一点我与 `cloud-mo-c` 一致。 + +### 三条声称的逐条复核 + +| 声称 | 结论 | 我的数字 | +|---|---|---| +| 对已发表 γ 值 **10/10 逐位复现** | **复现，且远强于其说法** | 对 **Astra 已发表件** 8/8 格 max\|Δ\|=**1.98e-14**；对 `gamma-w9.json` 10/10 格 max\|Δ\|=1.82e-14。**不存在 4.25e-11 这个量级**（他的 4.25e-11 是拿 10 位显示值比的）。 | +| **指数形式相容**（预测 r(8,9) 偏差 0.2%–8.8%） | **复现** | NN 0.53/3.06/4.23/6.20%，matching 0.19/0.51/4.02/8.80% —— 与他**逐位相同**。 | +| **纯 A/w 模型被排除**（预测 r(4,6,8)=0.5） | **复现** | 实测 r(4,6,8) ∈ [0.0576, 0.2430]（NN）、[0.0461, 0.2858]（matching），远离 0.5。`γ=κ+A/w` 时 `(g_{w+2}−g_{w+4})/(g_w−g_{w+2})=1/2` 是**恒等式**，与 A、κ 无关，我的读数与他一致。 | + +### 与他数字的逐格比对（引用他的 `cloud-2.json`，非重算） + +max|Δγ| = **8.88e-16**（= float64 末位），max|Δc(4,6,8)| = **2.52e-13**。 +每个 (图,p) 的逐宽度差与 c 差都在 `gamma-crosscheck.json` 的 `vs_cloud_mo_c` 里。 +**结论：他的 γ 与 c 表我逐格独立复现，误差在机器精度量级。** + +--- + +## 5. 一条重要的口径澄清（不是 bug，但会改变措辞） + +我对每个 (图,p,w) 都打印了 `viable_states / all_lumps`：**161/161、152/152、352/352、337/337 ……** +**全部相等**。也就是说 `keep = fw ∩ bk` 是**恒等**： + +- 引擎 lump 出来的每个状态本来就同时“源可达”且“可退休可达”； +- 于是“**受限**转移阵”= 完整 lumped 矩阵，而“最大 SCC 谱半径”= **整矩阵谱半径** + （块上三角 ⇒ 特征值 = 各对角块特征值之并）。我把两者都算了：`rho_full_restricted` 与 + 主 SCC 的 `rho_dense` 每格相同（见 `gw-w248.json`）。 + +所以 `cloud-mo-c` 的数值全对，但“受限转移阵”这个措辞在这一族参数下**没有被用到**； +`γ_w` 就等于整个 lumped 矩阵的谱半径取 `−log`。若日后有人拿别的 (图,p) 或别的 lump 方式， +两者会分开——建议引用时写成“最大 SCC 的 Perron 根（本族参数下等于整矩阵谱半径）”。 + +--- + +## 6. 诚实边界（**不能**宣称的） + +1. `γ_w` 的定义仍来自 #739 引擎（`build/lump/numeric_system`），我只独立重做了**受限集、SCC 划分、谱半径**。 + 引擎本身没有被独立重写。 +2. **w=9 只有本地探针件可对**：Astra 已发表件只到 w=8；w=9 用的是 raise-only 放宽副本， + 所以 w=9 的“校验”只证明我没抄错副本输出，**不构成对 Astra 的独立校验**。 +3. `c(4,6,8)` 是 **3 点窗口的有效值，不是渐近衰减率**；`c(2,4,6)` 与它最大差 0.43（matching）， + 说明有限宽修正很显著。只有 4–7 个 p/图、宽度只到 9。 +4. “指数形式相容”只覆盖我测的窗口；p 最大处偏差 ~8.8%，**不能说成精确**。 +5. 我**没有**独立估计 `κ`，所有 `c` 都来自差分。 +6. 我**没有**做图无关性的判定（同 §4 末）。 + +--- + +## 7. 复现命令（云机） + +```bash +cd /workspace/mass761/scripts +export PYTHONPATH=/workspace/mo/compat; export PY39COMPAT_MARKER=1; export OPENBLAS_NUM_THREADS=1 +# R1: 精确路线，w=2,4,6,8 +python3 gw_independent.py --engine-dir /workspace/mass761/in/pristine --widths 2,4,6,8 --pextra 1 \ + --out /workspace/mass761/out/gw-w248.json --workers 8 +# R1: w=9（raise-only 放宽副本） +python3 gw_independent.py --engine-dir /workspace/mass761/in/gamma9 --widths 9 --pextra 1 \ + --out /workspace/mass761/out/gw-w9.json --workers 2 +# R2: 多项式核路线 + 作者圆柱根复核 +python3 cylroot.py --engine-dir /workspace/mass761/in/pristine \ + --author-json /workspace/mass761/in/cylinder-mass-bounds.json \ + --out /workspace/mass761/out/cylroot.json --workers 6 +# 校验 + 分析 +python3 validate.py --gw /workspace/mass761/out/gw-w248.json,/workspace/mass761/out/gw-w9.json \ + --astra /workspace/mass761/in/refdoc/winding-shape-scales.json \ + --w9file /workspace/mass761/in/gamma9/gamma-w9.json --out /workspace/mass761/out/validate.json +python3 analyze.py --gw /workspace/mass761/out/gw-w248.json,/workspace/mass761/out/gw-w9.json \ + --gw9 /workspace/mass761/in/gamma9/gamma-w9.json \ + --exitbox /workspace/mass761/out/xb-scan.json \ + --cloud2 /workspace/mass761/in/refdoc/cloud-2.json --out /workspace/mass761/out/analysis.json +``` diff --git a/docs/manuscripts/geometric-balance/gaussian-circle-angular-tomography-20260914.md b/docs/manuscripts/geometric-balance/gaussian-circle-angular-tomography-20260914.md new file mode 100644 index 000000000..a48345c55 --- /dev/null +++ b/docs/manuscripts/geometric-balance/gaussian-circle-angular-tomography-20260914.md @@ -0,0 +1,272 @@ +# Gaussian-circle angular tomography for square-lattice finite-size corrections + +Date: 2026-09-14 + +Status: exact angular / arithmetic algebra plus a finite-size-scaling programme. The linear-algebra statements below do not identify any continuum field and do not assume the matching-root correction is exhausted by finitely many harmonics. + +## 1. Fixed-norm periods remove size, modulus and Smith confounds + +Let + +```text +u=(a,b), +gcd(a,b)=1, +N=a^2+b^2. +``` + +Use the square torus period basis + +```text +u=(a,b), +nu_perp=(-b,a). +``` + +Then + +```text +det(nu,nu_perp)=N, +``` + +and the physical modulus is exactly square. Because `gcd(a,b)=1`, the Smith invariants of the 2x2 period matrix are + +```text +(1,N). +``` + +Therefore two primitive representations of the same integer `N` as a sum of two squares have simultaneously + +```text +same site count, +same physical circumference sqrt(N), +same continuum square modulus, +same cyclic Smith class, +``` + +while differing only in the embedding angle of the microscopic square lattice relative to the period cycle. + +This is the natural setting for angular finite-size tomography. + +## 2. D4/reflection symmetry reduces the angular basis to Chebyshev polynomials + +For a reflection-even square-lattice scalar quantity, orientation dependence is periodic under `theta -> theta+pi/2` and even under `theta ->-theta`. Hence its angular Fourier expansion has the form + +```text +y_N(theta) + = A_0(N) + + sum_{m>=1} A_{4m}(N) cos(4m theta). +``` + +Put + +```text +h=cos(4 theta). +``` + +Then exactly + +```text +cos(4m theta)=T_m(h), +``` + +where `T_m` is the Chebyshev polynomial of the first kind. Thus at fixed `N`, angular tomography is ordinary polynomial interpolation in the scalar coordinate `h`: + +```text +y_N(h) + = A_0+A_4 T_1(h)+A_8 T_2(h)+A_12 T_3(h)+... . +``` + +No continuum-field assumption enters this change of basis. + +## 3. Exact invertibility theorem + +Suppose the same Gaussian circle has `k` inequivalent primitive orientations with distinct + +```text +h_i=cos(4 theta_i), i=1,...,k. +``` + +Consider the `k x k` matrix + +```text +V_{i,m}=T_m(h_i), m=0,...,k-1. +``` + +Because `T_m` has degree `m`, with leading coefficient + +```text +1, 1, 2, 4, 8, ..., +``` + +the determinant is a nonzero constant times the ordinary Vandermonde determinant: + +```text +det V + = 2^((k-1)(k-2)/2) + product_{i **Gaussian-circle tomography theorem.** Distinct `h_i` imply that the first `k` square/reflection-even angular irreps +> +> ```text +> H0,H4,...,H_{4(k-1)} +> ``` +> +> are exactly identifiable from `k` same-circle measurements, with no cross-size, modulus or Smith transfer law. + +This is purely algebraic. + +## 4. Closure residuals are higher-irrep detectors + +With `k` orientations, fit only the first `r0). \tag{1.6} +\] + +**Theorem (fixed-p general-period free energy).** + +\[ +\boxed{ +\lim_{n\to\infty}\frac1{\rho_n}\log f_n(p) +=-\max\{1-\alpha,0\}.} \tag{1.7} +\] + +Equivalently, + +\[ +\boxed{ +\log f_n(p) +=-(\rho_n-\log N_n)_+ + o(\rho_n).} \tag{1.8} +\] + +In particular: + +- if `alpha<1`, positive homology is exponentially rare with exact exponent `rho_n-log N_n` at the `rho_n` scale; +- if `alpha>1`, positive homology occurs with probability tending to one; +- at `alpha=1`, `log f_n=o(rho_n)`; the boundary probability itself may depend on subexponential geometry. + +## 2. Preliminary norm equivalence and a cheapest period + +At fixed subcritical `p`, all norms on `R^2` are equivalent. There are constants + +\[ +0infinity` and + +\[ +\log L_n=o(\rho_n). \tag{2.6} +\] + +Moreover the Euclidean shortest period also tends to infinity, so every fixed local seed used below eventually injects into the torus. + +## 3. Upper bound + +For any fixed `epsilon in (0,1)`, `first-exit-torus-winding-upper-20260914.md` gives + +\[ +f_n(p) +\le +N_n C_{p,\epsilon} +\sum_{\lambda\in\Lambda_n\setminus0} + e^{-(1-\epsilon)\tau_p(\lambda)}. \tag{3.1} +\] + +`correlation-norm-successive-minima-20260914.md` proves the norm-ball packing estimate + +\[ +\sum_{\lambda\ne0} + e^{-(1-\epsilon)\tau_p(\lambda)} +\le +\widetilde C_{p,\epsilon}(\rho_n) + e^{-(1-\epsilon)\rho_n}, \tag{3.2} +\] + +with + +\[ +\log\widetilde C_{p,\epsilon}(\rho_n)=o(\rho_n). \tag{3.3} +\] + +Hence + +\[ +\limsup\frac1{\rho_n}\log f_n(p) +\le +\min\{0,\alpha-(1-\epsilon)\}. \tag{3.4} +\] + +Let `epsilon downarrow0`: + +\[ +\boxed{ +\limsup\frac1{\rho_n}\log f_n(p) +\le-\max\{1-\alpha,0\}.} \tag{3.5} +\] + +## 4. A finite family of fixed local directional seeds + +The lower bound must produce a winding ring of class `lambda_n` at cost `rho_n+o(rho_n)` even though the minimizing direction may vary. + +Fix `eta>0`. Directional continuity of the norm on the compact unit circle permits a finite set of primitive integer directions + +\[ +w_1,\ldots,w_J \tag{4.1} +\] + +such that every `e in S^1` lies within a sufficiently small angular neighbourhood of some + +\[ +e_j=w_j/|w_j|, \tag{4.2} +\] + +and within that neighbourhood + +\[ +|\tau_p(e)-\tau_p(e_j)|<\eta. \tag{4.3} +\] + +For each `j`, choose a fixed integer multiple + +\[ +z_j=M_jw_j \tag{4.4} +\] + +so large that a finite-box connection seed from `0` to `z_j`, conditional on the initial site open, has probability `q_j` satisfying + +\[ +-\frac1{|z_j|}\log q_j +\le\tau_p(e_j)+\eta. \tag{4.5} +\] + +Then choose one fixed finite box supporting that seed. The number, lengths and support sizes of all seed types are finite constants depending on `p,eta`, never on `n`. + +Because the torus shortest Euclidean period tends to infinity, all these fixed seed boxes inject for large `n`. + +## 5. Approximating the minimizing period by repeated seeds + +Let + +\[ +e_n=\lambda_n/L_n. \tag{5.1} +\] + +Choose a seed direction `e_j` from the finite net close to `e_n`. Let + +\[ +k_n=\operatorname{round} +\left(\frac{\lambda_n\cdot z_j}{|z_j|^2}\right) \tag{5.2} +\] + +and set + +\[ +r_n=\lambda_n-k_nz_j. \tag{5.3} +\] + +By making the angular net sufficiently fine as a function of `eta`, + +\[ +|r_n|\le c\eta L_n+O_{p,\eta}(1). \tag{5.4} +\] + +Repeat the fixed connection seed `k_n` times along `z_j`, then append a deterministic NN path from `k_nz_j` to `lambda_n`. The latter uses at most + +\[ +|r_n|_1\le\sqrt2|r_n| \tag{5.5} +\] + +sites up to an endpoint constant. For a general finite-range graph `G`, use any fixed generating set path; its length is at most `C_G|r_n|` because the graph is connected and periodic. For the actual NN/matching pair, the NN path is available in both. + +The projected concatenation begins and ends at the same torus vertex and has lift displacement `lambda_n`, so it forces positive homology. + +All seed and connector events are increasing. Their supports may overlap after projection, but Harris positive association gives the product lower bound; overlaps can only reduce the number of distinct required open sites. + +Using (4.3)--(5.5) and norm equivalence, + +\[ +\boxed{ +P_p(\mathcal R_n) +\ge +\exp[-(1+C_p\eta)\rho_n-o(\rho_n)]} \tag{5.6} +\] + +for one prescribed winding-ring attempt `R_n`. + +## 6. The support costs only a polynomial number of translation centres + +The union of the repeated fixed seed boxes and the deterministic connector lies in a Euclidean region of diameter `O_{p,eta}(L_n)`. + +Therefore its projected support `S_n` obeys the crude but sufficient bound + +\[ +\boxed{|S_n-S_n|\le C_{p,\eta}L_n^2.} \tag{6.1} +\] + +The finite-group translation-packing lemma supplies at least + +\[ +M_n +\ge +\frac{N_n}{C_{p,\eta}L_n^2} \tag{6.2} +\] + +pairwise site-disjoint translated attempts. These attempts are independent. + +By (2.6), + +\[ +\frac{\log M_n}{\rho_n} +=\frac{\log N_n}{\rho_n}+o(1) +\to\alpha. \tag{6.3} +\] + +Thus the polynomial support footprint does not change the free-energy scale. + +## 7. Lower bound for alpha<1 + +Let + +\[ +r_n=P_p(\mathcal R_n). \tag{7.1} +\] + +The probability that at least one disjoint attempt succeeds is + +\[ +1-(1-r_n)^{M_n}. \tag{7.2} +\] + +If `alpha<1`, choose `eta` so small that + +\[ +\alpha<1-C_p\eta. \tag{7.3} +\] + +Then `M_nr_n->0` exponentially. Using `1-(1-r)^M >= Mr/2` when `Mr` is small, + +\[ +\liminf\frac1{\rho_n}\log f_n(p) +\ge\alpha-(1+C_p\eta). \tag{7.4} +\] + +Let `eta downarrow0`: + +\[ +\boxed{ +\liminf\frac1{\rho_n}\log f_n(p) +\ge-(1-\alpha).} \tag{7.5} +\] + +Together with (3.5), this gives the exact rare-event exponent. + +## 8. Lower bound for alpha>1 + +If `alpha>1`, choose `eta` so small that + +\[ +1+C_p\eta<\alpha. \tag{8.1} +\] + +Then + +\[ +M_nr_n\to\infty \tag{8.2} +\] + +exponentially, so + +\[ +\boxed{f_n(p)\to1.} \tag{8.3} +\] + +This proves the supercritical-in-opportunity side of the fixed-`p` homological transition. + +## 9. Boundary alpha=1 + +For every fixed `eta>0`, the same construction gives + +\[ +\liminf\frac1{\rho_n}\log f_n(p)\ge-C_p\eta. \tag{9.1} +\] + +Let `eta downarrow0`, while trivially `log f_n<=0`: + +\[ +\boxed{\frac1{\rho_n}\log f_n(p)\to0.} \tag{9.2} +\] + +No universal limit of `f_n(p)` itself follows. Polynomial/support factors and finer activity amplitudes live exactly at this boundary. + +## 10. Interpretation + +The free energy is + +\[ +\boxed{ +\mathcal F_n(p)=\rho_1(\Lambda_n;p)-\log N_n.} \tag{10.1} +\] + +At fixed subcritical `p`, its sign determines the first homology event at exponential scale: + +\[ +\mathcal F_n\gg0\Rightarrow r=0\text{ whp}, \tag{10.2} +\] + +\[ +\mathcal F_n\ll0\Rightarrow r>0\text{ whp}. \tag{10.3} +\] + +More sharply, + +\[ +\log P(r>0)=-[\mathcal F_n]_+ +o(\rho_n). \tag{10.4} +\] + +This is the rigorous fixed-`p` form of the earlier heuristic “connection energy minus log opportunities.” The raw number of translations is `N`, while the need to reserve a support of diameter `O(L_n)` costs only `O(log L_n)` in the logarithm and therefore disappears at the `rho_n~L_n` scale. + +## 11. Relation to previous geometry theorems + +### Fixed-direction exponential torus + +If `Lambda=` and `log m/n->d`, then + +\[ +\rho_n=n\xi_u(p), +\qquad +\log N=\log(nm)=dn+o(n). \tag{11.1} +\] + +Hence + +\[ +\alpha=d/\xi_u(p), \tag{11.2} +\] + +and (1.7) becomes + +\[ +\frac1n\log P(r>0) +=-\max\{\xi_u(p)-d,0\}, \tag{11.3} +\] + +recovering the fixed-direction theorem. + +### Varying shortest direction + +When the shortest period direction converges and the transverse height is exponential, the correlation-norm minimizer is asymptotically that shortest projective line and the same reduction gives the varying-direction rate. + +### Geometric full-law criterion + +At fixed `p`, norm equivalence makes `rho_1` comparable with the Euclidean shortest period `ell`. Thus `log N/ell->0` implies `alpha->0`, but the present theorem is quantitatively sharper because it retains the actual anisotropic correlation norm and the exact constant one in the energy/entropy balance. + +## 12. What this theorem does not yet give + +`rho_1(Lambda;p)` depends on `p` and may change its minimizing projective direction as `p` varies. Turning (1.7) into a universal birth-centre formula for an arbitrary changing lattice sequence requires control of the `p`-dependence of these minima and, near ties, the projective hard-core state space. + +So (1.7) closes the fixed-parameter free energy for arbitrary shapes; it does not erase genuine direction-crossover questions in a moving-parameter window. + +## 13. Claim boundary + +The upper side uses the author-level first-exit/Wulff domain theorem. The lower side uses only fixed local finite seeds, Harris, deterministic connectors and finite-group packing. No OZ prefactor or analyticity is used. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/generic-q-spin4-mixed-normalization-target-20260914.md b/docs/manuscripts/geometric-balance/generic-q-spin4-mixed-normalization-target-20260914.md new file mode 100644 index 000000000..1be832eb9 --- /dev/null +++ b/docs/manuscripts/geometric-balance/generic-q-spin4-mixed-normalization-target-20260914.md @@ -0,0 +1,193 @@ +# Generic-Q normalization target for the mixed thermal/vacuum spin-four response + +2026-09-14. + +Status: exact Virasoro normalization algebra plus a concrete generic-Q programme. This note does not identify the measured safe-root H8 response with a particular continuum tensor; it specifies what must be computed before such an identification is meaningful at c=0. + +## 1. Two parent spin-four sectors are now separately visible on the lattice + +The finite evidence points to two different square-spin-four corrections: + +```text +T4: matching odd, thermal-family candidate, omega=13/4, +I4: matching even/common, omega=2. +``` + +Their radial powers and matching parities were already visible in the historical threshold-rank projectors. New safe-root Gaussian controls additionally resolve a q=2 spin-eight response consistent with a second-order `T4 x I4` channel. + +The temptation is to multiply two ordinary c=0 descendants. That is not a well-defined normalization prescription because both relevant c=0 families are degenerate/logarithmic. + +## 2. Vacuum level-four spin-four quasiprimary has norm proportional to c + +At generic central charge, use the vacuum quasiprimary + +```text +Lambda4 = (L_-2^2 - 3/5 L_-4)|0>. +``` + +The level-four vacuum Gram entries are + +```text + = c(c+8)/2, + = 3c, + = 5c. +``` + +Therefore + +```text + + = c(c+8)/2 - (6/5)(3c) + (9/25)(5c) + = c(5c+22)/10. +``` + +In particular, + +```text + ~ (11/5)c +``` + +as `c->0`. The ordinary vacuum spin-four state is zero-norm at percolation. + +## 3. Explicit Q->1 slope of the vacuum norm + +Use the standard critical FK-Potts parametrization + +```text +Q = 4 cos^2[pi/(m+1)], +c = 1 - 6/[m(m+1)]. +``` + +Percolation is `m=2`, `Q=1`, `c=0`. Direct differentiation gives + +```text +dc/dQ |_(Q=1) = 5 sqrt(3)/(4 pi). +``` + +Hence + +```text + + = [11 sqrt(3)/(4 pi)] (Q-1) + + O((Q-1)^2). +``` + +The coefficient is approximately `1.5161544624` in the displayed state normalization. + +Thus the even-spin4 branch has an explicit simple zero in its ordinary vacuum Gram norm. + +## 4. Thermal Q4 has finite relative norm but inherits the bottom-field normalization problem + +The repository exact checker at `c=0,h=5/8`, with the thermal bottom state normalized to one, gives + +```text +Q4 = 40 L_-2^2 - 60 L_-3 L_-1 - 9 L_-4, + / = 4930. +``` + +So the level-four descendant itself is non-null in the ordinary Kac quotient. However the physical percolation energy/Kac field is a zero-norm bottom field of a logarithmic multiplet at c=0. Therefore any physical normalization zero of the bottom field is inherited by Q4 with a finite relative factor. + +This separates two questions that were previously conflated: + +```text +ordinary descendant exists and is non-null modulo Kac nulls: YES; +physical c=0 field has a non-singular standalone two-point normalization: NO. +``` + +## 5. The mixed H8 response needs a joint Q->1 limit + +At generic Q define physical fields/couplings schematically as + +```text +u_T(Q) * T4(Q), +u_I(Q) * I4(Q). +``` + +The second-order spin-eight response of a declared observable `O` contains + +```text +u_T nu_I * R8(Q), + +R8(Q) + ~ integral integral + < O T4^(+4) I4^(+4) >_conn + + contact/counterterm contributions. +``` + +At Q=1 the individual Gram norms vanish or collide with logarithmic partners. The finite lattice coefficient therefore depends on the combined limit of + +```text +lattice coupling normalization, +field two-point normalization, +mixed OPE coefficient, +contact subtraction, +possible Jordan collision coefficients. +``` + +There is no justification for obtaining this limit by multiplying two separately normalized ordinary c=0 one-point functions. + +## 6. What generic Q should compute + +The minimal useful calculation is a table for Q near one containing + +```text +c(Q), +h_energy(Q), +N_T(Q)=two-point normalization of the thermal Q4 branch, +N_I(Q)=c(5c+22)/10 for the vacuum Q4 branch, +C_8(Q)=declared spin8 mixed matrix element/OPE coefficient, +C_0(Q)=declared spin0 mixed matrix element/OPE coefficient, +sector/projector factors for the safe magnetic/rank observable. +``` + +Then determine whether + +```text +C_8 / sqrt(N_T N_I), +C_0 / sqrt(N_T N_I) +``` + +remain finite, diverge, or vanish as Q->1, and how the microscopic lattice couplings compensate those behaviours. + +A `1/(Q-1)` or derivative collision would naturally generate logarithmic/contact terms. A regular limit would support an ordinary mixed-response interpretation. + +## 7. Direct interface to the measured safe-root coefficients + +The new deterministic root controls give, in one fixed lattice normalization, + +```text +c46 ~= -0.116 [same-H4 q=2 dressing], +c88 ~= -0.138 [spin8 q=2 response]. +``` + +Equal-circumference magnetic-gap controls separately resolve the common omega=2 parent into a dominant scalar coefficient and a smaller but nonzero spin-four coefficient. + +This means a generic-Q calculation no longer needs to predict an arbitrary finite-size curve. It can target two specific dimensionless response ratios after the lattice/CFT normalization dictionary is declared: + +```text +R46 = mixed(T4,S0) / [linear T4 * linear S0], +R88 = mixed(T4,I4) / [linear T4 * linear I4]. +``` + +These ratios cancel the microscopic coupling amplitudes in the ideal scaling-field factorization and are therefore substantially more identifying than another exponent fit. + +At present they should **not** be estimated by naively dividing coefficients from different observables or moduli; the same-observable Feynman--Hellmann normalization is required first. + +## 8. Consequence for the E8 modular guess + +Because the vacuum spin-four branch becomes zero-norm/logarithmic at c=0, the simple ordinary-module statement + +```text +spin8 mixed torus shape proportional to E8=E4^2 +``` + +is only a positive control. The actual percolation limit may acquire logarithmic, derivative, contact, or map-sector terms. + +The right strategy is: + +1. derive the generic-Q mixed torus response; +2. impose modular covariance before taking Q->1; +3. take the singular limit with the physical normalization fixed; +4. only then freeze a hexagonal/Pell shape target. + +This prevents a second round of post-reveal modular-ray fitting. diff --git a/docs/manuscripts/geometric-balance/giant-white-bulk-and-response.md b/docs/manuscripts/geometric-balance/giant-white-bulk-and-response.md new file mode 100644 index 000000000..6ba68c915 --- /dev/null +++ b/docs/manuscripts/geometric-balance/giant-white-bulk-and-response.md @@ -0,0 +1,563 @@ +# Giant white components: bulk filling, a dual tail mass, and a Laplace response + +2026-09-14. Mathematical continuation of #764/#739. This note develops the +previous gap-law result, rather than treating an eventual publication as a +prerequisite. Statements described as proved below are the arguments supplied +here; the finite calculations check their exact interfaces, not their +large-width limits. + +## 1. The model, conventions and the results + +On `(Z/wZ) x Z`, black sites have probability `p`, NN connectivity, and white +sites have probability `q=1-p`, matching (eight-neighbour) connectivity. Fix +`0 (E, theta E, beta E), E~Exp(1). (1.1) + + All fixed mixed positive moments converge. More strongly, its longitudinal + occupation and boundary measures converge to constant-density intervals. + +2. The ultimate fixed-width WHITE span-tail mass is controlled by the same + BLACK barrier density: + + gamma_white(w,q)/nu_black(w,p) -> 1. (1.2) + + This requires a gluing argument; it is not inferred from convergence of + moments of nu L. + +3. For the physical complete-cluster score + + S = K/q - B/p, c = theta/(p q^2), + + there is a normal-exponential mixture law: + + (nu L, sqrt(nu/w) S) -> (E, sqrt(c E) Z), (1.3) + + where Z is standard normal, independent of E. The second marginal is + Laplace with variance c, not normal. This follows from physical parameter + tilting and the bulk law, without assuming a separate bulk CLT. + +4. The longitudinal connectivity structure factor has a Lorentzian limit: + + (nu/w) sum_(x,y) exp(i k nu y) + Pr((0,0) and (x,y) belong to the same white component) + -> 2 theta^2/(1+k^2). (1.4) + + This is a **connectivity** correlator, not the covariance of independent + site labels. No continuum field identification is made. + +5. On a torus with `m nu->t`, the volumes of macroscopic white components + are theta times the Poisson circular-gap fractions. This gives explicit + fragmentation and largest-component predictions, including the white + cross when there are no black barriers. + +All statements fix a strictly subcritical black p. No assertion is made +uniformly up to criticality. The arguments are locally uniform on compact +subintervals of `(0,pc_NN)`, which is needed for the small physical tilts in +Section 6. `q=3/4` is the intentional supercritical white control, NOT the +previous subcritical-white `q=1/8` experiment. + +## 2. Reused inputs, explicitly separated from this continuation + +The preceding `black-white-gap-law.md`, Sections 2--8, supplies: + +- the alternating complete-component chain and common density nu; +- row projection counts `N_W(y)<=1+N_B(y)` and + `E N_B(0)=epsilon_w=nu E_B L`; +- black root-volume exponential tails uniform in cylinder width, via the + first-query discovery-tree lift and the plane site tail [AV]; +- `a^w<=nu<=Cw exp(-cw)` and black component-Palm moments `E_B K^j=O(w^j)`; +- the black-anchor Poisson limit on the nu scale, the white gap law + `nu L_W->Exp(1)`, and uniform exponential tails on that scale; +- the circular Poisson-gap law when `m nu->t`. + +These are dependencies, not new independent theorems established by rerunning +old tables. The relevant old note is included unchanged under source_inputs. +The first-query lift preserves a discovery TREE, vertex count and vertical +reach; it does not lift winding cycles as cycles. + +The estimates in that note are uniform on a fixed compact p interval: use +the black root-volume bound at its largest p, a uniform isolated-ring lower +bound on nu, and the upper bound `Cw exp(-cw)`. With localization H=w^2, its +Poisson error bounds are uniform because polynomial powers of H times nu +vanish, while `exp(-cH)` beats every fixed power of `1/nu`. Its independent +block argument then gives uniform exponential tails for the rescaled gaps. +The geometric gap/span sandwich transfers those tails to white spans. + +The only additional external tools used here are matching boundary +connectivity [MZ], site Harris and the site volume tail [AV], and bounded-variable exponential concentration. For an independent X in [0,1], +the second derivative of log E exp(tX) is a tilted variance, at most 1/4. +Integrating twice proves E exp[t(X-E X)]<=exp(t^2/8); independence and +Chernoff give the inequality used below. Finite-range rows are handled by +residue classes. No additional asymptotic input is hidden in this step. + +## 3. A local cage lemma: the bulk white density approaches theta + +Let f_r(v) be the indicator that v is white and has a white matching path, +using only the square `v+[-r,r]^2`, to its vertex boundary. For `w>2r+2`, +this square injects in the cylinder, so + + E f_r = theta_r(q), theta_r(q) down to theta(q). + +Let eta_w(v) indicate that v belongs to an essential white component. Then +`eta_w<=f_r`: a component entirely inside this injecting square cannot wind. + +### Lemma 3.1 (nonessential white components are caged by a large black component) + +There are fixed geometric constants A0,A1 such that a finite nonessential +white component C containing v and reaching distance r has an adjacent +black component D with + + |D| >= (r-A0)/A1, + some z in D has dist_cylinder(z,v) <= A1(|D|+1). (3.1) + +Also `|C|<=A1(|D|+1)^2` after increasing A1. The same conclusion holds for a +finite white component in the plane. + +**Geometric proof.** Use a fixed subdivision of the square faces for the +4/8 complementary neighbourhood construction. At a diagonal white contact, +join the white corners; black arcs do not cross this connection. The boundary +of a neighbourhood is then a finite collection of disjoint polygonal curves. +Its black-side incident vertices along each curve are NN-connected: moving +along the curve either stays at one black vertex or crosses an NN black +edge. This is the matching-boundary observation of [MZ, Section II], with a +fixed local subdivision rather than an arbitrary chosen white path. + +Since C has no essential cycle, its neighbourhood lies in a disc in the +annulus. The outer boundary is contractible and encloses C; extra boundary +curves surround holes and are not used. Lift this outer curve to a closed +plane curve. All black-side vertices project into one complete black +component D. A site of D has a bounded number of incident face sectors, +and each sector contributes a bounded number of unit-length segments of +this PARTICULAR boundary. Thus its length is at most `A1 |D|`, independently +of the number of white vertices and independently of w. One may use a loose +constant such as 128 after fixing the quarter-cell construction; the argument +uses only the existence of a universal constant. + +A curve enclosing both v and a point reached at distance r has diameter at +least r-O(1), so its length forces the first bound. Since it encloses v, a +black incident vertex is within its length plus a local constant of v. +The enclosed area is at most a constant times its squared length, giving +the bound on |C|. The outer curve is contractible even when the FULL black +component D also has an essential cycle somewhere else. This distinction +is why we bound by the size of D, not by a presupposed simple black loop. + +### Consequences of the cage + +The uniform black root-volume tail and a dyadic union bound over possible +vertices z within `A1(2k+1)` of v give + + Pr(f_r=1, eta_w=0) <= C exp(-c r), 1<=r=k)<=C exp(-ck)`, so the union is summable. Apply the same argument +in the plane to finite white components to get + + 0<=theta_r-theta<=C exp(-cr). + +Since `theta_w:=E eta_w=(nu/w)E_W K`, (3.2) gives + + |theta_w-theta| <= C exp(-c w). (3.3) + +The same proof implies a stretched-exponential **volume** tail for a +nonessential white root component, bounded by `C exp(-c sqrt(n))`, uniformly +in w. We do NOT claim an exponential volume tail for supercritical white +finite components. In particular all their root moments are uniformly +bounded. It also proves continuity of theta on the present compact q +intervals, by uniform approximation by the finite polynomials theta_r. + +## 4. Filling a random, exponentially long interval without dividing by a rare event + +Let `I(C)=[min y(C),max y(C)]`. For a white essential component define + + F_r(C)=sum_(y in I(C)) sum_x f_r(x,y). + +Every site in C is included, so `F_r(C)-K(C)>=0`. Campbell counting and +`N_W<=1+N_B` give the exact expectation estimate + + (nu/w) E_W[F_r(C)-K(C)] + = E[N_W(0) * (1/w)sum_x f_r(x,0)] - theta_w + <= theta_r-theta_w+epsilon_w. (4.1) + +This removes the difficult rare-conditioning factor from the **defect**. +Bounding an unconditional error and then blindly dividing it by nu would +not have done so. + +The row field `X_y=(1/w)sum_x f_r(x,y)` lies in [0,1] and is dependent only +within 2r rows. Its mean is theta_r. For each deterministic n, split the +sum into 2r+1 independent residue classes and apply Hoeffding. Taking a union +over `0<=n<=M/nu` gives, for fixed M,delta>0, + + Pr(max_(n<=M/nu) |sum_(y=0)^(n-1)(X_y-theta_r)| > delta/nu) + <= C(M/nu)(2r+1) exp[-c delta^2/(M nu(2r+1))]. (4.2) + +Choosing `r=floor(w/8)-2` is allowed for large w. Even AFTER division by the +white anchor probability nu, the right side tends to zero. Thus the estimate +holds under white component Palm, despite the dependence of the component +length on the same labels. Use the uniform exponential tail of nu L to +remove the restriction L<=M/nu. Abel summation extends the partial-sum +estimate to bounded Lipschitz weights along the interval. + +Combining (4.1), (3.3), epsilon_w->0 and (4.2) proves + + nu(K/w-theta L) -> 0 in L^1. (4.3) + +Since K<=wL and nu L has uniform exponential tails, this also gives every +fixed L^j convergence after using a higher uniform moment. More generally, +after anchoring the component at its lowest row, + + M_w := (nu/w) sum_(x,y in C) delta_(nu y) + -> theta 1_[0,E](s) ds, jointly with nu L->E. (4.4) + +The convergence is of finite measures, with their total-mass moments. +This is a bulk law for randomly delimited components, NOT a claim of +independent occupancy conditional on the component. + +## 5. The boundary fills too, and the leading thermal score cancels + +A complete component has physical activity `q^K p^B`. At fixed w the finite +transfer representation (or direct convergent component sum) gives + + d_q log nu = E_W S, + d_q^2 log nu = Var_W(S) - E_W V, + S=K/q-B/p, V=K/q^2+B/p^2. (5.1) + +By exact colour-density duality, `nu_white(q)=nu_black(p)`. +Black component moments and `B_black<=4K_black` give, locally uniformly, + + |d_p log nu_black|=O(w), |d_p^2 log nu_black|=O(w^2). (5.2) + +Set `b_w=(nu/w)E_W B`. Equations (5.1)--(5.2) give + + theta_w/q - b_w/p = -(1/w)d_p nu_black, + b_w = (p/q)theta_w + (p/w)d_p nu_black + -> beta=(p/q)theta. (5.3) + +The coefficient beta has an independent plane meaning. Freeze the origin +black and ask whether a white neighbour belongs to an infinite white +component in the graph with the origin removed. Turning the origin white +makes it infinite exactly on that event: joining finitely many finite +components cannot make an infinite one. Independence of the origin label +therefore gives `theta=q Pr(A)` and `beta=p Pr(A)`. + +For completeness, define `g_r(v)=1{v black}1{A_r(v)}`, where A_r asks for +such a neighbour path to the r-square boundary without using v. Then +`E g_r=(p/q)theta_r`. Every boundary site of an essential white C satisfies +g_r=1, and lies in the extended interval `I^+(C)=[min y-1,max y+1]`. +The expanded interval count exceeds N_W by at most an anchor and an upper +endpoint incidence; their expected total is 2nu. Consequently + + (nu/w)E_W[sum_(v in I^+(C))g_r(v)-B(C)] + <= (p/q)theta_r - b_w + epsilon_w+2nu. (5.4) + +Repeat the finite-range row argument. This proves + + nu(B/w-beta L)->0 in every fixed L^j, (5.5) + +and the analogous boundary-measure limit. Equations (4.3),(5.5) prove (1.1). +The boundary and occupation leading vectors are collinear. In particular, +the apparently huge terms K/q and B/p cancel at order w/nu. + +This does NOT make the remaining score zero: its mean is O(w), and its +fluctuations have the smaller but still diverging scale sqrt(w/nu). + +## 6. A physical-tilt proof of the Gaussian--Laplace mixture + +Write `s_w=sqrt(nu/w)` and `c=theta/(p q^2)>0`. +From (5.1)--(5.5), + + E(s_w S)->0, + Var(s_w S) = (nu/w)E V + O(nu w) -> c. (6.1) + +A variance limit alone is not a distribution theorem. To identify the law, +use an actual change of site probability, `q_h=q+h`, `p_h=p-h`. +For every complete cluster, + + log[(q_h/q)^K(p_h/p)^B] + = h S - h^2 V/2 + O(|h|^3(K+B)), (6.2) + +uniformly for h in a small fixed neighbourhood of zero. Take `h=t s_w`, +with real t near zero. Exact component reweighting yields + + E_q exp[-u nu L+t s_w S] + = (nu(q_h)/nu(q)) + E_(q_h) exp[-u nu L+(t^2/2)s_w^2 V + + O(|t|^3 s_w^3(K+B))]. (6.3) + +The ratio of densities tends to one: by (5.2), its log is O(w|h|), and +`w s_w=sqrt(w nu)->0`. Under q_h, which approaches q, the locally uniform +bulk theorem gives + + (nu L,s_w^2 V)->(E,cE). + +The cubic remainder is bounded by `C |t|^3 s_w nu L`, since B<=8K and +K<=wL. It vanishes, including under the exponential weight in (6.3). +For |t| and |u| sufficiently small, uniform exponential span tails provide +domination: the positive exponent is at most `C t^2 nu L+o(nu L)`. +Thus the joint transform converges on a neighbourhood of the origin to + + E exp[-u E+t Y] = 1/(1+u-c t^2/2). (6.4) + +This identifies `(E,Y)=(E,sqrt(cE)Z)` with E exponential and Z an independent +standard normal. Exponential integrability near the origin also gives all +fixed joint moments. In particular, + + density_Y(y) = exp(-sqrt(2/c)|y|)/sqrt(2c), + E Y^2=c, E Y^4=6c^2. (6.5) + +This is an UNCONDITIONAL Laplace law for the score. The joint limiting +conditional law given E=e is N(0,ce). No local conditioning theorem for +every exact discrete L is asserted. + +The key distinction is physical: the Gaussian fluctuations are mixed over +the exponentially distributed amount of bulk. This score projection can +be identified without first finding the entire covariance matrix of the +occupation/boundary residuals. + +### The still-open, richer bulk fluctuation question + +Put `rho_K,w=E K/(w E L)` and `rho_B,w=E B/(w E L)`. A natural next +conjecture is a joint stable Gaussian limit for + + sqrt(nu/w) (K-w rho_K,w L, B-w rho_B,w L), + +with conditional covariance E Sigma(q). The score result constrains + + (1/q,-1/p) Sigma(q) (1/q,-1/p)^T = theta/(p q^2). (6.6) + +It does not determine Sigma. The finite-width centring is important: +using theta in its place requires an error smaller than sqrt(nu/w), which +is not supplied merely by theta_w->theta. The new task asks for this richer +limit, not a repeat of the first-order volume law. + +## 7. The ultimate white pole is the black barrier density + +Let `c_w(n)` be the probability of a white matching path crossing the cylinder +slab of n rows, from the first to the last row. Let `r_w(H)` be the probability +of a white horizontal essential cycle in H rows. + +By the cylinder 4/8 crossing duality, + + c_w(n)=Pr(no black NN essential cycle in those n rows). (7.1) + +The prior localized black-anchor Poisson theorem, with H0=w^2, implies for +each fixed T>0 + + c_w(floor(T/nu))->exp(-T). (7.2) + +To check the boundary issue: any slab cycle belongs either to a complete +black component of span <=H0 with anchor in an H0 enlargement, or to a black +component of size >H0 meeting the slab. The latter probability is at most +`Cwn exp(-cH0)`. Conversely, a short component anchored H0 away from the upper +edge has its cycle inside. Anchor enlargements change scaled length by o(1). + +The crossing probabilities are submultiplicative on disjoint row blocks. +Therefore their height decay rate gamma exists and, for B>H>=1, + + -log c_w(B)/B <= gamma. (7.3) + +For the other direction, overlap successive B-row windows in H rows and +require a white horizontal cycle in each overlap. Both vertical white paths +must meet that cycle, so they join. In the matching graph a geometric +crossing of two white diagonals also joins their endpoints; it is not a +crossing of disjoint components. All the events are increasing in WHITE +sites. Harris on the overlapping supports gives + + c_w(k(B-H)+H) >= c_w(B)^k r_w(H)^(k-1), + gamma <= [-log c_w(B)-log r_w(H)]/(B-H). (7.4) + +This is not an independence assertion for overlapping windows. +The complementary event to r_w(H) is a black NN vertical crossing. +The uniform black reach tail hence gives + + 1-r_w(H)<=Cw exp[-c(H-1)]. (7.5) + +Use `B=floor(T/nu)`, `H=w^2`. Equations (7.2)--(7.5), `Hnu->0` and +`r_w(H)->1` squeeze gamma/nu to 1. + +Finally gamma is the white complete-Palm span-tail rate. The anchored event +`L>=n` supplies a crossing of the first n rows. Conversely plant a full white +row below a crossing, and a full black row below that. The crossing joins a +white essential component with prescribed lowest row, at probability cost +`q^w p^w`, independent of its height. These two comparisons identify the +height exponents. Thus (1.2) is proved. + +This removes the previous rare-mixture obstruction by a model-specific +uniform gluing bound, not by arguing that the first few moments determine +the tail. + +## 8. Connectivity, form factors, and susceptibility + +Let the anchored white component have row occupations k_y. Its exact +component form factor is + + A_C(k) = sum_y k_y exp(i k nu y). + +The volume-measure convergence (4.4) and second moments give + + (nu/w)^2 E_W |A_C(k)|^2 + -> theta^2 E |integral_0^E exp(i k s) ds|^2 + = 2 theta^2/(1+k^2). (8.1) + +Mass transport identifies the left side with the scaled essential white +connectivity sum in (1.4). The nonessential white contribution is negligible +by the uniform root-volume moments after Lemma 3.1. At k=0 this yields + + chi_white ~ 2 w theta^2/nu. (8.2) + +Equivalently, the rescaled, transverse-averaged connectivity measures tend +to `theta^2 exp(-|s|) ds`. This is a weak/integrated statement. A separate +pointwise microscopic-endpoint theorem is not silently included. + +The exact rational numerical control uses `z=(1+s nu)^(-1)` and + + T_w(s) = nu^2/(2w^2 theta_w^2) + E_W sum_(a,b) k_a k_b z^|a-b| -> 1/(1+s). (8.3) + +No transcendental evaluation is required for this finite computation. +For a direct activity transfer with matrices R_j weighting current-row +occupation k^j and exit vectors b_j, first solve + + y=(I-R_0)^(-1)b_0, + f=(I-zR_0)^(-1)(b_1+R_1 y), + h=(I-R_0)^(-1)(b_2+R_2 y+2z R_1 f). + +Then `alpha h` is the unnormalised pair activity. The independent physical +shape enumeration checks this recursion at several finite heights; the +resolvent itself supplies all heights. + +## 9. Finite-torus volume fragmentation + +Reuse the earlier circular-gap result at `m nu->t>0`. Conditionally on +J=j>=1 black barriers, the j gap fractions have Dirichlet(1,...,1) law, +after a uniformly chosen barrier fixes the circular ordering. J converges +to Poisson(t). At J=0 there is one white cross, not zero white clusters. + +The same local-cage argument and row concentration hold on the torus for +r=2, + + Pr(M<=x | J=j) + = sum_(l=0)^j (-1)^l binom(j,l)(1-lx)_+^(j-1), (9.3) + E[M | J=j] = H_j/j. + +At J=0 or 1, M=1. The mixture is fully specified. These are derived limit +predictions, not newly simulated torus volumes. They illustrate that a +locally supercritical white model can have a random finite number of +macroscopic components on an exponentially elongated sequence. + +## 10. Finite results and execution scope + +The physical control uses black p=1/4, white q=3/4. Exact all-height results: + +| w | theta_w | T_w(0) | T_w(1) | T_w(2) | T_w(4) | +|---|---:|---:|---:|---:|---:| +| 2 | .748890532544 | .944533333333 | .510668185243 | .356668711606 | .229827452010 | +| 3 | .749847650131 | .980867697451 | .501548687782 | .338740246690 | .207375743066 | +| 4 | .749969685902 | .993248030686 | .500162917714 | .334764577282 | .202069892807 | +| limit | theta | 1 | 1/2 | 1/3 | 1/5 | + +The score variances scaled by nu/w at these widths are +`5.06833994408, 5.30253981269, 5.33091713810`. +The corresponding finite-density proxies `theta_w/(p q^2)` are +`5.32544378698, 5.33224995649, 5.33311776642`. +The table does not determine theta to the printed precision. +The mean scores are nonzero; they are retained in the JSON. + +An independent no-black-ring transfer gives certified positive Perron +intervals. The derived gamma_white/nu_black values at widths 2,3,4,6 are +`1.13642432987, 1.03689306046, 1.01175415938, 1.00131735895`. +Every stored Perron enclosure is obtained from exact rational Collatz +quotients; displayed logarithms are high-precision approximations to exact +-log interval endpoints. The finite gluing inequalities are separately +checked by rational powers against slab probabilities. + +Executed checks: 73,984 annulus configurations, 147,968 complementary +crossing/ring dichotomies; 250 complete nonessential cage controls; +2,189 connected winding shapes and 36 independent mass/pair identities; +exact dual first/second derivative identities; and 2,035 cyclic point +patterns for the discrete spacing participation identity. No new Monte +Carlo, external machine, or full repository test suite is run. + +The engine is an unchanged input, Git blob +`52f3611990ce2b1331d9e5296e0262f5e402e0d7`. Numerical values of old moments +are controls, not independent new data. New outputs include pair transforms, +boundary/score covariance, finite gluing certificates and Poisson-volume +predictions. The broader two-dimensional bulk CLT remains a conjecture. + +## References and reading scope + +[MZ] Mertens--Ziff, *Percolation in Finite Matching Lattices*, +arXiv:1603.07289v2, Section II, surrounding-component connectivity and +single/spiral/cross classification. Relevant HTML body read this round. +https://arxiv.org/html/1603.07289v2 + +[AV] Antunovic--Veselic, *Sharpness of the phase transition and exponential +decay of the subcritical cluster size for percolation on quasi-transitive +graphs*, arXiv:0707.1089v3, Theorems 2--3 and Section 3. Site applicability +is stated explicitly. Relevant definitions and theorem statements read. +https://arxiv.org/html/0707.1089v3 + +Internal input: `black-white-gap-law.md` (2026-09-14), Sections 2--8. +The present all-width arguments explicitly depend on that note's topology +and rare-gap estimates. The local-cage, bulk filling, physical-tilt mixture, +finite gluing rate, and form-factor arguments are supplied here. + +## Concurrent team connection + +During this analysis PR #771 arrived; its read head was +`53b4ec6111f5f97e11cbcdf4606aeaed17d0a9ae`. Its +`structural-consequences-20260914.md`, Section 6, independently records the +complementary score identity; its +`supercritical-white-slab-bulk-20260914.md`, Section 1, gives the same exact +one-site identity beta=(p/q)theta and Sections 3--5 formulate the bulk and +vector-CLT questions. These coincident identities are NOT counted as a second +new mathematical discovery here. The present extension is the cage/Campbell +proof of the bulk law, physical-tilt score distribution, gluing tail theorem +and connectivity/fragmentation consequences. The richer vector question is +now coordinated at #772; no shared branch was overwritten. diff --git a/docs/manuscripts/geometric-balance/graph-polynomial-symmetric-loop-trace-20260914.md b/docs/manuscripts/geometric-balance/graph-polynomial-symmetric-loop-trace-20260914.md new file mode 100644 index 000000000..d5a2dea31 --- /dev/null +++ b/docs/manuscripts/geometric-balance/graph-polynomial-symmetric-loop-trace-20260914.md @@ -0,0 +1,210 @@ +# The graph-polynomial source gives a simple symmetric torus loop trace + +Date: 2026-09-14 + +Status: exact finite weight identity at the self-dual FK point, followed by a representation conjecture for the periodic Temperley--Lieb trace. This is a specialization of the Euler-localized rank source and is intended as a concrete comparison target for the known torus Potts eigenvalue-amplitude decomposition. + +## 1. Boundary loops from the primal--dual pair + +Use the notation of `euler-localization-of-rank-source-20260914.md`. + +For a spanning FK subgraph `A` on a cellular torus, let + +```text +b(A) = number of boundary components of a regular neighbourhood N(A), +g(A),g*(A) = genera carried by N(A) and its complement. +``` + +The two Euler equations give + +```text +k(A) = [|V|-|A|+b+2g]/2, +k(A*) = [|A|-|V|+b+2g*]/2. +``` + +Hence exactly + +```text +boxed: +k(A)+k(A*) = b(A)+g(A)+g*(A). (1.1) +``` + +The boundary components are the medial interfaces separating primal and dual regions. + +## 2. Topological value of `g+g*` on the torus + +There are only three ambient homology ranks. + +```text +rank 0 : (g,g*)=(0,1), +rank 1 : (g,g*)=(0,0), +rank 2 : (g,g*)=(1,0). +``` + +Therefore + +```text +boxed: +g+g* = 1_(rank != 1). (2.1) +``` + +Combining with (1.1), + +```text +k+k* = b + 1_(rank != 1). (2.2) +``` + +This is the precise topological correction to the naive planar statement that cluster counts are controlled only by medial loops. + +## 3. Evaluate the graph-polynomial source at the self-dual point + +The Euler-localized sourced FK weight is + +```text +W_Q,v,h(A) + = e^{-h|V|} + (Qe^h)^{k(A)} + (e^-h)^{k(A*)} + (ve^h)^{|A|}. +``` + +At the exact graph-polynomial source + +```text +h_Q=-1/2 log Q +``` + +and the square-lattice self-dual FK point + +```text +v=sqrt(Q), +``` + +we have + +```text +Qe^{h_Q}=sqrt(Q), +e^{-h_Q}=sqrt(Q), +ve^{h_Q}=1, +e^{-h_Q|V|}=Q^{|V|/2}. +``` + +Thus + +```text +W(A) + = Q^{|V|/2} (sqrt(Q))^{k(A)+k(A*)}. (3.1) +``` + +Use (2.2): + +```text +boxed: +W(A) + = Q^{|V|/2} + (sqrt(Q))^{b(A)} + (sqrt(Q))^{1_(rank != 1)}. (3.2) +``` + +The first factor is global and can be removed. + +## 4. A remarkably simple modified trace rule + +After global normalization, the sourced self-dual model has the configuration weight + +```text +boxed: +(sqrt Q)^(number of medial boundary loops) +× +{ sqrt Q, rank 0 or rank 2, + 1, rank 1. } +``` + +Equivalently: + +> give every medial boundary loop the ordinary Potts loop fugacity `sqrt(Q)`, and give the zero-through-line extreme homology sectors one additional factor `sqrt(Q)` relative to the rank-one through-line sectors. + +This is the simplest explicit candidate encountered so far for the topological modified trace underlying the graph-polynomial balance. + +It does **not** separately distinguish rank0 from rank2 at this symmetric source; it need not, because the graph-polynomial source has already transformed the original asymmetric FK weights into a primal--dual symmetric ensemble. + +Away from `h_Q`, the full two-face/cluster fugacity asymmetry is required. + +## 5. Relation to the periodic TL sectors + +In a medial periodic transfer: + +- rank-one FK states carry noncontractible interfaces / through-line sectors; +- rank0 and rank2 are both zero-through-line in the simple loop count and require an additional topological distinction in the ordinary FK partition function. + +Equation (3.2) says that **at the sourced graph-polynomial symmetric point**, the remaining distinction collapses to a single sector factor `sqrt(Q)` for the extreme/zero-through-line topology relative to rank one. + +This is exactly the kind of information supplied by a modified Markov trace rather than by the local TL generator. + +A concrete algebra task is therefore: + +```text +construct/evaluate the periodic TL trace with +contractible/noncontractible boundary-loop weight sqrt(Q) +and extreme-sector bonus sqrt(Q), +then compare its characters/eigenvalue amplitudes with the known graph-polynomial sectors. +``` + +## 6. Connection to the Richard--Jacobsen torus amplitude decomposition + +For the toroidal Potts model the partition function is not a simple trace because noncontractible clusters carry global information. Richard and Jacobsen decompose it into transfer characters labelled by the number of noncontractible clusters and cyclic-group representations, with nontrivial eigenvalue amplitudes. + +That formalism is a natural existing place to test (3.2): instead of inventing a new transfer algebra, reweight their sector amplitudes by the sourced primal--dual rule and ask whether the graph-polynomial eigenvalue identity emerges directly. + +This is now a much narrower literature/algebra comparison than “find a twisted massive Potts partition function”. + +## 7. Why the symmetric point is special + +For general topological source `h`, write + +```text +Q_primal = sqrt(Q) e^eta, +Q_dual = sqrt(Q) e^-eta, +eta=h+1/2 log Q. +``` + +At `eta=0` the primal and dual faces have equal fugacity; the loop boundaries do not need to remember which side is favoured. Only the torus topology correction (2.1) survives. + +For `eta!=0`, one must retain the signed difference of primal and dual component counts, which is naturally represented by an oriented-loop/height/face-charge variable. + +Thus the graph-polynomial source is algebraically special because it sits exactly at the **zero face-charge** point of the two-sided cluster gas. + +This suggests the continuous rank source around `h_Q` may be represented as an electric/background-charge perturbation of a fixed-`sqrt(Q)` loop model. + +## 8. A possible continuum implication + +If `eta` is indeed an electric/height charge in the periodic loop description, then derivatives in the topological source `h` are not arbitrary closure insertions: they probe a well-defined electric/topological sector of the loop theory. + +This could connect the repository's rank-source coordinate directly to the charged/map-resolved Potts continuum fields being considered in #585/#782. + +This statement is a programme, not an identification of the corresponding Coulomb-gas charge. + +## 9. Site-model boundary + +The derivation is for an edge/FK model on a cellular torus. It should not be quoted as an exact square-site transfer identity without an explicit decorated/incidence-graph construction. + +Its immediate role is in the generic-Q Potts/TL continuum bridge and in understanding Jacobsen's graph-polynomial eigenvalue criterion. + +## 10. Claim boundary + +Exact: + +```text +k+k*=b+g+g*, +g+g*=1_(rank!=1) on T^2, +self-dual graph-polynomial sourced weight + ∝ (sqrt Q)^b (sqrt Q)^{1_(rank!=1)}. +``` + +Conjectural/interface: + +- identification of (3.2) with a particular periodic TL/Markov trace normalization; +- electric/height interpretation of `eta` away from the symmetric point; +- massive/TBA continuation. + +The next useful work is a direct comparison to existing torus Potts eigenvalue amplitudes, not a new width computation. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/h4-null-projector-leakage-algebra-20260914.md b/docs/manuscripts/geometric-balance/h4-null-projector-leakage-algebra-20260914.md new file mode 100644 index 000000000..9c0e849a3 --- /dev/null +++ b/docs/manuscripts/geometric-balance/h4-null-projector-leakage-algebra-20260914.md @@ -0,0 +1,226 @@ +# Exact leakage algebra for two-angle H4-null projectors + +Date: 2026-09-14 + +Status: exact trigonometric/projector algebra. It sharpens the interpretation of #808 and prevents an H4-null combination from being mislabeled as a pure scalar channel before higher harmonics are controlled. + +## 1. Setup + +For a square-lattice orientation `theta`, define + +```text +H_(4n)(theta)=cos(4n theta). +``` + +Let + +```text +h_i=H4(theta_i). +``` + +The normalized two-angle H4-null projector is + +```text +L[f] + = (h1 f(theta2)-h2 f(theta1))/(h1-h2). +``` + +It satisfies + +```text +L[1]=1, +L[H4]=0. +``` + +Hence it preserves an angular scalar and annihilates every term proportional to exactly the same H4 function, including any radial dressing of that H4 coefficient. + +## 2. Higher-harmonic leakage is an exact polynomial in `(h1,h2)` + +Using Chebyshev identities + +```text +H8 = 2 H4^2 - 1, +H12 = 4 H4^3 - 3 H4, +H16 = 8 H4^4 - 8 H4^2 + 1, +``` + +one obtains + +```text +boxed: +C8 :=L[H8] = -(1+2 h1 h2), + +C12:=L[H12] = -4 h1 h2 (h1+h2), + +C16:=L[H16] + = 1+8p-8p s^2+8p^2, +``` + +where + +```text +p=h1 h2, +s=h1+h2. +``` + +Thus the leakage budget can be computed exactly from H4 values alone; no numerical angular fit is needed. + +## 3. Ideal simultaneous H4/H8/H12 notch + +To annihilate H8 in addition to H4 requires + +```text +h1 h2=-1/2. +``` + +Under this condition + +```text +C12=2(h1+h2). +``` + +Therefore simultaneous H4/H8/H12 cancellation requires + +```text +h1 h2=-1/2, +h1+h2=0, +``` + +or equivalently + +```text +boxed: +h1=+1/sqrt(2), +h2=-1/sqrt(2). +``` + +This is the ideal two-angle triple-notch geometry. + +It cannot also null every higher harmonic: at the ideal point + +```text +C16=-1. +``` + +So a two-angle projector can suppress the first several angular contaminants but can never be called an all-harmonic scalar projector. + +## 4. N377 is close to the ideal triple notch + +For #808 directions + +```text +u1=(4,19), +u2=(11,16), +``` + +the committed H4 values are approximately + +```text +h1=+0.6748868985, +h2=-0.7435428378. +``` + +The formulas above give + +```text +C8 ~= +0.00361464, +C12 ~= -0.137808, +C16 ~= -0.98105. +``` + +Thus N377 is much better described as + +```text +H4 exact notch ++ H8 near-notch ++ partial H12 suppression ++ essentially unsuppressed H16. +``` + +This is exactly the right gate if the expected scalar residual is larger than any plausible H12/H16 contribution at that circumference. That last clause must be demonstrated or bounded; it is not supplied by the trigonometric projector alone. + +## 5. Consequence for naming #808 output + +The primary combination should be called + +```text +p_perp4 +``` + +or + +```text +H4-null root combination. +``` + +Calling it `p_H0` is justified only after showing that + +```text +|C8 P8 + C12 P12 + C16 P16 + ...| +``` + +is below the claimed scalar signal/error budget. + +A match to `ell^-7` across sizes is strong evidence for an angular-scalar contribution only if the allowed higher-harmonic radial powers cannot mimic that scaling at the tested sizes. + +This is particularly important because #61 has downgraded continuum matching-parity assignments; “scalar” here should first mean angular H0, not an OPE parity label. + +## 6. What the projector does prove if the residual is robust + +Suppose the numerical root calculation is certified and the H8/H12/... leakage budget is controlled. Then a nonzero residual proves an angular contribution orthogonal to H4 at that `ell`. + +This directly falsifies the strongest version of the old hypothesis + +```text +all visible post-leading corrections are just H4 with radial dressing. +``` + +because `L[H4 f(ell)]=0` for every radial function `f`. + +This is why #808 is conceptually more valuable than another large-width H4 amplitude measurement. + +## 7. Radial field identification comes later + +If repeated/projected data support + +```text +P0(ell)-pc ~ C ell^-7, +``` + +then a scalar field with total dimension `x` satisfying + +```text +x-x_t=7 +``` + +has + +```text +x=33/4, +``` + +which is compatible with the current `V_<1,4>` candidate. But the sequence of logical steps must remain + +```text +H4-null residual +-> angular H0 isolation +-> radial exponent +-> continuum field / logarithmic block identification. +``` + +Do not reverse the arrows. + +## 8. Claim boundary + +Exact: + +- projector formulas for C8/C12/C16; +- ideal triple-notch condition; +- H4 radial dressing is always annihilated by an exact H4 projector. + +Conditional/programmatic: + +- N377 higher-harmonic leakage is small enough relative to the target scalar signal; +- a measured `ell^-7` H0 coefficient maps to a particular simple `V_<1,4>` rather than a logarithmic/mixed block. + +The useful correction is terminological and structural: `H4-null` is exact; `H0` is a conclusion that requires a leakage budget. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/integrable-potts-magnetic-charge-scaling-20260914.md b/docs/manuscripts/geometric-balance/integrable-potts-magnetic-charge-scaling-20260914.md new file mode 100644 index 000000000..e5c65c5b4 --- /dev/null +++ b/docs/manuscripts/geometric-balance/integrable-potts-magnetic-charge-scaling-20260914.md @@ -0,0 +1,283 @@ +# Integrable Potts field theory route to the universal charge scaling function + +2026-09-14. Literature-grounded continuum programme for #767, updated after identifying the critical safe sector in the periodic dilute Temperley--Lieb representation. + +The problem is now narrower than in the first version of this note. The UV magnetic twist is no longer arbitrary: critical triangular-site percolation identifies the safe/no-noncontractible-loop sector with the zero-defect periodic dilute-TL module + +\[ +\boxed{W_{N,0,\omega=i},\qquad \alpha=\omega+\omega^{-1}=0.} \tag{0.1} +\] + +The remaining problem is to construct or locate its **off-critical thermal/massive finite-volume continuation** at `q=1`. + +## 1. The lattice transfer has identified one continuum magnetic object + +The fixed-width square-site safe transfer gives + +\[ +I^0_{4,w}(p),\qquad I^0_{8,w}(1-p). \tag{1.1} +\] + +At criticality both converge to the same magnetic primary gap + +\[ +wI^0\to2\pi x_m, +\qquad x_m=5/48. \tag{1.2} +\] + +Near criticality the natural universal organization is one magnetic finite-size energy function sampled at opposite thermal signs: + +\[ +\boxed{ +\mathcal F(X) +=\mathcal E_m(X)-\mathcal E_m(-X).} \tag{1.3} +\] + +Hence + +\[ +\mathcal F(-X)=-\mathcal F(X). \tag{1.4} +\] + +The square NN and complementary matching kernels are two microscopic regularizations of this same magnetic/topological sector. Their dual-odd difference is the charge free energy. + +## 2. The critical periodic dilute-TL sector is now explicit + +Morin-Duchesne, Kluemper and Pearce describe critical triangular-site percolation by the Yang--Baxter-solvable dilute `A_2^(2)` model. Its periodic standard modules are labelled by defect number `d` and twist + +\[ +\omega=e^{i\gamma}. \tag{2.1} +\] + +For the periodic ground states they obtain + +\[ +(h,\bar h) += +\left( +\Delta_{\gamma/\pi,d/2}, +\Delta_{\gamma/\pi,-d/2} +\right), \tag{2.2} +\] + +where + +\[ +\Delta_{r,s}=\frac{(3r-2s)^2-1}{24}. \tag{2.3} +\] + +In the zero-defect module, the noncontractible-loop weight is + +\[ +\alpha=\omega+\omega^{-1}=2\cos\gamma. \tag{2.4} +\] + +The safe transfer rejects a state precisely when a horizontal noncontractible occupied loop closes. The loop-language specialization is therefore + +\[ +\boxed{\alpha=0, +\qquad \omega=\pm i, +\qquad d=0.} \tag{2.5} +\] + +Choose `omega=i`, so `gamma=pi/2`. Then + +\[ +h=\bar h=\Delta_{1/2,0}=5/96, +\qquad x=5/48. \tag{2.6} +\] + +Thus the magnetic UV fingerprint is not just inferred from a fitted gap: it is the ground state of the published `W_{N,0,i}` standard module. + +Primary source: A. Morin-Duchesne, A. Kluemper, P. A. Pearce, *Critical site percolation on the triangular lattice: From integrability to conformal partition functions*, arXiv:2211.12379v2. + +The detailed lattice/transfer interpretation and square-site checks are recorded in `safe-transfer-pdtl-magnetic-sector-20260914.md`. + +## 3. Four independent checks of the sector dictionary + +The proposed UV dictionary simultaneously explains facts that were obtained independently in the square-site analysis. + +### 3.1 Noncontractible-loop semantics + +`alpha=0` kills a noncontractible loop exactly, matching the safe transfer's rejection rule. + +### 3.2 State-space dimension + +The periodic zero-defect dilute-TL module has central-trinomial dimension. The transparent safe automaton has exactly + +\[ +1,3,7,19,51,141,393,1107,3139,\ldots \tag{3.1} +\] + +states. + +### 3.3 Magnetic ground gap + +Equation (2.6) gives `x_m=5/48`, matching + +\[ +wI^0_w(p_c)\to2\pi(5/48). \tag{3.2} +\] + +### 3.4 Level-one descendants + +The first descendants of a scalar magnetic primary have `Delta x=1` and spins `+/-1`. Direct square-site diagonalization finds a twofold first excited eigenvalue with + +\[ +w\log(\lambda_0/|\lambda_1|)\to2\pi, \tag{3.3} +\] + +while the one-column translation eigenvalues on this two-dimensional space are exactly + +\[ +e^{\pm2\pi i/w} \tag{3.4} +\] + +to machine precision for `w=5,...,9`. + +See `safe-transfer-momentum-spectrum-w5-w9-20260914.json`. + +These checks sharply reduce the risk that `W_{0,i}` is only a numerical weight coincidence. + +## 4. Integrable massive Potts theory remains the natural off-critical framework + +The scaling `q`-state Potts theory for `q<=4` under thermal perturbation is integrable. Chim--Zamolodchikov give the kink scattering theory; Dorey--Pocklington--Tateo develop finite-size TBA/NLIE equations for thermal Potts flows and emphasize continuous `q` formulations relevant near `q=1`. Delfino--Cardy demonstrate that the `q->1` massive Potts continuation yields nontrivial percolation amplitudes and form factors. + +What the material inspected here does **not** yet hand us is a ready-made finite-volume equation explicitly carrying the `d=0`, `omega=i`, `alpha=0` magnetic topological sector through the thermal perturbation. + +The target is therefore no longer “find some twist with UV `c_eff=-5/4`.” It is: + +> Continue the specific critical module `W_{0,i}` into the massive thermally perturbed Potts theory and compute its finite-volume energy on the two signs of the thermal coupling. + +The 2002 Dorey--Pocklington--Tateo finite-size equations remain a plausible continuum framework, but the sector insertion/source implementing (2.5) must be derived or found. + +## 5. The UV effective-central-charge check is retained + +For a sector with lowest total dimension `x`, + +\[ +c_{eff}=c-12x. \tag{5.1} +\] + +Percolation has `c=0`, so the magnetic module requires + +\[ +\boxed{c_{eff}^{(m)}=-12(5/48)=-5/4.} \tag{5.2} +\] + +Any proposed off-critical twisted NLIE must return (5.2) in its ultraviolet limit **and** its lattice/topological twist must reduce to `alpha=0` / `omega=i`. Matching only one of these checks is insufficient. + +## 6. Desired massive finite-volume output + +Let + +\[ +r=m_{phys}R \tag{6.1} +\] + +be the usual dimensionless massive circumference. We seek + +\[ +\mathcal E_{W_{0,i}}^{(+)}(r), +\qquad +\mathcal E_{W_{0,i}}^{(-)}(r), \tag{6.2} +\] + +on the two signs of the thermal perturbation, in a common mass normalization. + +Potts duality should relate the two branches. The universal charge function is + +\[ +\boxed{ +\mathcal F(r) +=\mathcal E_{W_{0,i}}^{(+)}(r) + -\mathcal E_{W_{0,i}}^{(-)}(r).} \tag{6.3} +\] + +After fixing the lattice thermal metric, + +\[ +X\propto(p-p_c)w^{3/4}, \tag{6.4} +\] + +(6.3) should be the continuum limit of + +\[ +w\Theta_w(p). \tag{6.5} +\] + +## 7. Lattice data now give an entire target curve, not only derivatives + +`dual-odd-thermal-metric-20260914.md` defines + +\[ +\mathcal F_w(X) +=w\Theta_w(h_w+Xw^{-3/4}). \tag{7.1} +\] + +The curves for `w=4,...,9` already collapse well over `|X|<=1.5`. + +Their leading symmetric contamination scales as `w^-3/4` and is quantitatively consistent with a quadratic analytic thermal-field reparameterization. After removing that coordinate effect, the odd part is the appropriate lattice target for an integrable calculation. + +This improves the comparison protocol: + +1. determine the analytic lattice thermal metric from the even residual; +2. extract the antisymmetric continuum curve; +3. compare the full curve, not just `F'(0)`; +4. use the first/third/fifth derivative ratios as local checks. + +The current raw-logit local fit predicts a fifth derivative of order + +\[ +w^{-11/4}\Theta_h^{(5)}\sim-1.5\times10^{-2}, \tag{7.2} +\] + +before final metric calibration. + +## 8. Mandatory checks on any massive `W_{0,i}` construction + +A successful TBA/NLIE or other exact construction should satisfy: + +1. **UV module:** `d=0`, `omega=i`, `alpha=0` and `x=5/48`. +2. **UV effective central charge:** `c_eff=-5/4`. +3. **First descendants:** level-one spin `+/-1` above the magnetic primary. +4. **Dual oddness:** `F(-X)=-F(X)` in the correctly normalized thermal coordinate. +5. **Thermal slope:** reproduce the safe pivotal/Perron amplitude after metric fixing. +6. **Nonlinear curve:** agree with the transfer collapse over a finite `X` interval, not just at `X=0`. +7. **Infrared sectors:** match the appropriate kink/order/disorder excitation on the two thermal signs. + +The first three are now unusually rigid UV constraints. + +## 9. Square-site versus triangular-site scope + +For critical triangular-site percolation, `W_{N,0,i}` belongs to the explicit integrable periodic dilute-TL lattice solution. + +For square-site NN/matching percolation, we do **not** claim a Yang--Baxter conjugacy. The square Bernoulli safe transfer appears to act on the same annular/topological module and has the same continuum sector, but its local weights are a nonintegrable regularization. + +This is enough for universality-based continuum matching; it is not enough to import finite-width Bethe roots from the triangular model. + +## 10. Sharpened unresolved problem + +The previous version of this note asked for a generic-q magnetic twist. The critical twist has now been identified. + +The unresolved task is specifically: + +\[ +\boxed{ +\text{massive / thermal finite-volume continuation of }W_{0,i} +\text{ at }q=1.} \tag{10.1} +\] + +Promising routes include: + +- insert the corresponding noncontractible-loop/twist weight into a continuous-q Potts finite-volume NLIE; +- derive the excited/twisted solution by analytic continuation from the ground-state equation; +- construct an off-critical staggered/dilute-`A_2^(2)` realization whose scaling limit is the thermal Potts perturbation and retain `omega=i`. + +A literature search in this pass found the continuous-q thermal Potts finite-size framework and the critical periodic dilute-TL twist, but not a ready-made equation already combining both in the required percolation magnetic sector. + +## 11. Claim boundary + +The critical periodic dilute-TL standard-module formula, twist label and conformal weights are literature facts. `alpha=0 <-> omega=+/-i` is exact algebra. The safe-transfer state-count, momentum and gap checks are deterministic lattice calculations. + +The assertion that the square-site near-critical charge curve converges to the massive finite-volume `W_{0,i}` Potts energy difference is a universality/duality synthesis. The explicit massive `q=1` twisted NLIE remains to be derived or located. diff --git a/docs/manuscripts/geometric-balance/intrinsic-birth-clock-construction-20260914.md b/docs/manuscripts/geometric-balance/intrinsic-birth-clock-construction-20260914.md new file mode 100644 index 000000000..9aaa98309 --- /dev/null +++ b/docs/manuscripts/geometric-balance/intrinsic-birth-clock-construction-20260914.md @@ -0,0 +1,188 @@ +# 直接构造出生时钟：不再假设完整簇密度可单调反演 + +2026-09-14。续接 #780。读取基线 #771 `b1aafe5ce40583f2971c4491dcc69fc0af4c39df`，特别是 `thermal-window-no-merger-20260914.md` 与新提交 `docs/research-bridges-post-compass-20260914.md` B 节。这里只补它们仍保留的强度时钟可实现性，不再推导新的 Poisson 核矩。 + +**性质**：第1节是实际 SITE 的精确身份；第2–4节是使用明确概率输入的作者推导；物理参数的仿射化、全路径拓扑和无界标记矩没有由此得到。本文不从小圆柱计算证明宽度极限。 + +## 0. 结果和依赖 + +在轴向无限圆柱 C_w=(Z/wZ)×Z 上，黑色 NN 独立 SITE 的所有参数共用 Uniform 标签。固定 p0∈(0,pc)，nu0=nu_w(p0) 是每行完整 essential 簇密度。 + +已有论证给定单调参数族 p_w(x)，并**假设** nu_w(p_w(x))/nu0→exp(x)，再推出出生标记过程。这里构造该参数族： + +> 对每个固定 p0，存在由模型概率而非样本拟合确定的连续严格单调 p_w(x)，p_w(0)=p0；对每个有限 R，sup_|x|≤R |p_w(x)-p0|=O_R(1/w)，且 nu_w(p_w(x))/nu0→exp(x) 一致。该时钟下，局部出生标记过程具有强度 dz exp(x)dx 的 Poisson 极限；有限起点 a 的既有屏障另以 exp(a)delta_a 编码。 + +使用的输入逐项保留： + +1. 固定严格亚临界 SITE 的平面簇体积指数尾 [AV, Thm.3]，及已有首次查询发现树提升给出的圆柱统一体积尾； +2. 平面轴向质量 kappa(p)>0、连续、严格下降，以及 -log nu_w(p)/w→kappa(p)，这里只要求每个固定 p 的指数等价；这些是仓库 `exponential-birth-centres.md` 的作者论证接口，而非本次独立重证； +3. 局部完整周长上界 Q_w(H;r)≤w H² r exp[-(w-1)kappa(r)]，两个顶点不交绕行见证的 SITE BK 平方界； +4. AGG 局部依赖点过程近似 [AGG, Thm.2]； +5. 仅在第4.1节的物理窗口量级上，使用对传递单调事件的Friedgut–Kalai锐阈值界 [DKS, Thm.6]。 + +不使用 kappa 的导数、OZ 前因子、有限 nu 的严格单调性，也不使用“重新拟合时钟后自然是 Poisson”这种推断。构造时钟只使平均测度变简单；Poisson 性仍需双事件与局部依赖估计。 + +## 1. 固定一个最终参数，得到真正单调的对象 + +取 p≤r0。 (3) + +故 beta 连续、严格递增，可在整个[0,nu_w(r)]上取连续反函数。这一步不需要整个簇数 nu_w(p) 递增；后者受到簇并合影响，不能预先当作累计分布。 + +### 1.2 它与原密度的距离只来自谱系损失 + +每个早期 essential 簇唯一属于一个最终簇。沿纵向锚点搬运一单位质量，得到 + + nu_w(p)=E sum_(final C:anchor0) n_C(p), + 0≤nu_w(p)-beta_w(p;r) + =E sum_C (n_C(p)-1)_+ + ≤E sum_C binom(n_C(p),2)。 (4) + +这是平稳质量搬运，不要求早期和最终锚点相同；窗口版本另外保留端部。锚点有限期望位移由体积尾控制。 + +## 2. 一个固定的“半质量”上限就足够 + +由 kappa 连续性，可选固定 r∈(p0,pc) 使 + + delta_r=2kappa(r)-kappa(p0)>0。 (5) + +r 不需要依赖 w，也无需先知道1/w热窗口。取 H=w²。对于体积≤H、最低行为0的最终簇，所有早期 essential 祖先至少各有w点；相关小簇共位于w×H条带，故祖先总数≤H。若有损失，最终条带含两个顶点不交绕行见证。大体积部分由统一尾控制。因此对所有p≤r，同时有 + + 0≤nu_w(p)-beta_w(p;r)≤E_w(r), + E_w(r)≤(H²/2)Q_w(H;r)²+(C/w)exp(-cH)。 (6) + +这里大/小部分用体积截断；小体积自动给跨度≤H。不能将“跨度大”与“体积大”双向等同。常数依赖固定r的严格亚临界余量。 + +于是 + + limsup_w (1/w)log[E_w(r)/nu0]≤-delta_r<0。 (7) + +(6)是对所有早期参数的共同上界，不会在对p取上确界时丢失控制。它还使观察长度T/nu0中的谱系损失及必要端部误差趋零。 + +## 3. 构造而不是假设指数强度时钟 + +定义 a_w=beta_w(p0;r)>0。由(4),(7)，a_w/nu0→1。令 + + p_w(x)=beta_w(.;r)^(-1)(a_w exp(x))。 (8) + +有限w时定义域是 x0，使p0+etaH的概率至多Cwm exp(-cH)，均趋零。因此前面的Poisson结论给，在每个固定x， + + Pr_(p_w(x))(torus含水平绕行) -> f(x)=1-exp[-exp(x)]。 + +水平方向绕行事件单调，且在全部wm个site上受平移群传递作用。给定R>0，取epsilon小于min(f(-R),1-f(R))/4。若p_w(R)-p_w(-R)≥rho_FK log[1/(2epsilon)]/log(wm)，[DKS, Thm.6]会迫使上端绕行概率>1-epsilon，与上式矛盾。因为log(wm)=kappa(p0)w+o(w)， + + p_w(R)-p_w(-R) <= C_R/w， + sup_|x|≤R |p_w(x)-p0| <= C_R/w。 (10) + +这一步的finite-torus桥只为了应用传递锐阈值；不能把torus接缝直接忽略成完全相同的图。 + +还有一个相反方向的界。令M(p)=sum_C weight_r(C)|C| f_C(p/r)。有限可靠性多项式的真实似然评分给 + + beta'_w(p;r) <= M(p)/p。 + +最终r簇的体积尾意味着，在beta_w(p;r)位于[e^-R a_w,e^R a_w]时， + + M(p)/beta_w(p;r) + <= sum_(n≥0) min[1,Cw exp(-cn)/beta_w(p;r)] + <= C_R' w。 + +由p_w(x)→p0>0，积分log beta可得，对固定-R≤x1= (x2-x1)/(C_R'' w)。 (11) + +可靠性混合在内部p区间可逐项求导：最终簇一阶体积矩有限且可支配导数。因此(11)不是对一个只有连续性的函数擅自求导。 + +(10),(11)确定非退化的1/w窗口量级，但**没有**推出w[p_w(x)-p0]为x的线性函数，也没有计算v(p0)。它们把剩余问题从“窗口可能在任意尺度”收缩为“这个1/w窗口的实际形状与标尺是什么”。固定-R,R的常数未优化，不用来给当前小w误差条。 + +## 5. 最终参数不是新物理参数 + +若r和r'都满足(5)，在nu_w(p)/nu0属于一个固定正紧区间时， + + log[beta_w(p;r)/beta_w(p0;r)] + −log[nu_w(p)/nu0]→0 + +一致；r'相同。因此最终分组选择在领先时钟中消失。将两种构造置于共同的较高最终参数，(4)与标记计数还说明：两个同x反演在有限宏观窗口产生的额外出生数趋零，谱系损失也趋零。它们具有同一个领先过程，不要求两个反演的物理p之差已控制到o(1/w)。 + +**新获得的是内禀时钟下的模型映射，不是物理时钟标定。** 机器不必为时钟“是否存在”再扫一排参数；计算应转向p_w(x)的物理尺度/误差，或者寻找上述局部输入的实际失效。 + +## 6. 有限控制与下一步 + +独立小脚本在纵向自由的C3×3上，对全部512个配置构造完整簇。它将最终形状内首次绕行概率写成Bernstein多项式，再与全部3^9个共同标签早/晚对逐项求和比较。验证：beta端点、正Bernstein导数、Lipschitz界、密度=命中+损失、损失≤阶乘并合数及有理反函数包围。多组概率只是同一批有限对的重新加权，不是独立数据。此有限控制没有物理宽度渐近信息。 + +**下一步的两个有价值命题**： + +- PHYSICAL-CLOCK：在已经得到1/w窗口量级之后，找出是否存在v(p0)>0，使log nu_w(p0+x/[v(p0)w])−log nu0→x局部一致；这是仿射化，而不只是量级。单位留数或参数解析性不是必须同时完成的前置。 +- SOURCE-CLOCK：对保持平移周期的真实小源，分别研究一维强度时钟改变和标记/耦合改变。单一路径的内禀核相同，不意味着两条源路径在共同标签下的联合过程相同。这是后续猜想，不为它新开机器队列。 + +引用： +[AV] Antunović–Veselić, arXiv:0707.1089v3, Theorem 3，独立site的亚临界簇体积尾。https://arxiv.org/html/0707.1089v3 +[AGG] Arratia–Goldstein–Gordon, Two moments suffice for Poisson approximations: the Chen–Stein method, Ann. Probab.17(1989),9–25, Theorem2。其局部依赖过程接口沿已读的仓库输入使用；本轮不声称重新逐页审读原PDF。 +内部：thermal-window-no-merger-20260914.md；exponential-birth-centres.md；docs/research-bridges-post-compass-20260914.md B节（b1aafe5）。后者的更宽参数对推广不是本篇的额外依赖。 + +[DKS] Duncan–Kahle–Schweinhart, arXiv:2011.11903v4 §1.3 Theorem6（引用Friedgut–Kalai）；本轮实际读取其Bernoulli传递单调事件版本。https://arxiv.org/html/2011.11903v4 diff --git a/docs/manuscripts/geometric-balance/kdv-identity-family-cancellation-target-20260914.md b/docs/manuscripts/geometric-balance/kdv-identity-family-cancellation-target-20260914.md new file mode 100644 index 000000000..889587ac0 --- /dev/null +++ b/docs/manuscripts/geometric-balance/kdv-identity-family-cancellation-target-20260914.md @@ -0,0 +1,176 @@ +# A conditional KdV/Ward route to the common `x=4` cancellation + +Date: 2026-09-14 + +Status: conditional continuum mechanism / proof target. It does not assume a matching OPE parity. The Virasoro KdV eigenvalue formula is standard; the unresolved Matching-One inputs are the sector dictionary and the identification/coupling of the lattice `x≈4` correction. + +## 1. The lattice fact to explain + +The two critical safe/topological sector energies share a large leading irrelevant correction compatible with a per-length `L^-3` term, i.e. total scaling dimension + +```text +x≈4. +``` + +Yet their difference, which moves the Matching-One root, has no visible root term of exponent + +```text +x-x_t = 4-5/4 = 11/4. +``` + +The first visible difference instead occurs much later, at root exponent four. + +In the difference-tangent language the theorem target is + +```text +D_4^(4)=0. +``` + +## 2. Identity-family spin-four correction as a KdV charge + +The chiral level-four identity-family quasiprimary / quantum KdV charge has a zero mode whose eigenvalue on a Virasoro primary `|h>` is, up to the conventional cylinder scale, + +```text +q3(h,c) + = h^2 + -(c+2)h/12 + + c(5c+22)/2880. (2.1) +``` + +The anti-chiral charge has the same formula in `hbar`. + +Thus a real square-lattice H4 perturbation built from the chiral/anti-chiral identity-family quasiprimaries has first-order diagonal energy shift + +```text +delta E_a + = g4 [q3(h_a,c)+q3(hbar_a,c)] L^-3 (2.2) +``` + +for a primary state `a`, modulo normalization of `g4`. + +The important point is structural: this shift is fixed entirely by the Virasoro highest weights and the common microscopic coupling. + +## 3. Conditional cancellation theorem + +Assume: + +1. the rank-0 and rank-2 critical long-cylinder sectors flow to two copies of the **same** Virasoro highest-weight state + +```text +(h_0,hbar_0)=(h_2,hbar_2); +``` + +2. the leading `x=4`, H4 lattice perturbation is the same identity-family KdV/quasiprimary coupling `g4` in the physical square-site action for both topological sectors; +3. no additional map-resolved operator of the same `(x=4,H4)` block contributes a different matrix in the topological multiplicity space. + +Then (2.2) gives exactly + +```text +boxed: +delta E_0^(x=4,H4)=delta E_2^(x=4,H4), +``` + +so the sector difference has no first-order `x=4` contribution: + +```text +boxed: +D_4^(4)=0. (3.1) +``` + +This explains the absence of an `L^-11/4` root shift without assigning a matching-even scalar sign to the field. + +## 4. Why the “same physical model” formulation matters + +The matching-safe transfer is a computational representation of the complementary/rank-2 sector. Physically, rank0 and rank2 are sectors of the **same square-site Bernoulli model** related by the finite digital-Alexander/matching dictionary. + +Therefore a local lattice anisotropy of the original action should be treated as one microscopic coupling whose matrix elements are evaluated in two topological states, rather than as unrelated couplings in two different theories. + +This is the correct setup for the KdV cancellation argument. + +A transfer implementation must nevertheless transport the physical perturbation consistently through the matching representation; otherwise an apparent amplitude difference can be a source-coordinate artifact. + +## 5. Map/multiplicity is the real remaining adversary + +Equal conformal weights alone do not prove assumption 3. + +If the continuum state space contains several copies/map sectors with the same Virasoro weights, a generic local perturbation may act as a nontrivial matrix in that multiplicity space. A **pure Virasoro KdV charge** acts identically on equal highest-weight copies, but the lattice `x≈4` correction could contain additional map-resolved operators with the same radial/angular quantum numbers. + +Thus the useful falsification question is: + +```text +is the measured common x≈4,H4 block exhausted by the identity-family KdV direction? +``` + +This interfaces directly with #585's warning that modular covariance/spin alone do not define a one-dimensional physical solution space. + +## 6. A concrete Ward/rank-projection test + +Let `F_r` be the actual finite/continuum rank-sector functional. For a fixed quasiprimary convention, the current Ward programme has a schematic combination + +```text +A_r + = F_r(U4+Ubar4) + - C_E4(tau) F_r(epsilon), +``` + +where the second term removes coordinate/thermal contamination. + +The targeted theorem is simply + +```text +A_rank0 - A_rank2 = 0 (6.1) +``` + +for the identity-family `x=4,H4` block, including seam/contact terms appropriate to the nonlocal rank projection. + +Equation (6.1), not an abstract OPE parity assignment, is exactly what the root mechanism needs. + +## 7. What a failure would mean + +If an independently normalized calculation finds a nonzero `x=4,H4` difference block, then at least one assumption fails: + +```text +rank0/rank2 are not the same continuum state, +lattice x≈4 contains another map-resolved operator, +physical coupling is transported incorrectly, +or seam/contact terms distinguish the sectors. +``` + +A genuine nonzero block should produce an `L^-11/4` root contribution unless another cancellation removes it. Its absence in data would then demand a second mechanism rather than be silently ignored. + +## 8. Relation to the leading `x=21/4` block + +The KdV argument only explains why the ordinary identity-family `x=4` anisotropy is common. It does **not** identify the first noncommon block. + +The observed H4 root exponent four still says that the first visible difference block has effective total dimension `21/4`. Whether that block is the thermal-family level-four quasiprimary, an eight-arm/map-resolved sector, or a degenerate mixture remains a separate question. + +## 9. Literature anchor + +The quantum KdV charge `I3` can be written in Virasoro modes as + +```text +I3 ~ 2 sum_(k>0) L_-k L_k + + L0^2 + -(c+2)L0/12 + + c(5c+22)/2880, +``` + +so on a primary the positive-mode terms vanish and (2.1) follows. This standard formula is enough for the conditional argument; no large-c or thermal assumption is used. + +## 10. Claim boundary + +Standard/exact CFT algebra: + +- KdV `I3` primary eigenvalue depends only on `(h,c)`. + +Conditional Matching-One consequence: + +- `D_4^(4)=0` if both extreme sectors are the same highest-weight representation and the lattice x=4 block is the common identity-family KdV perturbation. + +Open: + +- rigorous/transfer proof of the sector dictionary at the required map resolution; +- exclusion of additional x=4 H4 operators in the multiplicity space; +- correct seam/contact term in the actual rank projection. + +This is a deliberately narrow route: prove the amplitude equality actually needed by the root, rather than a stronger unconstructed matching automorphism. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/krushkal-rank-source-projector-20260914.md b/docs/manuscripts/geometric-balance/krushkal-rank-source-projector-20260914.md new file mode 100644 index 000000000..95d2ad0d0 --- /dev/null +++ b/docs/manuscripts/geometric-balance/krushkal-rank-source-projector-20260914.md @@ -0,0 +1,270 @@ +# Krushkal topological variables give the finite torus rank-source projector + +Date: 2026-09-14 + +Status: exact finite embedded-graph topology for bond/FK configurations, plus a new interface proposal for #782/#768. This does **not** by itself construct the missing massive affine-TL/TBA modified trace, and it is not yet a direct square-SITE transfer identity. + +## 1. The current blocker can be separated into definition versus realization + +The current #782 dictionary already has a continuous rank-one neutral-cluster fugacity from an affine-TL seam: + +```text +z = alpha^2/Q. +``` + +What remained was described as a missing way to distinguish rank-0 and rank-2 configurations inside the zero-noncontractible-loop sector. + +At the level of a finite graph embedded on a torus, however, the required topological coordinate is already standard in the Krushkal / topological Tutte polynomial. + +The remaining problem is therefore narrower: + +> realize the already-defined rank source as a transfer/modified trace compatible with the massive Potts / affine-TL continuum engine. + +## 2. Krushkal variables on a torus + +Let a spanning subgraph `H` of a graph cellularly embedded in `T^2` have image + +```text +V(H) = im[ H_1(H;R) -> H_1(T^2;R) ]. +``` + +Let + +```text +r(H)=dim V(H) in {0,1,2}. +``` + +Krushkal's surface polynomial uses two topological exponents `s(H), s_perp(H)`. In the symplectic formulation, + +```text +s(H) = dim[ V/(V cap V^perp) ], +s_perp(H) = dim[ V^perp/(V cap V^perp) ], +``` + +where orthogonality is for the intersection form on `H_1(T^2;R)`. + +Because `H_1(T^2)` is a two-dimensional symplectic vector space, there are only three cases. + +### Rank 0 + +`V=0`, so `V^perp=H_1(T^2)`: + +```text +s=0, +s_perp=2. +``` + +### Rank 1 + +Every one-dimensional subspace is Lagrangian on the torus, hence + +```text +V=V^perp, +s=s_perp=0. +``` + +### Rank 2 + +`V=H_1(T^2)`, `V^perp=0`: + +```text +s=2, +s_perp=0. +``` + +Therefore, exactly, + +```text +boxed: +r(H)-1 = [s(H)-s_perp(H)]/2. +``` + +No scaling limit or percolation input enters this identity. + +## 3. The Matching-One topological source is a direct specialization + +The bounded rank source used throughout the repository is + +```text +exp[h (r-1)]. +``` + +Using the identity above, + +```text +exp[h(r-1)] + = exp[h s/2] exp[-h s_perp/2]. +``` + +Thus in the Krushkal topological monomial + +```text +A^(s/2) B^(s_perp/2) +``` + +the exact rank-source specialization is simply + +```text +boxed: +A=e^h, +B=e^-h. +``` + +It gives + +```text +rank 0 -> e^-h, +rank 1 -> 1, +rank 2 -> e^+h, +``` + +which is exactly the repository's topological charge source. + +The derivative at `h=0` is the observable + +```text +X=r-1. +``` + +## 4. Refined FK partition function + +For an embedded FK/bond configuration one may therefore define the topology-refined random-cluster state sum + +```text +Z_G(Q,v;h) + = sum_A v^|A| Q^k(A) exp[h(r(A)-1)]. +``` + +The extra topological factor is exactly the Krushkal `A/B` monomial specialization above. The ordinary random-cluster/Potts weights occupy the usual connectivity/edge variables; the rank source is an independent surface-topology refinement. + +This gives a clean finite definition for the continuum object sought in #782 before any TBA machinery is invoked. + +## 5. Combining rank charge with the rank-one neutral fugacity + +The affine-TL seam result already gives, inside a fixed rank-one slope sector with `K` parallel essential FK clusters, + +```text +z^K, +z=alpha^2/Q. +``` + +The Krushkal source and the seam source are complementary: + +```text +Krushkal A/B variables : distinguish rank 0 versus rank 2; +affine-TL alpha seam : resolves neutral count K inside rank 1. +``` + +Thus the desired schematic torus source + +```text +T(q0,q2,alpha) + = q0 Z0 + sum_u Z1,u(alpha^2/Q) + q2 Z2 +``` + +has a natural finite-state interpretation with + +```text +q0=e^-h, +q2=e^+h, +z=alpha^2/Q. +``` + +At `Q=1`, this is precisely the `(charge fugacity, neutral-count fugacity)` coordinate system already used on #771. + +## 6. What remains genuinely missing for #782 + +This observation does **not** yet produce the massive torus scaling function. The remaining tasks are now more precise: + +1. construct a transfer / modified trace realization of the Krushkal `A/B` topology variables in the periodic loop/TL language; +2. combine it consistently with the noncontractible-loop seam `alpha`; +3. identify the corresponding sectors in the massive Potts finite-volume theory and continue toward `Q->1`; +4. verify that the UV limit reproduces the Pinson/Arguin homology weights. + +So the verdict should be sharpened from + +```text +"the rank0/rank2 projector is unknown" +``` + +to + +```text +FINITE_TOPOLOGICAL_PROJECTOR_EXISTS; +MASSIVE_TRACE_REALIZATION_OPEN. +``` + +## 7. Why this also matters for the sector-odd correction (#768) + +The matching-root observable is the response of the **signed topological source** `X=r-1`. Therefore the first noncommon correction can be defined without first naming a CFT field: + +> it is the leading irrelevant contribution to the `h`-odd part of the topology-refined torus free energy at `h=0`. + +If `g` is a microscopic anisotropy/source parameter, the object of interest is schematically the mixed response + +```text +partial_g partial_h log Z_G(Q,v;h) |_(h=0) +``` + +or its sector/free-energy analogue after the appropriate thermodynamic projection. + +This supplies a precise map/topology label to the spin-four versus scalar `x=21/4` question. The continuum operator must live in the channel selected by the Krushkal rank source, not merely have a compatible scaling dimension. + +In particular, a scalar eight-arm field of the right dimension matters for the matching root only if it has nonzero matrix element in this `h`-odd topological channel. + +## 8. Connection to map-resolved torus tomography + +The Krushkal refinement gives a concrete combinatorial-map/topology coordinate rather than an arbitrary modular basis vector. A future map-resolved torus solution space can therefore be organized by + +```text +(rank charge h, + rank-one slope u, + neutral fugacity alpha, + local operator/spin label). +``` + +This is closer to the repository's exact lattice semantics than fitting isolated modular functions without a source dictionary. + +## 9. Site-model boundary + +The exact derivation above is for spanning subgraphs / bond-FK configurations on a graph embedded in the torus. Matching One's primary numerical model is square **site** percolation with the matching graph. + +The site model already has the exact digital-Alexander rank variable `r` and source `exp[h(r-1)]`, so the observable itself is unambiguous. What is not asserted here is that the square-site transfer is literally a Krushkal polynomial specialization without a decorated/incidence-graph construction. + +For the massive Potts/FK continuum route of #782, however, the bond/FK formulation is exactly the relevant setting. + +## 10. Literature interface + +Krushkal's graph-on-surface polynomial introduces variables that record the genus of a regular neighborhood of a spanning subgraph and of its complement, with a duality motivated in part by Potts/Tutte statistical mechanics. Equivalent information is carried by the Bollobas--Riordan polynomial for ribbon graphs. + +The torus identity in Section 2 is a direct specialization of those definitions to the two-dimensional symplectic homology of `T^2`; it is not an additional literature theorem. + +Primary references to inspect for the trace realization rather than the state-sum definition: + +- V. Krushkal, *Graphs, links, and duality on surfaces*, arXiv:0903.5312; +- the Bollobas--Riordan / Krushkal ribbon-graph equivalence and subsequent transfer/algebra realizations; +- toroidal Potts/TQFT work where twisted sectors are required in partition functions. + +## 11. Claim boundary + +Exact: + +```text +r-1=(s-s_perp)/2 on T^2, +A=e^h, B=e^-h realizes exp[h(r-1)] in the Krushkal topological monomial. +``` + +Strong interface conclusion: + +```text +the finite rank0/rank2 topological projector already exists as a standard surface-graph state-sum coordinate. +``` + +Open: + +```text +a local affine-TL/modified-trace realization compatible with massive Potts and Q->1, +and the resulting continuum matrix elements/scaling functions. +``` + +This turns an undefined-projector problem into a representation/transfer-realization problem. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/literature-calibration-frontier-20260914.md b/docs/manuscripts/geometric-balance/literature-calibration-frontier-20260914.md new file mode 100644 index 000000000..4c875038d --- /dev/null +++ b/docs/manuscripts/geometric-balance/literature-calibration-frontier-20260914.md @@ -0,0 +1,373 @@ +# lit-note-C.md —— 长程前沿文献定标（条目 1–8，完整版） + +**作者**：lit-frontier-longb（长程检索专员）　**日期**：2026-09-14 +**范围**：只做检索与写作。**未做任何计算、未做任何仓库写操作**（GitHub 只读）。不需要云机。 +**继承**：`lit-note-B.md`（R/J/U）、`lit-note-B2.md`（1a/2/3/J(d)/5）、`round45-out/arms-audit.md`。 +本文只补**新条目**与本轮**新发现**；冲突处**显式标出**，不重做已覆盖部分。 + +**格式纪律**：每条给可检索标识（arXiv id / DOI / 期刊卷页）；找不到写「未找到，已检索关键词：…」。 +**加引号 = 直接引用；否则 = 转述**。「本项目内部笔记」**不作为文献证据**。 + +--- + +## 0. 总表 + +| 条目 | 已有结果 | 出处（可检索标识） | 关系 | 证据强度 | +|---|---|---|---|---| +| **1** square-site matching function `M_L` 的有限尺寸修正指数 | **有两份已发表测量，结论相反**。①根位移指数 `Δ₁ = 4.000 1(2)`（= 本项目「根指数 4」）②`M_L(p_c)~L^{2−x}` → `2−x = −3.42`；`p_L^⋆−p_c` 斜率 `−4.07`（推得 `w = −4.17`） | ① **Jacobsen 2015**, *J. Phys. A* **48** 454003, arXiv:`1507.03027`（§7.1，式 (36)(40)）② **Mertens & Ziff 2016**, *Phys. Rev. E* **94** 062152, arXiv:`1603.07289`（式 (38)(39)，图 6/7） | **①支持 `x=21/4 ⇒ h=21/8`；②冲突** | ① 直接数值（n ≤ 21，TM 特征值法）＋CFT 佐证；② 直接数值（L=3–7 精确 + 16/24/32/48 MC，作者自认系统太小） | +| **1′** `alpha_j=(j²−1)/12` 的出处 | **已发表**：`x_ℓ^P = (ℓ²−1)/12`（ℓ = 非重叠 path/臂数），显式「extends to half-integers the Saleur–Duplantier exponents」 | **Aizenman, Duplantier, Aharony 1999**, *Phys. Rev. Lett.* **83** 1359–1362, arXiv:`cond-mat/9901018`（式 (1)(2)） | **已蕴含**（arms-audit §1.3 的裁定获原文确认） | 已证（PRL） | +| **2** 判别臂数的 `c_slope` 渐近先例 | **未见完全对应量**。最接近的方法学先例 = **对数导数 / 导数比 FSS 估计子**（其极值随 `L^{1/ν}` 标度）＋「有效比值函数」逼近极限 | 对数导数 FSS：U(1) gauge–Higgs 双模拟论文（*Nucl. Phys. B* 通篇；「logarithmic derivatives of moments … their maxima also scale with」）；Ferrero et al. arXiv:`0803.3339`（`s^{1/ν} = 1 + x ∂_x F_ξ`）；有效比值函数：Shchur–Berche–Butera, *Nucl. Phys. B* **811** (2009) 491–518, DOI `10.1016/j.nuclphysb.2008.10.024` | **未见同口径量；方法学相邻** | 方法学先例充分，**对象不同** | +| **3** 环面按**同调秩**分辨的占据计数 `C[L,j,k]` | 已发表的拓扑分类是**绕行方向型**（`Z_0/Z_1/Z_2`），**不是**占据子图的**同调秩**；同调分辨**存在**但只在**高维** | `Z_·`：**Scullard–Jacobsen / arXiv:`2010.02887`**（式 (6) 上下文）；同调分辨：**Duncan–Kahle–Schweinhart**, arXiv:`2011.11903` (math.PR)；环面精确多项式到 L≤12：**Akhunzhanov–Eserkepov–Tarasevich**, arXiv:`2204.01517`, J. Phys. A **55** 204004, DOI `10.1088/1751-8121/ac61b8` | **口径相邻（绕行型 vs 同调秩）、对象不同** | 已发表，但**不是**本项目那张双下标整数表 | +| **4** `√(cE)·Z` ⇒ Laplace（四阶矩 6） | **命名恒等式**：**Gauss–Laplace transmutation**；标准表述「Gaussian × exponential-variance mixture = Laplace」 | **Ding & Blitzstein**, arXiv:`1510.08765`（§1）；更早 **Andrews & Mallows 1974**、**West 1987**；应用 **Park & Casella 2008**（Bayesian Lasso） | **已蕴含（恒等式，无条件）** | 已证（命名恒等式） | +| **5** 稀有大簇的稳定 CLT / 随机区间加性泛函 | **框架齐备**（marked point process CLT/stable、stabilizing functionals、Gibbs 条件原理、Palm 渐近正态），但**「随机（指数长）区间上的加性泛函 + 完整 component-Palm 锚点」的具体陈述未见** | Penrose, arXiv:`math/0410021`（stabilizing 泛函 CLT，**应用含「critical percolation 簇计数」的白噪极限**）；Basrak–Wintenberger–Zugeč, arXiv:`1903.09387`（marked Poisson cluster：CLT 或**无穷方差稳定律**）；Onaran–Bobrowski–Robert, *Electron. J. Probab.*（动态点过程**局部加性泛函 CLT**）；Kipnis–Varadhan 1986；Ferré–Hervé–Ledoux 2012（arXiv:`1201.4579`）；Prokešová–Jensen（Palm 似然渐近正态，DOI `10.1007/s10463-012-0376-7`） | **技术空白（非原理空白）** | 框架 = 已证；具体陈述 = 未见 | +| **6** 孔洞场 / cavity field / `9/8` 极点 | (a) 「纵向步长至多 1」长步机构**未见**；相邻=**长程渗流**（power-law 长边，非「步长 ≤1」）。(b) **孔洞/内边界**在渗流文献**有**，且**孔洞 ↔ 互补（白）簇的对偶**是标准的。(c) **fugacity-倾斜的活动矩阵 + 谱端点 → 极点**有样板：**R-矩阵 / Yang–Lee 零点**，`ρ(z)=Σ_λ ψ_1(λ)²/(z^{−1}+λ)`，奇异点 = 谱端点。(d) `1/9` / `9/8` **未见** | (b) **Isichenko**, *Rev. Mod. Phys.* **64** (1992) 961–1043（内 hull ↔ 互补簇）；Hu 等的 hole 幂律 + 「largest hole」分布（*Physica A* 2021, DOI `10.1016/j.physa.2021.125847`）(c) **arXiv:`0907.4037`**（R-矩阵谱倒数 = Yang–Lee 零点）；Collatz–Wielandt / Perron–Frobenius（Meyer, *Matrix Analysis*, ch. 8） | (b)(c) **方法学已蕴含**；(a)(d) **未见** | (b)(c) 已发表；(a)(d) 空白 | +| **7** first-exit 盒与向量指数矩域 | **未见**「向量指数矩域给方向范数内外界」的现成结果。可引用的样板：**Collatz–Wielandt 显式向量界**（非负矩阵 Perron 根的双侧夹逼）＋「inclusion interval」的严格理论；first-exit 尾律的标准工具 = Freidlin–Wentzell LDP | **「Upper bounds on the growth rates of hard squares and related models」**, *DMTCS*（Lemma 2 = Collatz–Wielandt：`min_i (Ax)_i/x_i ≤ λ ≤ max_i (Ax)_i/x_i`）；**Oepomo**, *Electron. J. Linear Algebra* **10** (2003) 31–45（Zbl 1022.15018，Collatz 特征值 inclusion interval）；**Lifshits–Shi**, *Bernoulli* **8** (2002) 745–765（Zbl 1018.60084，first exit + LDP 尾律）；Freidlin–Wentzell | **构件齐备，组合未见** | 构件已证；组合 = 未见 | +| **8** `#757` 投稿先例矩阵（低优先） | 几何/渗流**精确判据**文献充分：临界多项式（任意周期 basis、任意 2D 周期格）、`P_B(p)=R_2−R_0`、`P(A,B,C)=P(Ā,B̄,C̄)`、site 渗流经 covering lattice 归约；且**「full scaling law」被该领域明确列为 OPEN** | **Scullard–Jacobsen**, arXiv:`1209.1451`（J. Phys. A **45** 494004）；arXiv:`1207.3340`（deletion–contraction）；**is a graph invariant**：PRE 2012, PubMed `23214553`；**Jacobsen–Scullard 2019/2020**, arXiv:`1910.12376`（摘要明列 open question「the full scaling law」） | 「任意周期 / 精确判据」**已有**；「两次出生 / balance root / matching root」**未找到** | 部分已发表；术语未见 | + +--- + +## 1. 条目 1（最高优先）：`h=21/8` 的独立数值证据**存在，但是「一支持一冲突」** + +### 1.1 被引文献自己的数（**已逐式核对，直接引用**） + +**Mertens & Ziff 2016**（*Percolation in finite matching lattices*, **Phys. Rev. E 94, 062152**, DOI `10.1103/PhysRevE.94.062152`, arXiv:`1603.07289`；下称 **MZ16**）。 + +MZ16 把 Sykes–Essam 关系做**有限尺寸推广**，定义 **matching function** +``` +M_L(p) = N_L(p) − N̂_L(1−p) − L²χ(p) （MZ16 式 (15)） +``` +并证明它**精确等于**若干**绕行概率之差**（式 (20)）： +``` +M_L(p) = R_L^x(p) − R̂_L^x(1−p), x ∈ {c(交叉绕行), b(两向), e(任一方向), h(横向)} +``` +（式 (17) 特别给出 `M_L(p) = R_L^b(p) − R̂_L^b(1−p)`。） + +**关键直接引用（MZ16 式 (38)(39) 及其后正文）**： +> `M_L(p) = A₂ L^{2−x} + 2B₁ b L^{1/ν}(p−p_c) + C₂ L^{2−y}(p−p_c)² + 2D₁b³L^{3/ν}(p−p_c)³ + …`　(38) +> 「That is, `M_L(p_c)=A₂L^{2−x}`, … In Fig. 6, using exact and Monte-Carlo data, we plot these quantities vs. `L` on a log-log plot. **These plots give `2−x = −3.42`** and `2−y = 0.705`.」 +> `p_L^⋆ − p_c ∼ L^{2−x−1/ν}`　(39) +> 「The numerical value for `x` implies that this exponent has the value **`w = 2−x−1/ν = −3.42−3/4 = −4.17`**, somewhat larger than the value `4` suggested by Jacobsen [Jacobsen 2015]. In Fig. 7 we show the results for `p_L^⋆−p_c` … and find a **slope of `−4.07`** for this criterion.」 +> 「If we assume that `w` is exactly `−4`, then **`x−2 = 3.25` exactly**.」 + +**对照 arms-audit 的链条**（`x = 2 − y_t + |根指数|`，`y_t = 3/4`；`omega = x − 2`）： + +| 输入 | `x` | `h = x/2` | `24h+1` | 完全平方? | +|---|---|---|---|---| +| MZ16 实测 `2−x = −3.42` | 5.42 | 2.71 | 66.04 | ❌ | +| MZ16 实测斜率 `−4.07` | 5.32 | 2.66 | 64.84 | ❌ | +| **`21/4`（目标）** | 5.25 | **21/8** | **64** | ✅ | + +→ **arms-audit §3 的数逐字复核无误**。MZ16 自己写「Presumably, larger systems are needed to find the true behavior」——**他们没有宣称 `−3.42/−4.07` 是渐近值**。 + +### 1.2 ⚠️ 本轮新发现（arms-audit 未覆盖）：**存在一份同模型、支持 `4` 的独立数值测量** + +**Jacobsen 2015**（*Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras*, **J. Phys. A: Math. Theor. 48 (2015) 454003**, arXiv:`1507.03027`；下称 **J15**）。 + +J15 **§7.1 标题即 "Site percolation on the square lattice"**（= 本项目**完全相同的模型**；同为 §7.2 的才是 kagome bond）。**直接引用**： +> `p_c(n) = p_c + Σ_{k≥1} A_k n^{−Δ_k}, with 0 < Δ_1 < Δ_2 < ⋯`　(34) +> 有效指数由逐差对数导数给出：`Δ_1(n) = [log δp_c(n) − log δp_c(n−1)] / [log n − log(n−1)]`　(35) +> **`Δ_1 = 4.000 1(2).`**　(36)　「This agrees well with the value `w = 4.03 ± 0.01` … **It appears inevitable to admit that `Δ_1 = 4` exactly.**」 +> `Δ_2 = 6.00(1)`　(39)　「we henceforth admit that `Δ_2 = 6` exactly.」；`Δ_3 = 8.0(5)`，「we conjecture that `Δ_3 = 8`」。 +> `p_c(n) = p_c + Σ_{k≥1} A_k / n^{2(k+1)}`　(40)（即 **`Δ_k = 2(k+1)`**） +> 规模 **n ≤ 21**；`p_c = 0.592 746 050 792 10(2)`。 + +**CFT 依据（直接引用，§8）**： +> 「In the continuum limit there is no difference between whether the propagating cluster is an FK cluster or a dual FK cluster. Therefore `f_open(n)` and `f_closed(n)` both determine the **same critical exponent, namely `x_m`**, and they both scale like `f_1(n)` in (47).」 +> 「The values `c = 0` and **`x_m = 5/48`** are of course known」 +> `p_c(n) − p_c = O(n^{−4})`　(50)　「and moreover the corrections appear to be `O(n^{−6})`, `O(n^{−8})`, and so on.」 + +**为什么关键**：J15 的 `Δ_1` 与 MZ16 的 `w` **是同一个物理指数**（临界多项式/匹配函数的根位移指数）。而 +``` +w = x − 2 + 1/ν +J15: w = 4 ⟹ x = 4 + 2 − 3/4 = 5.25 = 21/4 ⟹ h = 21/8 ✅ +MZ16: w = −4.17 ⟹ x = 5.42 ≠ 21/4 ⟹ 21/8 不成立 ❌ +``` +MZ16 自己承认这个冲突：「somewhat larger than the value 4 suggested by Jacobsen」。 + +### 1.3 判定（条目 1） + +> **问题**：有没有任何**已发表的、独立的**对 square-site matching polynomial（或 `M_L=P2−P0` / matching function）**有限尺寸修正指数**的测量或证明？ + +**答：有，且有两份，结论相反。** + +| # | 被测量 | 数值 | 出处 | 对 `21/8` 的含义 | +|---|---|---|---|---| +| A | 根位移指数 `w`（= 项目「根指数」） | **4.000 1(2)**（n ≤ 21，TM 特征值法，**方形格点 site 渗流**） | **J15** §7.1 式 (36)(40)，arXiv:`1507.03027` | **支持** `x=21/4 ⇒ h=21/8` | +| B | 直接测 `M_L(p_c)~L^{2−x}` 的 `2−x` | **−3.42**（L=3–7 精确 + 16/24/32/48 MC） | **MZ16** 式 (38)，图 6 | **冲突**（`21/4` 要求 `−3.25`） | +| C | `p_L^⋆−p_c` 拟合斜率 | **−4.07**（同 B 数据）；由 B 推 `w=−4.17` | **MZ16** 式 (39)，图 7 | **冲突**（`21/4` 要求 `−4`） | + +**三条并存事实**： +1. **A 与 C 是同一个指数**（都等于根位移/阈值估计子收敛指数），**A 说 4，C 说 −4.07**。二者不可能同时渐近成立。 +2. **没有任何第二份**对 `M_L(p_c)` 本身指数（`2−x`）的独立测量 —— 只有 MZ16 的 **B**。所以 `−3.25`（`21/4` 的直接后果）**从未被直接测过**。 +3. **两测法内核不同**：J15 用**转移矩阵特征值相等**（半无限圆柱周长 n ≤ 21，**非 Monte Carlo**）；MZ16 用**环面**精确枚举 + MC（L ≤ 48）。**几何不同**（圆柱 vs 环面），按普遍性该指数应相同。 + +**诚实结论**：`21/8` 这条链**不是**「无独立证据」，而是「**有一份支持（J15 `Δ_1=4`，同模型、精度高，n ≤ 21）、一份冲突（MZ16 `2−x=−3.42` / 斜率 `−4.07`，系统小、作者自认不够）**」。**判定生死需第三方**： +- 用 J15 的特征值法把 `p_c(n)−p_c` 的 `Δ_1` 推到 n > 21，看是否稳定在 4； +- 或用 MZ16 的框架把 `M_L(p_c)` 推到 L ≳ 100，看 `2−x` 趋向 `−3.25` 还是 `−3.42`。 + +**⚠️ 对 arms-audit §3 措辞的最小修正**：不要写「被引文献自己的数字让 `21/8` 消失」，应写「**MZ16 的小尺寸数据与 `21/4` 不符，但同一指数的独立测量 J15（同模型、n≤21）给出 `Δ_1=4.0001(2)`，与 `21/4` 一致；两者冲突，`21/8` 的生死未被这两篇判定**」。 + +### 1.4 附带确认：`alpha_j = (j²−1)/12` 是**已发表公式** + +**Aizenman, Duplantier, Aharony 1999**（**Phys. Rev. Lett. 83 (1999) 1359–1362**, arXiv:`cond-mat/9901018`）**直接引用**摘要与式 (1)(2)： +> 「2D Percolation path exponents `x_ℓ^P` describe probabilities for traversals of annuli by **`ℓ` non-overlapping paths** … whose exponents, believed to be exact, yield **`x_ℓ^P = (ℓ²−1)/12`**. **This extends to half-integers the Saleur–Duplantier exponents for `k = ℓ/2` clusters**, yields the exact fractal dimension of the external cluster perimeter, `D_EP = 2 − x_3^P = 4/3`…」 +> 式 (1)：`x_k^C = x_{ℓ=2k}^{O(N=1)} = (4k²−1)/12`（k = 簇数，ℓ = 2k = 线/臂数）；式 (2)：`D_H = 2 − x_1^C = 7/4`。 + +**含义**：`alpha_j=(j²−1)/12` **不是本项目自造**，是 **ADA 1999 / Saleur–Duplantier** 的已发表 path/arm 指数公式（`j` = 臂/线数）。与 arms-audit §1.3 的裁定完全一致：它与 `c=0` spinless Kac `x_bulk(k)=(k²−1)/12` **是同一个二次族**，故「8-arm ⇒ `x=21/4` ⇒ `h=21/8`」是**恒等式而非独立证据**。**arms-audit 该裁定成立，已由原始文献逐字确认。** + +--- + +## 2. 条目 2：判别臂数的 `c_slope` 渐近先例 + +**待定标量**：`c_slope = h1 = (log c)_z`，其中 `c = C_L(b)` 是自归一化曲线，`b = (1/2) log(P0/P2)`。 + +### 2.1 最接近的方法学先例（**三族**） + +**(i) 对数导数 / 导数比的 FSS 估计子（最标准的一族）** +FSS 里常用「可观测量对控制参数的对数导数」作 `1/ν` 的估计子；其**极值随 `L^{1/ν}` 标度**。直接引用一例（U(1) gauge–Higgs 双模拟，θ = π 临界端点）： +> 「we also determine ν by studying observables that have the same scaling behavior as U, for instance the **logarithmic derivatives of moments** of the topological charge. In particular we study the derivatives (44) **which have maxima that also scale with** `L^{1/ν}`. Thus these derivatives can again be fit as described in (43) and allow for an **independent determination of ν**.」 +同族的显式 FSS 函数导数式（Ferrero et al., 4D Ising spin glass, arXiv:`0803.3339`）： +> `s^{1/ν} = 1 + x ∂_x F_ξ(x, s)|_{x=x_ξ(L^{-ω})}`；以及 `s^{1/ν} = 1 + g ∂_g F_g(g, s)|_{g=g(L^{-ω})}` + +**(ii) 「有效比值函数」（effective ratio functions）逼近极限** +**Shchur–Berche–Butera**, *Numerical revision of the universal amplitude ratios for the two-dimensional 4-state Potts model*, **Nucl. Phys. B 811 (2009) 491–518**, DOI `10.1016/j.nuclphysb.2008.10.024`。直接引用摘要： +> 「we estimate ratios of critical amplitudes, constructing **effective ratio functions**, and computing their **limiting values at the critical point**.」 +→ 即「构造一个 L-依赖的有效比值，再研究它随 L 的**逼近速率**」，与 `c_slope` 的「随尺寸的渐近」在**方法学上同型**（但对象是临界振幅比，非自归一化曲线）。 + +**(iii) 有限尺寸修正系数的普适比（Izmailian–Hu 型）** +**Okabe–Kawashima**, *Universal relations in the finite-size correction terms of two-dimensional Ising models*, arXiv:`cond-mat/0107514`（直接引用，转述其引用的 Izmailian–Hu, PRL **86** (2001) 5160）： +> `N(f_N − f_∞) = Σ_{k≥1} a_k/N^{2k−1}`，`ξ_N^{−1} = Σ_{k≥1} b_k/N^{2k−1}`，且 **`b_k/a_k = (2^{2k}−1)/(2^{2k−1}−1)`** 普适；`a_1 = cπ/6`、`b_1 = 2π x_H`。 +→ 「两个级数系数之比是普适的」，是「比值型观测量」的**精确**样板。 + +### 2.2 判定（条目 2） + +**未找到完全对应的已发表量**：「**自归一化曲线对平衡变量** `b=(1/2)log(P0/P2)` **的对数导数 `(log c)_z` 随尺寸的渐近**」—— +**已检索关键词：** `self-normalized observable`, `log-derivative of balance observable`, `finite-size asymptotics of derivative ratios`, `ratio observable finite-size scaling`, `effective ratio function critical amplitude`, `universal amplitude ratio finite-size correction coefficients`, `logarithmic derivative finite-size scaling estimator`, `Binder cumulant derivative pseudo-critical scaling`, `finite-size scaling derivative of FSS function correlation length exponent`, `modulus/plateau approach exponent`. +**最接近的先例 = (i) 对数导数/导数比 FSS 估计子**（其极值随 `L^{1/ν}` 标度；出处见 §2.1(i)），以及 **(ii) 有效比值函数**。**投稿时建议**：把 `c_slope` 明确定位为「(i) 的一例」，并说明与 (i) 的差别在于**对数导数取自平衡变量 `b` 而非控制参数**。 + +--- + +## 3. 条目 3：环面按**同调秩**分辨的占据计数是否已有人发表 + +### 3.1 已发表的是**「绕行方向型」三分**，不是**「同调秩」** + +**直接引用**（**arXiv:`2010.02887`**, *Critical polynomials in the nonplanar and continuum percolation models*）： +> 「All the configurations `{C}` on the torus are classified into three types as `{Z_0}`, `{Z_1}`, and `{Z_2}` according to their topological properties. … a configuration `C` belongs to `{Z_2}` if it **wraps along two different directions**, to `{Z_1}` if it **wraps along one and only one direction**, and to `{Z_0}` if it **does not wrap**. `R_2, R_1, R_0` … generally the critical polynomial is defined as `P_B ≡ R_2 − R_0`.」 + +同一分类亦见 **MZ16**（`R_L^e / R_L^h / R_L^s(螺旋) / R_L^b / R_L^1 / R_L^c(交叉绕行)`，arXiv:`1603.07289` §II 条目列表）。 + +**口径差（关键）**：`Z_0/Z_1/Z_2` 判的是**配置中是否存在某个簇的绕行**（winding 非零），即**存在性/方向**；而 `#775` 的 `j = r_black(ω)` 是**占据子图的环境同调秩** `r = dim H_1 ∈ {0,1,2}`。二者**不同**： +- 「横向与纵向由**不同簇**分别绕行」⇒ `Z_2` 成立，但**无单簇 cross-wrap**； +- `r = 2` 需两个**独立**非可缩圈，可由**两个不同簇**提供。 +- 故 **`R_j`（绕行型概率）≠ `Pr[r_black = j]`（同调秩分布）**，不能把 `Z_·` 计数当 `C[L,j,k]`。 + +### 3.2 环面**同调分辨**研究**存在**，但对象是**高维** + +- **Duncan, Kahle, Schweinhart**, *Homological percolation on a torus: plaquettes and permutohedra*, **arXiv:`2011.11903`** (math.PR, v1 2020-11-24, v4 2023-09-29)。**直接引用摘要**： + > 「We study higher-dimensional homological analogues of bond percolation on a square lattice and site percolation on a triangular lattice. … finite cell complexes … with the topology of the torus `T^d`. When random subcomplexes induce nontrivial `i`-dimensional cycles in the homology of the ambient torus, we call such cycles **giant**. We show that for every `i` and `d` there is a sharp transition from nonexistence of giant cycles to giant cycles spanning the homology of the torus. … we prove that `p_c = 1/2` in the case of middle dimension `i = d/2` for both models. This gives finite-volume high-dimensional analogues of Kesten's theorems…」 + → **同调分辨**（giant cycle = 非平凡同调类）**是**已发表概念，且就在**环面**上；但为**高维 plaquette / permutohedral**，2D 退化为经典 Kesten。**未给**「按 `r_black ∈{0,1,2}` 与 `|ω|=k` 双分辨的精确整数表**」。 + +### 3.3 判定（条目 3） + +> **问题**：是否有人发表过环面上按 cycle rank / homology rank 分辨的占据子集计数？ + +**答**： +- 「环面 + 同调分辨」研究**存在**（Duncan–Kahle–Schweinhart, arXiv:`2011.11903`），但为**高维同调渗流的相变/阈值**，**不是** 2D square-site 的 `C[L,j,k]`。 +- 标准**「拓扑三分」**（`Z_0/Z_1/Z_2`）是**绕行方向型**，**口径不同于同调秩**；`#775` 要的按 `r_black`（`0/1/2`）双下标整数表——**未找到已发表版本**。 + **已检索关键词：** `rank generating polynomial`, `Tutte polynomial torus`, `cycle rank distribution`, `homology resolved counting`, `Aizenman Duplantier Aharony`, `Akhunzhanov Eserkepov Tarasevich`, `wrapping polynomial`, `topological sector counting`, `Betti number distribution subgraph`, `random subgraph homology`, `critical polynomial Z_0 Z_1 Z_2 torus`, `homological percolation torus giant cycles`, `exact percolation probabilities torus cylinder plane polynomial`, `simplicial homology random configuration torus`, `cycle rank distribution random subgraph`. +- **`#752` 的 wrapping polynomial 覆盖哪一部分**：`R_2 − R_0`（= 临界多项式 `P_B`）与 `R_j`（`j=0,1,2`，**绕行型**）已被 **Scullard–Jacobsen / arXiv:`2010.02887`** 与 **Akhunzhanov–Eserkepov–Tarasevich**（*Exact percolation probabilities for a square lattice: Site percolation on a plane, cylinder, and torus*, **arXiv:`2204.01517`**, J. Phys. A **55** (2022) 204004, DOI `10.1088/1751-8121/ac61b8`；**torus L ≤ 12 的精确多项式**）覆盖。**但它们分的是「绕行方向型」，不是「同调秩」**。若 `#752` 指这套，则**不覆盖** `C[L,j,k]`，**口径差 = 绕行型 vs 同调秩**。 + +--- + +## 4. 条目 4：`normal × exponential` 混合 = Laplace —— **命名恒等式** + +### 4.1 标准出处（**直接引用**） + +**Ding & Blitzstein**, *Representation for the Gauss–Laplace Transmutation*, **arXiv:`1510.08765`**（§1）： +> 「**The Gauss–Laplace transmutation** states that `V ∼ 2Exp(1), L|V ∼ N(0,V) ⟹ L ∼ Laplace`, or equivalently, if `Exp ∼ Exp(1)` is independent of `Z ∼ N(0,1)`, then **`L = √(2Exp)·Z ∼ Laplace`**.」 +> 「Some proofs of the Gauss–Laplace transmutation exist in the literature (**Andrews and Mallows, 1974; West, 1987**)…」 +> 「…crucial to efficiently simulate posterior distribution of the **Bayesian Lasso (Park and Casella, 2008)**, which imposes **Laplace priors**…」 + +同义表述（转述，佐证非唯一出处）：「compounding a Gaussian with **exponential** variance (or Rayleigh standard deviation) yields a **Laplace**」；属 **variance-gamma 族** gamma 形状参数 = 1 的特例。 + +### 4.2 与本项目写法的对应（**含 `c` 的版本**） + +`√(cE)·Z = √(c/2)·√(2E)Z = √(c/2)·L`，其中 `L` 为标准 Laplace。取 Laplace scale `λ = √(c/2)`，密度 `(1/(2λ))e^{−|y|/λ}` 即 **`(1/√(2c))e^{−√(2/c)|y|}`** —— 与本项目写法**逐字一致**。`c = 2` 即标准 Laplace（方差 2，四阶矩 24 ⇒ **标准化四阶矩 24/2² = 6** ✅）。 + +### 4.3 判定 + +**是已命名恒等式**：**Gauss–Laplace transmutation**（normal–exponential scale mixture）。**无需额外条件**，只要 `E ⊥ Z` 且 `E ∼ Exp`；换 `Exp(λ)` 只改 `c` 的标定。 +可检索标识：**arXiv:`1510.08765`**；渊源 **Andrews & Mallows 1974**、**West 1987**；应用 **Park & Casella 2008**。 +→ **此条不必作为新结果证明，引用即可。** + +--- + +## 5. 条目 5：稀有大簇的稳定 CLT / 随机区间上的加性泛函 + +**待定标物**（`#772` 路线 A）：supercritical site 的有限半径局部近似、**随机区间上的加性泛函**、rare barrier 的 marked point process、**稳定 CLT / 随机信息 LAN**；特别地「**完整 component-Palm 的锚点选择不能被无条件块 CLT 自动覆盖**」。 + +### 5.1 已有的现成框架（**逐条给标识**） + +| 需要的构件 | 已有结果 | 出处 | +|---|---|---| +| 可加泛函 CLT / 函数 CLT | 可逆 Markov 链上可加泛函 CLT + FCLT | **Kipnis–Varadhan 1986**, *Comm. Math. Phys.* **104** 1–19；**Ferré–Hervé–Ledoux 2012**, *Ann. I.H.P. B* **48**(2) 396–423, arXiv:`1201.4579` | +| **随机几何上的加性泛函 + CLT（含渗流应用）** | 「stabilizing 泛函」的 LLN/CLT，**应用明列「critical percolation 簇计数」的白噪极限** | **Penrose**, *Multivariate spatial central limit theorems with applications to percolation and spatial graphs*, arXiv:`math/0410021` | +| **marked point process 的 CLT 与稳定 CLT** | 「we find sufficient conditions under which the total claim amount satisfies the **central limit theorem** or alternatively tends in distribution to an **infinite variance stable random variable**」 | **Basrak–Wintenberger–Zugeč**, *On total claim amount for marked Poisson cluster models*, arXiv:`1903.09387` | +| **（动态）点过程局部加性泛函的有限维 CLT** | 「finite-dimensional central limit theorems for **local, additive, interaction functions** of temporally evolving point processes … via a distributionally equivalent **marked point process**」 | **Onaran–Bobrowski–Robert**, *CLTs for Local Functionals of Dynamic Point Processes*, *Electron. J. Probab.*（ISSN 1083-6489） | +| **随机区间上的加性泛函（古典）** | 随机区间装箱：`N(x)` 的 LLN + **Dvoretzky–Robbins CLT**（RSA / random interval packing） | 见 **arXiv:`1311.4967`** 引言对 Dvoretzky–Robbins 的转述引用 | +| **条件（rare-event）下的极限** | Gibbs 条件原理 → 指数倾斜的 Markov 过程（driven process） | **Csiszár 1984 / van Campenhout–Cover 1981 / Dembo–Zeitouni**；**Chetrite–Touchette**, arXiv:`1405.5157`, *Ann. Henri Poincaré* **16** (2015)（亦见 `lit-note-B2.md` §4） | +| **Palm 条件下的渐近正态 / 非退化性** | Palm 似然估计子的**强相合 + 渐近正态**（Neyman–Scott、log-Gaussian Cox）；Slivnyak 区分 reduced/non-reduced Palm | **Prokešová–Jensen**, *Asymptotic Palm likelihood theory for stationary point processes*, *Ann. Inst. Statist. Math.* (2013), DOI `10.1007/s10463-012-0376-7`；Palm 教程 arXiv:`1512.05871` | +| **临界点上的非高斯「分形 CLT」** | 「**Fractal Central Limit Theorem**」(long-range correlations) holds at the unstable, critical fixed point | *Stochastic renormalization group in percolation: I*, *Physica A*（ScienceDirect S0378437102012128） | + +### 5.2 判定（条目 5） + +- **框架齐备**：随机几何上的加性泛函 CLT 有 **Penrose arXiv:`math/0410021`**（且**直接应用在渗流的簇计数**上）；marked point process 的 CLT/**稳定律**有 **Basrak–Wintenberger–Zugeč arXiv:`1903.09387`**；rare-event 条件化有 **Gibbs 条件原理 / Chetrite–Touchette**；Palm 条件下的渐近理论与**非退化条件**（Fisher 信息 / 二阶矩）有 **Prokešová–Jensen**。 +- **未见**：「**随机（指数长度）区间上的加性泛函**」＋「**完整 component-Palm 的锚点选择**」的**具体联合陈述**（即「锚点选择」这一步在文献里通常由 **Slivnyak/Campbell 公式**处理，而**无条件块 CLT 不给锚点条件分布**——这与 spec 的判断一致）。 + **已检索关键词：** `additive functional random interval`, `marked point process CLT`, `stable CLT rare event`, `random information LAN`, `exploration process Poisson cluster`, `Fisher projection nondegeneracy`, `conditional CLT under conditioning`, `Palm conditioning`, `stabilizing functional CLT percolation`, `Palm likelihood asymptotic normality`, `fractal central limit theorem percolation`, `exponential length interval additive functional`. +- **裁定**：**技术空白（非原理空白）**。攻击路径 = 先证条件律 → 指数倾斜律（Gibbs），再对倾斜律用 Penrose 型 stabilizing 泛函 CLT；**锚点选择**那一步须用 **Palm/Slivnyak** 显式处理，不能由无条件块 CLT 自动给出。 + +--- + +## 6. 条目 6：孔洞场 / cavity field / `9/8` 极点 + +### 6.1 (a) 「纵向步长至多 1」型长步/长笼罩机构（issue 里称 KING） + +**未找到**该机制。相邻文献是**长程渗流（long-range percolation）**，即**边概率随距离幂律衰减** `P(r) ~ C r^{−s}`（**不是**「步长 ≤ 1」）： +- **Crawford–Sly**, *Simple Random Walk on Long Range Percolation Clusters II: Scaling Limits*, arXiv:`0911.5668`；后续 arXiv:`2403.18532`。**直接引用**（前者）：当 `s ∈ (d, d+1)`，无穷簇上简单随机游走的标度极限收敛到 **α-稳定 Lévy 过程**，`α = s − d`，quenched 与 annealed 皆成立。 +- Kesten 本人的长程渗流工作（Durrett–Kesten；Grimmett–Keane–Marstrand 的连通判据；Kesten 对 `Z^{d−e}×Z^e_+` 的**可和性充要条件**），见 **Grimmett**, *Harry Kesten's work in probability theory*, arXiv 版与 *PTRF* (2021) DOI `10.1007/s00440-021-01046-4`。 +**已检索关键词：** `Kesten long range percolation long step`, `long run mechanism renormalization`, `stretched cluster long step`, `directional step bound percolation mechanism`, `KING mechanism percolation`. + +### 6.2 (b) cavity field / hole field / 孔洞机制 + +**已有**（但**机制不同**）： +- **孔洞 ↔ 互补（白）簇的对偶**是标准的：**Isichenko**, *Percolation, statistical topography, and transport in random media*, **Rev. Mod. Phys. 64 (1992) 961–1043**。**直接引用**（转述自其 §2）：「an internal hull can be considered to be the **external hull of a complementary cluster of vacant sites** that fills up a hole in the original cluster」；并给出内/外 hull 的不同普适指数（外 hull `d_h = 7/4`，`D_H = 2 − x_1^C`；unscreened perimeter `d_u ≈ 1.343`）。 +- **孔洞尺寸分布**：Hu 等发现「`n_h ~ h^{−τ}`，`τ = 1 + d_f/d`」（hyperscaling），且 hole 是 **volatile fractal**；**largest hole** 满足 `h_max = ⟨C/L^d⟩ ≈ h_{max,0} + a L^{d_H − d}`（`d_H = 7/4`/`4/3`），见 *Size distributions of the largest hole in the largest percolation cluster and backbone*, *Physica A* (2021), DOI `10.1016/j.physa.2021.125847`（S0378437121000789）。 +- 教学式「above `p_c` 的簇像 **Swiss cheese**，洞的典型尺寸 = ξ」（见 Geometry of Clusters, Springer 2024, DOI `10.1007/978-3-031-59900-2_5`）。 +**但**：spec 描述的「**黑 NN（`p↓0`）的孔洞场拉长巨大白簇**」这一**具体机制未见**。 +**已检索关键词：** `cavity field percolation`, `hole field percolation`, `holes percolation cluster scaling`, `swiss cheese cluster holes`, `internal hull complementary cluster`, `volatile fractal holes backbone`, `cavity mechanism random cluster`. + +### 6.3 (c) fugacity 倾斜的活动矩阵（非行随机）+ 谱半径的极点/收敛半径判定 + +**有样板**：**R-矩阵 / Yang–Lee 零点**方法——**arXiv:`0907.4037`**（*Critical exponents from cluster…*）。**直接引用**： +> 「It is always possible … to define a tridiagonal symmetric R matrix which satisfies `(R^n)_{11} = (−1)^n (n+1)b_{n+1}` … `ρ(z) = Σ_{n≥1} n b_n z^n = z(I + zR)^{−1}_{11}`。」 +> 「Alternatively … `ρ(z) = Σ_λ ψ_1(λ)² / (z^{−1} + λ)` … **The reciprocals of the eigenvalues of this matrix are the Yang-Lee zeroes of the grand-canonical partition function.** … `ρ(z)` has two singular points at `z` values for which `−z^{−1}` coincides with the **spectrum edges** of the R matrix, leading to vanishing of the denominator.」 +→ 这正是「**fugacity `z` 依赖的矩阵谱 → 极点 → 收敛半径/奇异性**」的**严格样板**（`z^{−1}` 型极点，与 spec 的 `u + v = 1/z ≠ 1` 同型）。 +另：Perron 根的**双侧夹逼**与 **Collatz–Wielandt 公式**是标准工具（Meyer, *Matrix Analysis*, ch. 8；见 §7）。 +**已检索关键词：** `fugacity tilted transfer matrix`, `Perron root fugacity`, `activity transfer matrix percolation`, `Yang-Lee zero spectrum edge`, `radius of convergence activity generating function`, `singularity analysis generating function pole`. + +### 6.4 (d) 几何分布参数 `1/9` 或极点 `9/8` + +**未找到。** **已检索关键词：** `geometric distribution 1/9 percolation`, `pole 9/8 singularity percolation`, `9/8 exponent percolation`, `geometric parameter one ninth cluster`, `rational singularity 9/8 lattice model`. + +### 6.5 判定（条目 6） + +- **(b)(c) 方法学已蕴含**：孔洞/内 hull 的对偶（Isichenko RMP 1992；Hu 等 Physica A 2021）与「fugacity 倾斜矩阵谱端点 → 极点」（arXiv:`0907.4037`）都是**已发表样板**，**可直接引用**。 +- **(a) 「纵向步长至多 1」长步机构** 与 **(d) `1/9` / `9/8`** **未见**；`9/8` 若真出现，需自查是否为本项目**自建口径**产生。 + +--- + +## 7. 条目 7：`#761` / `#766` first-exit 盒与向量指数矩域 + +**待定标**：`#761` 独立质量区间、优化 first-exit 盒；`#766` first-exit 的**向量指数矩域**给方向范数提供**可计算内外界**。 + +### 7.1 现成构件 + +**(i) 非负矩阵 Perron 根的双侧夹逼（=「可计算内外界」的严格工具）** +**«Upper bounds on the growth rates of hard squares and related models»**, *DMTCS*。**直接引用 Lemma 2（Collatz–Wielandt）**： +> 「Let `A` be an irreducible square matrix with non-negative entries. Then for any vector `x > 0`, the largest eigenvalue of `A` (denoted `λ`) is real and positive and is bounded by `min_i (Ax)_i/x_i ≤ λ ≤ max_i (Ax)_i/x_i`。」 +且该文**正是**用它给**转移矩阵主特征值** `Λ_o(m)`（宽度 m 圆柱的列转移阵）做**上界**，且用 CTM 型向量逼近主特征向量以获得**紧界**： +> 「we do not compute the eigenvalue exactly. Instead we find upper bounds for `Λ_o(m)` using the **Collatz-Wielandt formula**。」 +→ **这就是 spec 要的「转移矩阵特征值 enclosure / Perron root 界」的可用先例**（它给上界；双侧界即 `min` 与 `max` 同时算）。 + +**(ii) 「inclusion interval」的严格理论** +**Oepomo**, *A contribution to Collatz's eigenvalue inclusion theorem for nonnegative irreducible matrices*, **Electron. J. Linear Algebra 10 (2003) 31–45**（Zbl `1022.15018`）。**直接引用（zbMATH 综述）**： +> 「the ‘coherence’ (i.e. simultaneous closeness) of the **Collatz–Wielandt lower and upper estimates** `m(x)` and `M(x)` of `Λ[A]` (**forming an “inclusion interval”**) for variable positive `x`'s … implying that the set of all the inclusion intervals forms a two-dimensional wedge-shaped domain.」 +→ 「**区间套收敛**」的严格结果，可直接支撑「**证书式内外界**」。 + +**(iii) first-exit 尾律的标准工具** +- **Lifshits–Shi**, *The first exit time of Brownian motion from a parabolic domain*, **Bernoulli 8 (2002) 745–765**（Zbl `1018.60084`）：用 **LDP（Schilder）+ 变分**给出 `lim T^{−(p−1)/(p+1)} log P(τ_D > T)`，其中把**加性泛函的 Biane–Yor 定理**用于求解变分问题（`d=a=1, p=2` 时 `−3π²/8`）。 +- **Freidlin–Wentzell** 框架（first exit / Arrhenius）：`lim_{ε→0} ε log E τ_D^ε = inf_{x∈∂D} V(x)`（见 arXiv:`2306.11418` 综述式 (7)）。 +- 跳跃扩散 first-exit 的 **MGF/均值 PDE–积分方程**：**Lefebvre**, *Similarity Solutions of PDIE from the Theory of Stochastic Processes*, *Symmetry* **17** (2025) 704, DOI `10.3390/sym17050704`（含 **moment-generating function of the first-passage time** 的 PDIE）。 + +### 7.2 判定（条目 7） + +- 「**Perron 根 / 转移矩阵主特征值的可计算内外界**」**已有严格样板**：Collatz–Wielandt（**DMTCS** 用它对**转移矩阵**做界）＋ inclusion interval 理论（**Oepomo EJLA 10 (2003) 31–45**）。**可直接引用，不必重造。** +- 「**first-exit 尾律**」的标准工具是 **Lifshits–Shi（Bernoulli 2002）** 与 **Freidlin–Wentzell**。 +- **未见**：「**向量**指数矩域给**方向范数**提供**内外界**」这一**组合**。 + **已检索关键词：** `first exit box`, `exponential moment domain`, `directional norm certificate`, `certified bounds mass cylinder`, `transfer matrix eigenvalue enclosure`, `Perron root interval arithmetic`, `Collatz bound`, `Collatz-Wielandt transfer matrix`, `first exit time large deviation tail`, `moment generating function first passage PDE`, `eigenvalue inclusion interval nonnegative matrix`. +- **裁定**：**构件齐备、组合未见**。**建议**：把 `#766` 明确定位为「(i)+(ii) 的向量化/方向化推广」，并说明与 Oepomo 的 inclusion interval 的差别（后者是**标量**谱半径，`#766` 要**方向范数**）。 + +--- + +## 8. 条目 8（低优先）：`#757` 投稿先例矩阵 + +**待定标**：几何论文的**逐定理先例矩阵**——任意整数周期、两次出生（two-birth）、周期簇。 + +**已有（几何/渗流精确判据）**： +- **Scullard–Jacobsen**, *Transfer matrix computation of generalised critical polynomials in percolation*, **arXiv:`1209.1451`**, *J. Phys. A* **45** (2012) 494004。**直接引用**：「the critical polynomial `P_B(p)` … may be defined on **any periodic lattice**. The polynomial depends on a finite subgraph `B`, called the **basis**, and the way in which the basis is **tiled** to form the lattice.」；`P(A,B,C) = P(Ā,B̄,C̄)`（式 (1)）即精确判据，`A,B,C` 为三角形**三边界顶点三连通/三不连通**概率；「we can also treat **site percolation** problems by reasoning on the **covering lattice** or by introducing **correlations**」。 +- **Scullard**, *The computation of generalized percolation critical polynomials by the deletion–contraction algorithm*, **arXiv:`1207.3340`**（deletion–contraction 定义；任意 2D 周期格）。**直接引用（式 (2)）**：hexagonal 的临界曲面 `H(p,r,s) ≡ prs − pr − ps − rs + 1 = 0`；并给出 `FE(p,r,s,t,u,v) = pA(r,s,t,u,v) + (1−p)H(s, ur, tv)` 型**一阶（first-order in each argument）**递推。 +- **Scullard**, *Percolation critical polynomial as a graph invariant*, **Phys. Rev. E** 2012, PubMed `23214553`（「the generalized critical polynomial can be viewed as a **graph invariant**, similar to the Tutte polynomial … can be found using the recursive **deletion–contraction** algorithm」）。 +- **Jacobsen–Scullard 2019/2020**, **arXiv:`1910.12376`**：给出两类有限尺寸修正指数（`Δ = 6,7,8` 与 `Δ = 4,6,8`），**并在摘要明列 open question**： + > 「We discuss the open questions related to the method, such as **the full scaling law**, as well as its potential for determining critical points of other models.」 + +**未找到**：「**两次出生（two-birth）**」、「**balance root**」、「**matching root**」、「**周期簇（periodic cluster）**」作为**已发表术语/定理**。 +**已检索关键词：** `arbitrary period percolation`, `born distribution percolation`, `two-birth percolation`, `periodic cluster percolation`, `balance root percolation`, `matching root percolation`, `critical polynomial full law`, `full-law criterion percolation`, `critical polynomial graph invariant deletion-contraction`, `basis tiling periodic lattice critical polynomial`. + +### 判定（条目 8） +- 「**任意整数周期**」（= 任意周期 lattice 与任意 basis tiling）与「**精确判据**」**在临界多项式文献中已有系统处理**，**可逐条引用**（arXiv:`1209.1451`、arXiv:`1207.3340`、PRE 2012 PubMed `23214553`、arXiv:`1910.12376`）。 +- 「**两次出生 / balance root / matching root / 周期簇**」**未见已发表术语**；建议在 `#757` 的投稿矩阵里**把这几项标为「未见，需自建定义并声明术语新」**，**不要**假定它们有先例。 +- **利好消息**：`1910.12376` 明确把「**the full scaling law**」列为**该领域 OPEN question** —— 若 `#757` 的「full-law criterion」正是这一条，则**领域承认它是开放的**，投稿定位反而更好。 + +--- + +## 9. 检索轮次日志（≥6 轮，每轮关键词） + +| 轮 | 目标条目 | 用过的关键词（原样） | +|---|---|---| +| R1 | 1 | `Mertens Ziff matching polynomial square lattice finite-size correction exponent`；`Jacobsen matching polynomial square lattice critical exponent 21/8`；`8-arm exponent 21/8 percolation matching c=0 Kac table` | +| R2 | 1 | `correction to scaling exponent L^-4 wrapping probability percolation torus irrelevant exponent`；`"matching polynomial" percolation finite size correction exponent W_4 W_8 winding number square lattice`；＋ 下载 arXiv:`1603.07289v2`（PDF）与 arXiv:`2204.01517`（PDF） | +| R3 | 1/3 | `Jacobsen 2015 percolation threshold correction exponent 4 critical polynomial "L^{-4}"`；`Scullard Jacobsen critical polynomial winding configurations Z_0 Z_1 Z_2 torus homology counting`；`cycle rank distribution random subgraph torus Tutte polynomial rank generating function counting` | +| R4 | 1/3/4 | `Jacobsen 2015 "critical polynomial" scaling exponents conformal field theory L^{-4} site percolation square lattice`；`Betti number distribution random subgraph torus homology rank cycle rank counting occupied sites`；`normal exponential scale mixture Laplace distribution variance gamma named identity` | +| R5 | 1/2/5 | `matching function exponent 3.42 OR 3.25 percolation Mertens Ziff finite-size correction measurement later`；`"self-normalized" observable logarithmic derivative finite-size asymptotic ratio of observables critical exponent estimate`；`central limit theorem additive functional random interval marked point process cluster Palm rare event` | +| R6 | 1/3/6 | `Aizenman Duplantier Aharony "wrapping" OR "topological" counting configurations cycle rank torus percolation exact enumeration`；`"cavity field" OR "hole field" percolation random cluster long range correlation white cluster mechanism`；＋ 下载 arXiv:`1507.03027` HTML、arXiv:`2011.11903` 摘要页并 grep §7.1/作者 | +| R7 | 2/5/6 | `universal amplitude ratio finite-size scaling derivative ratio observable plateau approach exponent logarithmic derivative order parameter`；`fugacity tilted transfer matrix Perron root spectral radius singularity activity generating function percolation`；`Kesten long range percolation long step mechanism stretched cluster renormalization "long run"` | +| R8 | 2/5/6 | `"logarithmic derivative" finite-size scaling estimator correlation length exponent pseudo-critical Binder cumulant derivative`；`stable central limit theorem infinite variance critical cluster exploration conditional invariance principle percolation`；`percolation activity transfer matrix fugacity "activity representation" Yang-Lee singularity radius of convergence cluster weight 1/z` | +| R9 | 7/8/6 | `Collatz-Wielandt bound Perron root enclosure interval arithmetic nonnegative matrix transfer matrix eigenvalue certified bounds`；`"first exit" box additive functional exponential moment generating function domain large deviation rate`；`matching polynomial root percolation threshold "arbitrary period" periodic cluster birth distribution balance root` | +| R10 | 8/6/5 | `"critical polynomial" percolation "full" law algorithm arbitrary lattice basis number of births exact thresholds Chen Li`；`holes in percolation clusters large hole scaling number of holes hull swiss cheese cluster topology`；`Palm conditioning central limit theorem nondegenerate Fisher information exploration process Poisson cluster functional` | + +共 **10 轮**、**30 次检索查询**＋ **4 次全文抓取/核对**（MZ16 PDF、Jacobsen 2015 HTML 逐式、homological percolation 摘要、arXiv:`2204.01517` 摘要）。 + +--- + +## 10. 对主代理的直接影响 + +**(A) 可以直接引用、不必再证的** +- **条目 4**：`√(cE)Z ∼ Laplace`（含 `c` 的标定）= **Gauss–Laplace transmutation**（Ding–Blitzstein, arXiv:`1510.08765`；渊至 Andrews–Mallows 1974 / West 1987）。**直接引，别当新结果。** +- **条目 1′**：`alpha_j = (j²−1)/12` 是 **ADA 1999 PRL 83 1359**（`x_ℓ^P = (ℓ²−1)/12`，`ℓ` = 臂/线数）的已发表公式。**arms-audit §1.3 的裁定成立且已由原文确认。** +- **条目 3 的 `Z_0/Z_1/Z_2` 与 `P_B = R_2 − R_0`**：**Scullard–Jacobsen / arXiv:`2010.02887`**。 +- **条目 6(b)(c)**：孔洞 ↔ 互补簇对偶（**Isichenko RMP 64 (1992) 961**；Hu 等 *Physica A* 2021）；fugacity 依赖矩阵谱端点 → 极点（**arXiv:`0907.4037`**，R-矩阵/Yang–Lee）。 +- **条目 7**：Collatz–Wielandt 对**转移矩阵主特征值**做界（**DMTCS** Lemma 2）＋ inclusion interval（**Oepomo EJLA 10 (2003) 31–45**）。 +- **条目 5**：随机几何加性泛函 CLT（**Penrose arXiv:`math/0410021`**，**应用含渗流簇计数**）；marked PP 的 CLT/稳定律（**arXiv:`1903.09387`**）；Palm 渐近正态（**Prokešová–Jensen**）。 +- **条目 8**：临界多项式（任意周期 basis / 任意 2D 周期格 / deletion–contraction / graph invariant）：**arXiv:`1209.1451`、arXiv:`1207.3340`、PRE 2012 PubMed `23214553`、arXiv:`1910.12376`**。 + +**(B) 必须改述的（重要）** +- **条目 1 的措辞**：**不能**写「文献数字让 `21/8` 消失」。正确表述： + - **MZ16**（PRE 94 062152）**直接测** `M_L(p_c)~L^{2−x}` 得 `2−x = −3.42`（`21/4` 要求 `−3.25`）；阈值估计子斜率 `−4.07`（要求 `−4`）；但**作者自认系统太小**（L ≤ 48），**未宣称渐近**。 + - **J15**（J. Phys. A 48 454003, arXiv:`1507.03027` §7.1，**同一方形格点 site 渗流模型**）用**转移矩阵特征值法**（圆柱周长 n ≤ 21，**非 MC**）测得 **`Δ_1 = 4.0001(2)`**，并**承认 `Δ_1 = 4` 精确**；CFT 依据 = 两拓扑扇区共享 `x_m = 5/48`。 + - `Δ_1` 与 MZ16 的 `w` **是同一个指数**。**A 说 4、C 说 −4.07，二者冲突**；MZ16 自己写「somewhat larger than the value 4 suggested by Jacobsen」。 + - **结论**：`21/8` **既未被证实也未被证伪**；`x=21/4 ⟺ w=4` **有一份高精度独立支持（J15）和一份低精度冲突（MZ16）**。 +- **条目 3 的措辞**：必须**显式区分** **「绕行方向型」`Z_0/Z_1/Z_2`** 与 **「占据子图同调秩」`r_black ∈{0,1,2}`**。二者**不等价**（可由两个不同簇分别绕行得到 `Z_2` 但无单簇 cross-wrap；反之 `r=2` 也可由两个不同簇提供两个独立非可缩圈）。**不能**把 `Z_·` 计数当 `C[L,j,k]`。 +- **条目 6(a)(d)**：`KING`/「纵向步长 ≤ 1」长步机构、`1/9` / `9/8` 极点 —— **未见**；若真要用，须自查是否为本项目**自建口径**的产物，投稿时**不得**暗示有先例。 + +**(C) 真空白 / 值得投入的（按价值排序）** +1. **`M_L(p_c)` 本身的指数 `2−x` 没有第二份独立测量** —— **这是唯一能判定 `21/8` 的靶子**。把 MZ16 的 `M_L(p_c)`（等价地 `R_L^b − R̂_L^b`）在**环面**上推到 `L ≳ 100`（或精确可算的最大 L），看 `2−x` 趋向 `−3.25` 还是 `−3.42`。**本轮最值得投入的真空白。** +2. **条目 3 的 `C[L,j,k]`（按**同调秩**分辨，而非绕行方向型）未找到已发表版本** —— 口径确实比 `Z_0/Z_1/Z_2` 更细。**但投稿必须在文中显式对照 `Z_·` 与 `R_j`**，否则会被认为与 Scullard–Jacobsen 重复。（可引 **Duncan–Kahle–Schweinhart arXiv:`2011.11903`** 作为「环面同调分辨」的最近邻，并说明它是**高维**、2D 退化为经典。） +3. **条目 5 的「随机区间加性泛函 + 完整 component-Palm 锚点」联合陈述未见** —— 技术空白、风险可控：攻击路径 = Gibbs 条件原理 → 指数倾斜律 → Penrose 型 stabilizing CLT，**锚点**用 Palm/Slivnyak 显式处理。 +4. **条目 7 的「向量指数矩域 → 方向范数内外界」组合未见**，但**构件齐备**（Collatz–Wielandt / Oepomo inclusion interval / Lifshits–Shi first-exit）。**投入产出比高**。 +5. **条目 2 的 `c_slope`**：**未见同口径量**，但**方法学先例充分**（对数导数 FSS 估计子）—— **定位为「(i) 的一例」即可，不必重造理论**。 + +**(D) 可信度标注** +- **均为直接引用（WebFetch / HTML 逐字核对）**：MZ16 式 (15)(17)(20)(38)(39) 及正文的 `2−x=−3.42`、斜率 `−4.07`、`w=−4.17`、`x−2=3.25`；J15 式 (34)(35)(36)(39)(40)(50) 及 `Δ_1=4.0001(2)`、`x_m=5/48`、§7.1 标题；ADA 式 (1)(2)；`2010.02887` 的 `Z_·` 与 `P_B`；`2011.11903` 摘要；`2204.01517` 摘要；Ding–Blitzstein §1；arXiv:`0907.4037` 的 R-矩阵/`ρ(z)` 公式；DMTCS Lemma 2；Oepomo 综述（zbMATH Zbl 1022.15018）；arXiv:`0911.5668` 摘要；arXiv:`1910.12376` 摘要（含「the full scaling law」）；Isichenko RMP 1992 的内 hull 对偶；Physica A 2021 的 largest-hole 标度。 +- **转述（非引号）**：Palm 教程 arXiv:`1512.05871`；Prokešová–Jensen；Lifshits–Shi（经 zbMATH Zbl 1018.60084）；Freidlin–Wentzell 综述（arXiv:`2306.11418`）；Lefebvre *Symmetry* 2025；arXiv:`1311.4967`（对 Dvoretzky–Robbins 的转述）。 +- **本项目内部算术（非文献）**：表格中「`21/4` 要求 `2−x = −3.25`」「`w = x−2+1/ν`」的换算（arms-audit §3 已给）。 +- **未找到的项**：§2.2、§3.3、§5.2、§6.1、§6.4、§7.2、§8 均已列出**本轮全部关键词**。 diff --git a/docs/manuscripts/geometric-balance/loop-branch-linear-tail-tests-20260914.md b/docs/manuscripts/geometric-balance/loop-branch-linear-tail-tests-20260914.md new file mode 100644 index 000000000..96491986b --- /dev/null +++ b/docs/manuscripts/geometric-balance/loop-branch-linear-tail-tests-20260914.md @@ -0,0 +1,132 @@ +# Linear-tail tests implied by the loop--branch variational candidate + +2026-09-14. Consequences of the **candidate** #758 rate function, not a proof that the actual SITE Palm span obeys that LDP. The purpose is to turn the variational proposal into sharp falsification tests that can reuse existing rare-component machinery. + +## 1. Candidate rate and its saturation point + +The proposed fixed-subcritical NN component-Palm rate is + +\[ +I_p(A)=\min_{0\le r\le A} +\{\tau_p(1,2r)-\kappa+\kappa(A-r)\}, \tag{1.1} +\] + +with `kappa=tau_p(1,0)=tau_p(0,1)`. + +As shown in `structural-consequences-20260914.md`, strict convexity of the directional norm implies a unique `r_*(p)>0`, independent of `A`, and + +\[ +I_p(A)= +\begin{cases} +\tau_p(1,2A)-\kappa,&A\le r_*,\\ +\kappa A+c_*,&A\ge r_*. +\end{cases} \tag{1.2} +\] + +Square symmetry gives the deterministic bound + +\[ +\boxed{0=1/2` is already in the saturated linear-branch regime of the candidate.** + +## 2. Tail ratios for any two macroscopic thresholds beyond 1/2 + +If the actual Palm span obeys the candidate LDP + +\[ +-\frac1w\log P_{Palm}(L\ge\lceil Aw\rceil)\to I_p(A), \tag{2.1} +\] + +then for any fixed + +\[ +A_2>A_1\ge1/2, +\] + +one must have + +\[ +\boxed{ +-\frac1w\log +\frac{P(L\ge\lceil A_2w\rceil)} + {P(L\ge\lceil A_1w\rceil)} +\to\kappa(p)(A_2-A_1).} \tag{2.2} +\] + +The unknown intercept `c_*`, the main-loop cost, and any overall Palm normalization cancel. + +In particular the #762 exploratory thresholds `A=1` and `A=2` satisfy the parameter-free exponent prediction + +\[ +\boxed{ +-\frac1w\log +\frac{P(L\ge2w)}{P(L\ge w)}\to\kappa(p).} \tag{2.3} +\] + +This is a cleaner first test of the mechanism than attempting to identify the complete rate curve from a few widths. + +## 3. Finite extra-span ratios inside the linear regime + +Suppose a sufficiently uniform local LDP/refined tail asymptotic holds in the saturated regime. Fix `A>1/2` and an integer `k>=0`. Since + +\[ +I_p(A+k/w)-I_p(A)=\kappa k/w, \tag{3.1} +\] + +the candidate predicts the local tail ratio + +\[ +\boxed{ +\frac{P(L\ge\lceil Aw\rceil+k)} + {P(L\ge\lceil Aw\rceil)} +\longrightarrow e^{-\kappa k}} \tag{3.2} +\] + +up to lattice-periodic/subexponential corrections not determined by the LDP alone. + +Thus after the macroscopic bulge has saturated, each additional **microscopic** unit of extreme span costs the ordinary axial mass `kappa`. A persistent effective rate below `kappa` would exhibit a cheaper network mechanism than the proposed one. + +Equation (3.2) is deliberately labelled a refined-tail prediction; (2.2) follows already at the LDP level. + +## 4. Morphology implication, stated only at the variational coordinate level + +For `A>r_*`, the minimizer of (1.1) remains exactly `r_*` as `A` increases. Therefore any operational morphology statistic that is rigorously shown to converge to the variational loop coordinate `r` must **saturate** for all `A>=1/2`. + +This gives the correct interpretation of the #762 `A=1,2` comparison: + +- if a certified winding-core observable is proved to represent the loop coordinate, its `w`-scaled value should be the same at `A=1` and `A=2` to leading order; +- the additional unit of macroscopic span must be carried by the branch part of the optimizer; +- if the same certified core coordinate instead grows proportionally with `A`, the candidate rate mechanism is falsified. + +The existing operational `winding core` in #762 is **not yet** proved to be the variational/OZ core. This note therefore does not equate `L_core` with `2r_*w` or any other particular formula. The earlier articulation counterexample is precisely why that map must be proved before using morphology to validate the LDP. + +## 5. The transition point is itself a directional-norm observable + +At differentiability, + +\[ +2\,\partial_y\tau_p(1,2r_*)=\kappa. \tag{5.1} +\] + +Thus `r_*` can be predicted from a directional mass certificate without rare-span sampling. Conversely, once a morphology variable is rigorously tied to `r_*`, it becomes an independent probe of the Wulff shape. + +The curvature relation + +\[ +D^{-1}=\partial_{yy}\tau_p(1,0) \tag{5.2} +\] + +controls only the small-`A` quadratic start of the rate; `r_*` probes the nonlinear directional norm farther away from the axis. These are complementary geometric quantities. + +## 6. Strong falsification outcomes + +The candidate (1.1) should be rejected or revised if any rigorous/asymptotic result shows one of: + +1. a large-`A` slope strictly below `kappa`; +2. a nonlinear rate persisting for some `A>=1/2`; +3. a certified loop-coordinate optimizer continuing to grow with `A>=1/2`; +4. a cheaper network topology with the same nonzero deck displacement and span. + +A finite-width deviation from (2.3) without controlled errors is not such a falsification. The useful experiment is an exponent/certificate comparison, not another unconstrained fit. diff --git a/docs/manuscripts/geometric-balance/matching-enhancement-mass-gap-20260914.md b/docs/manuscripts/geometric-balance/matching-enhancement-mass-gap-20260914.md new file mode 100644 index 000000000..ce3cf7c1d --- /dev/null +++ b/docs/manuscripts/geometric-balance/matching-enhancement-mass-gap-20260914.md @@ -0,0 +1,291 @@ +# Matching enhancement gives a strict inverse-correlation-length gap + +2026-09-14. Author-level derivation for the actual square-site NN/matching pair. This note combines Grimmett--Li's facial-site interpolation with the corrected two-dimensional enhancement rerouting theorem of Balister--Bollobas--Riordan. It should still receive independent proof review before publication, but the previous `Directional Enhancement Lemma` is closed here rather than left as a conjectural interface. + +## 1. Statement + +Let `G4` be nearest-neighbour square-site percolation and `G8` its matching graph. For subcritical parameters write + +\[ +\kappa_4(p)=-\lim_{n\to\infty}\frac1n\log P_p^{G4}(0\leftrightarrow ne_1), +\qquad +\kappa_8(p)=-\lim_{n\to\infty}\frac1n\log P_p^{G8}(0\leftrightarrow ne_1). +\] + +**Theorem (strict matching mass gap).** For every + +\[ +00` + +\[ +\boxed{c(d)1.} +\] + +This strict centre asymmetry is separate from the possible nondifferentiability exceptional set in the median-centred Gumbel theorem. + +## 2. Facial-site interpolation + +Following Grimmett--Li, add one facial site to every square face and join it to the four corner vertices; call the resulting bipartite augmentation `hat G`. Under `P_{p,s}`: + +- original square-lattice vertices are independently open with probability `p`; +- facial sites are independently open with probability `s`. + +For connectivity between original vertices: + +\[ +s=0\quad\Longleftrightarrow\quad G4, +\] + +and + +\[ +s=1\quad\Longleftrightarrow\quad G8. +\] + +The second equivalence is exact: a passage through an always-open facial site connects any two open corners of one square, which is precisely the clique/matching connectivity of that face; conversely every matching diagonal is represented by the two-edge path through its facial site. + +Grimmett--Li use the same interpolation in their equation (5.2). For the radial finite event `v_0 <-> partial Lambda_n`, their Lemma 5.4 constructs a bounded local map from an original pivotal vertex to a nearby pivotal facial site, yielding by Russo's formula + +\[ +\partial_p\theta_n(p,s) +\le g(p,s)\partial_s\theta_n(p,s). \tag{2.1} +\] + +The issue is whether the same uniform comparison holds for the two-point event defining `kappa`. On the square lattice, the corrected enhancement proof of Balister--Bollobas--Riordan supplies exactly the path-rerouting input needed for this adaptation. + +## 3. The two-dimensional rerouting theorem that closes the local geometry + +Balister--Bollobas--Riordan, *Essential enhancements revisited* (2014), isolate the flawed general Aizenman--Grimmett path lemma and replace it with a statement they prove for site percolation on `Z^2`. + +In their notation, for every fixed inner size `m` there exists a fixed `r=r(m,2)` such that two induced red/green paths that enter `B_{r+2}` from outside, end near the origin, and have no red--green adjacency can be modified **only inside `B_r`** so that they reach prescribed opposite inner terminals while remaining mutually nonadjacent. Equivalently, an induced path through the origin with two remote endpoints can be rerouted inside a bounded ball while keeping the same remote endpoints and replacing the central segment by a prescribed clean connector. Their Section 5.1 proves this statement for the square lattice. + +They then show how such a bounded path surgery produces a finite-to-one pivotal map: after clearing irrelevant local sites, a pivotal connection consists locally of exactly the two arms of a shortest/induced path. The rerouting inserts the enhancement gadget between those arms without creating an alternate red--green connection. Endpoint-near pivotal sites are moved a bounded distance into the bulk by another bounded modification. All constants are independent of the overall connection length. + +We use this **proved square-lattice rerouting statement**, not the false unrestricted lemma identified in that paper. + +## 4. Two-terminal pivotal comparison + +For integer `n` let + +\[ +A_n=\{0\leftrightarrow ne_1\text{ in }\widehat G\}. +\] + +Call an original vertex `p`-pivotal if flipping that original vertex changes `A_n`, and a facial site `s`-pivotal if flipping the facial site changes `A_n`. + +### Lemma 4.1 (bounded pivotal conversion) + +Fix a compact parameter rectangle `K subset (0,1)^2`. There are constants `R0. +\] + +Choose a small `L>0` so that + +\[ +\delta=\gamma L>0, +\qquad p+\delta0` is independent of `n`. + +## 6. Strict mass gap + +Take `-n^{-1}log` in (5.2) and send `n->infinity`: + +\[ +\kappa_8(p)\le\kappa_4(p+\delta). \tag{6.1} +\] + +The #739 branch already proves, for `p0. \tag{6.2} +\] + +Applying (6.2) with `q=p+delta` gives + +\[ +\kappa_8(p) +\le\kappa_4(p+\delta) +<\kappa_4(p), +\] + +which proves the theorem. + +Notice that no OZ prefactor, p-analyticity, directional differentiability, or numerical estimate of either mass is required. + +## 7. Strict separation of the two exponential-aspect centres + +Let + +\[ +a(d)=\kappa_4^{-1}(d), +\qquad +c(d)=\kappa_8^{-1}(d), +\qquad +b(d)=1-c(d). +\] + +If `a(d)=p_c(G8)`, then automatically + +\[ +c(d)0`, + +\[ +\boxed{c(d)1.} \tag{7.1} +\] + +In the dual-even/odd birth coordinates, + +\[ +C_\infty(d)=\frac{a(d)+b(d)-1}{2}>0. \tag{7.2} +\] + +So throughout the separated-window exponential-aspect regime the limiting two-atom law has a strictly positive complement-odd centre displacement. This statement is independent of whether the mass is differentiable at the inverse images used by the Gumbel scaling theorem. + +## 8. Quantitative compact-interval version + +The pivotal map changes a bounded number of coordinates. On a compact `I subset (0,p_c(G8))` and a fixed interior facial interval, its finite-energy constant is uniform. Therefore the construction gives one `delta_I>0` such that + +\[ +\kappa_8(p)\le\kappa_4(p+\delta_I),\qquad p\in I, +\] + +provided `delta_I` is reduced to keep `p+delta_I0, +\qquad p\in I.} \tag{8.1} +\] + +This is a qualitative rigorous certificate; the constants inherited from the local surgery are extremely conservative and are not proposed as a numerical mass bound. + +## 9. Claim boundary and references + +This is an author-level proof assembled from published enhancement machinery plus the #739 mass-monotonicity inequality. It has not received an independent line-by-line audit. In particular a reviewer should check the finite-to-one bookkeeping in Lemma 4.1 and the endpoint pivotal relocation, rather than merely accepting the analogy with the radial event. + +Primary inputs actually read: + +- G. Grimmett and Z. Li, *Percolation critical probabilities of matching lattice-pairs*, Random Structures & Algorithms 65 (2024), 832--856: facial-site interpolation, Lemma 5.4, equations (5.2)--(5.5). +- P. Balister, B. Bollobas, O. Riordan, *Essential enhancements revisited*, arXiv:1402.0834: the corrected enhancement argument, especially Conjecture 7 and its proof for `d=2` in Section 5.1, and the finite-to-one pivotal-map reduction preceding it. + +The older unrestricted Aizenman--Grimmett combinatorial lemma is **not** used without the Balister--Bollobas--Riordan correction. diff --git a/docs/manuscripts/geometric-balance/matrix-sewing-unit-residue-20260914.md b/docs/manuscripts/geometric-balance/matrix-sewing-unit-residue-20260914.md new file mode 100644 index 000000000..a27155f6d --- /dev/null +++ b/docs/manuscripts/geometric-balance/matrix-sewing-unit-residue-20260914.md @@ -0,0 +1,228 @@ +# Matrix sewing lemma: finite memory preserves the unit logarithmic residue + +2026-09-14. This is a self-contained analytic lemma for a finite-state Markov-additive/cyclic kernel. It generalizes the scalar renewal-loop calculation already on the branch and isolates what **cannot** be blamed on finite local memory in the still-missing SITE sewing theorem. + +## 1. Finite-state cyclic kernel + +Let + +\[ +A(z,y)=\sum_{x\ge1}\sum_{j\in\mathbb Z}A_{x,j}z^xy^j +\] + +be a `d x d` matrix Laurent polynomial with nonnegative real matrices `A_{x,j}` and finite support. The longitudinal increment `x` is strictly positive. Assume: + +1. for real `z>0,y=1`, the nonnegative matrix `A(z,1)` is irreducible in a neighbourhood of the critical point; +2. there is a unique `R>0` with Perron root + \[ + \rho(A(R,1))=1; + \] +3. this Perron eigenvalue is simple, all other eigenvalues of `A(R,1)` have modulus strictly below one, and the joint support is aperiodic so `(R,1)` is the only dominant singular point modulo the Fourier period; +4. transverse reflection symmetry gives zero mean transverse increment under the critical Perron tilt and an effective variance `sigma_eff^2>0`. + +Put + +\[ +\kappa=\log R. +\] + +Define the cyclic loop coefficient + +\[ +\boxed{ +L_w=w[z^wy^0]\{-\log\det(I-A(z,y))\}.} \tag{1.1} +\] + +The logarithm has the exact formal expansion + +\[ +-\log\det(I-A)=\sum_{n\ge1}\frac1n\operatorname{tr}A^n. \tag{1.2} +\] + +Thus (1.1) is the finite-memory analogue of the scalar cyclic-renewal object: the trace closes the internal state, `1/n` unmarks the cyclic transition index, and `w` restores longitudinal translation. + +## 2. Perron transform and the diffusion constant + +Let `r,l` be positive right/left Perron vectors of `A(R,1)`, normalized by `l^T r=1`. The critical tilted transition on an edge carrying increment `(x,j)` is proportional to + +\[ +(A_{x,j})_{ab}R^x\frac{r_b}{r_a}. +\] + +After normalization this is a finite-state Markov-additive chain. Let + +\[ +\mu=E X>0 +\] + +be its mean longitudinal advance and let `sigma_eff^2` be the asymptotic variance per transition of the accumulated transverse displacement, including the state correlations. Set + +\[ +\boxed{D=\sigma_{eff}^2/\mu.} \tag{2.1} +\] + +Equivalently, let `lambda(z,t)` be the analytic Perron eigenvalue of `A(z,e^t)` near `(R,0)` and write `z=Re^u`. Standard analytic perturbation of a simple eigenvalue gives + +\[ +\partial_u\log\lambda(0,0)=\mu, +\qquad +\partial_t\log\lambda(0,0)=0, +\qquad +\partial_t^2\log\lambda(0,0)=\sigma_{eff}^2. \tag{2.2} +\] + +The last derivative is the Green--Kubo/asymptotic variance of the additive functional, not merely a one-step variance when internal states carry memory. + +## 3. The unit-residue theorem + +**Theorem.** Under the assumptions above, + +\[ +\boxed{ +L_w=\frac{e^{-\kappa w}}{\sqrt{2\pi D w}} +\left(1+O(w^{-1})\right).} \tag{3.1} +\] + +In particular, a finite amount of Markov memory changes `kappa` and `D` but creates **no independent multiplicative residue** in the normalized cyclic object (1.1). + +### Proof + +Factor the determinant locally using the simple Perron band: + +\[ +\det(I-A(z,e^{i\theta})) +=(1-\lambda(z,i\theta))G(z,\theta), \tag{3.2} +\] + +where `G` is analytic and nonzero near `(R,0)`. Hence + +\[ +-\log\det(I-A) +=-\log(1-\lambda)+\text{analytic}. \tag{3.3} +\] + +The analytic term has a strictly larger radius in the dominant direction (after shrinking the neighbourhood and using the assumed global aperiodicity), so it is exponentially/subdominantly smaller than the Perron singularity in the coefficient under discussion. + +For small real `theta`, the implicit function theorem supplies the unique root `R(theta)` near `R` of + +\[ +\lambda(R(\theta),i\theta)=1. \tag{3.4} +\] + +Write `R(theta)=R exp u(theta)`. Expanding `log lambda` with (2.2) at imaginary transverse tilt gives + +\[ +0=\mu u(\theta)-\frac12\sigma_{eff}^2\theta^2+O(\theta^4), +\] + +therefore + +\[ +\boxed{ +\log R(\theta)=\log R+\frac12D\theta^2+O(\theta^4).} \tag{3.5} +\] + +Near its simple zero, + +\[ +1-\lambda(z,i\theta) +=C(\theta)(1-z/R(\theta))+O((1-z/R(\theta))^2) +\] + +with `C(theta) !=0`. The constant `C(theta)` enters only the **analytic** additive term `-log C(theta)`. The logarithmic singular part is exactly + +\[ +-\log(1-z/R(\theta)), +\] + +whose coefficient satisfies the identity + +\[ +w[z^w]\{-\log(1-z/R(\theta))\}=R(\theta)^{-w}. \tag{3.6} +\] + +This is the source of the unit residue: Perron left/right overlaps and the derivative of the eigenvalue do not multiply the logarithmic coefficient. + +Fourier inversion in the transverse displacement now gives + +\[ +L_w=\frac1{2\pi}\int_{-\pi}^{\pi}R(\theta)^{-w}\,d\theta ++\text{subdominant}. \tag{3.7} +\] + +Aperiodicity makes the integral away from `theta=0` exponentially smaller. Insert (3.5) in a neighbourhood of zero and apply the ordinary one-dimensional Laplace method: + +\[ +\frac{R^{-w}}{2\pi}\int_{\mathbb R} + e^{-Dw\theta^2/2}\,d\theta +=\frac{R^{-w}}{\sqrt{2\pi Dw}}. +\] + +The finite-support analytic expansion gives the stated `O(1/w)` correction. Since `R^{-w}=e^{-kappa w}`, (3.1) follows. `square` + +## 4. Directional curvature is automatically the inverse diffusion constant + +The same Perron cumulant also controls the open-endpoint large-deviation cost of the Markov-additive chain. For endpoint transverse slope `v`, the local rate expansion is the Legendre transform of the transverse cumulant per unit longitudinal distance: + +\[ +\tau(1,v)=\kappa+\frac{v^2}{2D}+O(v^4). \tag{4.1} +\] + +Therefore + +\[ +\boxed{\partial_{vv}\tau(1,0)=D^{-1}.} \tag{4.2} +\] + +Writing the homogeneous norm as + +\[ +\tau(r\cos\theta,r\sin\theta)=r\kappa(\theta) +\] + +and using reflection symmetry gives + +\[ +\boxed{D^{-1}=\kappa(0)+\kappa''(0).} \tag{4.3} +\] + +Thus the diffusion/Wulff-curvature relation highlighted in `structural-consequences-20260914.md` is not a separate miracle in a finite-memory sewing model: it is an automatic consequence of the same twisted Perron band. + +## 5. What this settles about `sewing-with-memory.md` + +The actual complete SITE activity has a three-column local weight and therefore microscopic memory. This lemma shows: + +> **Finite local memory, by itself, does not change the `w^{-1/2}` closure power and does not create an arbitrary residue.** + +If an exact percolation sewing theorem identifies complete components with (1.1) for a finite-state kernel satisfying the hypotheses, then automatically + +\[ +\beta=1/2,\qquad \zeta=1, +\] + +in the notation of the branch's HK/unit-residue conjectures. + +Therefore any non-unit `zeta` or different power for the actual component density must come from a failure of at least one of the following identifications: + +1. the complete-component class is not exactly the cyclic trace/log-determinant object; +2. the natural mark/cut law introduces an extra insertion rather than pure cyclic unrooting; +3. the effective state is genuinely infinite and lacks a simple isolated Perron band/quasi-compact reduction; +4. more than one soft transverse band contributes at the same exponential rate; +5. lattice periodicity/aperiodicity leaves multiple dominant saddles. + +This is much narrower than saying that the three-column boundary memory itself leaves the prefactor unknown. + +## 6. Countable-state extension: the actual next theorem + +The small-`p` programme in `research-frontier-20260914.md` should aim to replace the finite matrix by a positive operator on a weighted Banach space of column/connectivity states. A sufficient package would be: + +- quasi-compactness and an isolated simple Perron eigenvalue near the physical root; +- analytic dependence on longitudinal/transverse fugacities; +- an aperiodic Markov-additive Perron transform with finite exponential moments; +- trace/Fredholm-determinant control strong enough that the cyclic connected object has the same single logarithmic singularity. + +Under those hypotheses the proof above carries over with `det` replaced by a suitable Fredholm determinant. Establishing those operator hypotheses for actual SITE complete components on a nonempty small-`p` interval would close the prefactor problem there without importing a two-point OZ amplitude. + +## 7. Claim boundary + +The finite-matrix theorem is an analytic statement about the explicitly defined cyclic object (1.1), not yet an identification of that object with actual SITE complete components. It does, however, remove **finite Markov memory** from the list of possible reasons for an anomalous power/residue and supplies the exact curvature--diffusion relation once the common kernel is established. diff --git a/docs/manuscripts/geometric-balance/monotone-cut-mark-filtration-20260914.md b/docs/manuscripts/geometric-balance/monotone-cut-mark-filtration-20260914.md new file mode 100644 index 000000000..3df0b0a55 --- /dev/null +++ b/docs/manuscripts/geometric-balance/monotone-cut-mark-filtration-20260914.md @@ -0,0 +1,166 @@ +# 从静态屏障到整条占据率过程：切分、孔洞标记与可调的内部噪声 + +2026-09-14。继续 #764 / #777；不改用键模型。读取 #771 head `122bc328f292d686b799bf510421c1b9db5225cb`。已读团队新写入的 `barrier-cavity-resonance-20260914.md`：宽度八的单参数 Poisson–几何律及其评分结论不重新计作本轮成果。这里新增的是角色交换的宽度四、真正的共同标签参数过程，以及保持领先拓扑而改变内部标记的列场。 + +全文的“时间”都是单调增加的占据参数，不是标签反复刷新的 dynamical percolation。关于实际 site 的结论是以下给出的推导；有限控制仅核对其中有限图接口。 + +## 1. 两个实际匹配对与一个共同定义 + +在无限圆柱 `(Z/wZ) x Z` 上，给所有站点独立 `U_v~Uniform(0,1)`。固定正列幅度 `a_0,...,a_(w-1)`，在参数 t 时把 v=(x,y) 染黑，当且仅当 + + U_v <= epsilon t a_x. + +先固定 w、a 和任意 `0=b+1的黑簇接触窗口，在某个t<=T时出现) + <= C epsilon^-b (epsilon T max_x a_x)^(b+1) + = O(epsilon). + +这里直接看t=T的黑配置，利用单调性同时控制所有早期参数。不能为每个t分别证明o(1)后就擅自推出全路径结论。 + +在补事件中，所有相关 essential 黑簇就是最小模板，完成后不再合并或变形；所有查询孔洞就是最小围栏。八/四点围栏的中心若随后染黑，连通黑簇达到b+1点，也已经被排除。小于b的其他黑簇可以很多，但不改变粗尺度的 essential 分割或查询孔洞。 + +因此在一个固定宏观空间—参数窗口里，实际模型与有限模板过程有概率1-O(epsilon)的一致对应。白簇的端点与模板锚点只差有界行数；远端用“尚未看见包围标签的两道屏障”的空区间尾控制，先限制宏观窗口再扩大。这个论证不使用固定p的OZ前因子、Gumbel仿射化或大宽度CLT。 + +## 4. 完成时间是一个真正的时空Poisson标记 + +对模板S，定义其完成参数 + + T_S = max_(v in S) U_v/(epsilon a_(x_v)). + +每个最小模板大小为b；对固定t， + + P(T_S<=t)=epsilon^b t^b product_(v in S) a_(x_v). + +将纵坐标乘epsilon^b。不同平移模板的支撑若不相交，完成时间独立；相交而不相同的两个模板并集至少b+1点。因此任意有界宏观窗口内所有相邻双事件的总概率为O(epsilon)。同模板的时间标记只记一次，不能把不同时间箱当独立副本。 + +将参数区间切成有限个箱，对每个模板只保留它所属的一个时间箱。局部依赖Poisson过程定理[AGG, Thm2]，或相同的阶乘矩展开，给出时空点过程收敛；再细化时间箱。极限在紧空间—参数窗口里几乎必然只有有限个点，且无相同时间或位置的冲突，所以按时间切分区间的映射在这些点配置上连续。 + +令 + + A_B = sum_屏障模板 product a_x, + A_H = sum_孔洞模板 product a_x, + r=A_H/A_B, + tau=A_B t^b. + +则最终得到两份独立Poisson测度： + + cuts: dz d tau, + holes: r dz d tau. (4.1) + +这里z=epsilon^b y。cuts删除/切分白色区间；holes给仍在区间中的位置增加一个持久标记。切分会把已有孔洞分给子区间，不重新生成它们。 + +这是同一批site标签的整条单调过程极限，不是分别在每个p重新抽样后人为拼接的过程。实际白簇的局部区间和内部孔洞计数在避开切分时刻的有限维分布上收敛；在[t0,T]上，把每个微观端点替换为对应模板锚点后，也得到局部cadlag路径收敛。微观站点仍会不断变色；不能把“宏观区间没切分”说成完整站点集合一点没变。 + +均匀参数时： + + A: (A_B,A_H,r)=(1,8,8), + B: (A_B,A_H,r)=(19,4,4/19). + +### 有限重叠证书 + +按每行模板集合求和（有序重叠对；自对只在b1里），均匀黑概率p下： + + w4 B: b1=1315 p^8, + b2=152 p^5+428 p^6+712 p^7; + w8 A: b1=249 p^16, + b2=32 p^12+64 p^13+112 p^14+32 p^15. + +w4的收敛误差比w8的纯图案误差慢，并不影响极限独立性。真实簇替换还有第3节O(epsilon)误差。这些多项式不是完整有限p过程的精确误差等号。 + +## 5. 角色交换的静态预测现在是定理后果 + +在时钟tau固定时按每个完整白簇一次取样。令E=tau×区间长度，则 + + E~Exp(1), H|E=e~Poi(r e), + E[exp(-sE) z^H]=1/[1+s+r(1-z)]. (5.1) + +所以 B 的实际w4低黑密度极限为 + + (19 p^4 L,H) -> (E,H), H|E~Poi((4/19)E), + P(H=k)=(19/23)(4/23)^k, + P(H=0)=19/23. (5.2) + +这是前一轮待验证的角色交换预测。取样对象是完整component Palm，不是均匀站点。第一种白色matching、w8的r=8结果是团队已有结果，本轮不重复计新。 + +从固定纵向位置观察，左右屏障距离各为Exp(tau)，所以 E_tag=tau L~Gamma(2,1)。于是 + + P(H_tag=k)=(k+1) r^k/(1+r)^(k+2), + P(H_tag=0)=(1+r)^-2. (5.3) + +w4的随机位置零孔概率是(19/23)^2；w8为1/81。以上关于H的结论首先是弱收敛/有界变换收敛，所列均值与方差属于显式极限模型；不由弱收敛单独推出实际有限p高阶矩。 + +## 6. 保持领先拓扑不变，却能改变内部孔洞噪声 + +对列幅度a有以下精确模板强度： + + A_B^B=19 product_(x=0)^3 a_x, + A_H^B=sum_(x=0)^3 a_x^2 a_(x-1) a_(x+1), + +故 + + r_B=(a0/a2+a2/a0+a1/a3+a3/a1)/19 >=4/19. (6.1) + +等号当且仅当a0=a2且a1=a3。保持product a_x=1，就保持整个领先cuts过程不变，不仅保持某一个平均值。 + +A模型则有 + + A_B^A=product_(x=0)^7 a_x, + A_H^A=sum_(x=0)^7 a_(x-1)^3 a_x^2 a_(x+1)^3, + r_A>=8. (6.2) + +最后一步由8个单项式的AM-GM得到：每个a在它们的总乘积中恰出现8次。等号要求对数列向量满足3 l_(x-1)+2l_x+3l_(x+1)=sum l；在8周期上，非恒定Fourier模式的特征值2+6cos(2pi j/8)均非零，所以仅常数列场达到等号。 + +两个保持乘积1的有理实例： + +| 模型 | a | A_B | r | +|---|---|---:|---:| +| w4 B均匀 | (1,1,1,1) |19|4/19| +| w4 B变形 | (2,1,1/2,1) |19|25/76| +| w8 A均匀 | 全1 |1|8| +| w8 A变形 | (2,1,1,1,1/2,1,1,1) |1|45/2| + +所以相同的领先屏障过程可以带着不同的孔洞过程。它们不是依赖重加权的对照，而是真实独立Bernoulli列概率。四列交替a=(u,u^-1,u,u^-1)更特别：它同时保持A_B和r_B，因此这层联合极限也看不见它；有限epsilon及更高阶图案仍能不同。 + +均匀列场分别使孔洞/屏障比最小。这不是说均匀p使任意有限系统的孔洞数最小；结论限于这里的固定宽度低密度过程与product固定的列幅度。 + +## 7. 空间慢调制接口 + +若a_x依赖z=epsilon^b y，且有界、远离0并分段连续，同一局部模板证明给cuts强度A_B(z)b t^(b-1) dzdt、holes强度A_H(z)b t^(b-1) dzdt。在有限固定时间，不再有普遍的Euclidean指数间隔；正确的空区间概率是exp[-t^b integral A_B(z)dz]。 + +以s(z)=integral A_B(z)dz改坐标可以拉平cuts，但孔洞保留比值r(z)=A_H(z)/A_B(z)。因此一个空间坐标变化一般不能同时拉平两种对象。随机环境若在此宏观尺度保持随机，应得到条件Poisson/无条件Cox模型；随机环境版本这里作为后续目标，不假设遍历或混合条件已经满足。 + +## 来源、复现与范围 + +[AGG] Arratia–Goldstein–Gordon (1989), Two moments suffice for Poisson approximations, Ann. Probab.17,9–25, Thm2。作者公开PDF第11印刷页的过程界及TV约定已读图核对。https://dornsife.usc.edu/larry-goldstein/wp-content/uploads/sites/221/2023/06/AGG-1.pdf + +[MZ] Mertens–Ziff, arXiv:1603.07289v2 §II，4/8周界连接及单绕行配对；没有把其有限匹配恒等式称为这里的时空极限定理。https://arxiv.org/html/1603.07289v2 + +内部：#771 `barrier-cavity-resonance-20260914.md`；前轮查询孔洞说明。当前脚本独立重建19个最小matching屏障、四点孔洞、物理提升与不同参数共同标签概率；不导入旧转移引擎。 + +复现：`python scripts/dilute_fragmentation_clock.py --output results/geometric-consistency/dilute-fragmentation-clock.json`；`python -m unittest discover -s tests -p test_dilute_fragmentation_clock.py`。 diff --git a/docs/manuscripts/geometric-balance/near-critical-isotropic-loop-branch-rate-20260914.md b/docs/manuscripts/geometric-balance/near-critical-isotropic-loop-branch-rate-20260914.md new file mode 100644 index 000000000..14499bb26 --- /dev/null +++ b/docs/manuscripts/geometric-balance/near-critical-isotropic-loop-branch-rate-20260914.md @@ -0,0 +1,271 @@ +# Conditional near-critical isotropic loop/branch rate: an explicit universal target + +2026-09-14. A deliberately bold but sharply falsifiable consequence of the #758 loop/branch candidate under one extra universality hypothesis: restoration of rotational invariance of the normalized inverse-correlation norm near criticality. + +This is **not** asserted as a square-site theorem. Rotational restoration/Wulff-rounding is rigorous for triangular-lattice site percolation through its near-critical scaling limit, but is not certified here for the square-site NN/matching pair. + +## 1. Input hypotheses + +Let + +\[ +\kappa_p=\tau_p(1,0). \tag{1.1} +\] + +Assume the #758 candidate span rate + +\[ +I_p(A) +=\min_{0\le r\le A} +\left\{ +\tau_p(1,2r)-\kappa_p+\kappa_p(A-r) +\right\}. \tag{1.2} +\] + +Assume additionally the near-critical isotropy statement + +\[ +\boxed{ +\frac{\tau_p(x)}{\kappa_p}\longrightarrow |x| +\quad\text{locally uniformly in }x, +\qquad p\uparrow p_c.} \tag{1.3} +\] + +The local uniformity may be weakened to the compact set of directions/ratios used below. + +Then + +\[ +\frac{I_p(A)}{\kappa_p} +\longrightarrow +J(A) +:=\min_{0\le r\le A} +\left[\sqrt{1+4r^2}-1+A-r\right]. \tag{1.4} +\] + +## 2. The optimal bulge is explicit + +Let + +\[ +g(r)=\sqrt{1+4r^2}-1-r. \tag{2.1} +\] + +Then + +\[ +g'(r)=\frac{4r}{\sqrt{1+4r^2}}-1. \tag{2.2} +\] + +The unique stationary point is + +\[ +16r^2=1+4r^2, \tag{2.3} +\] + +so + +\[ +\boxed{ +r_*=\frac1{\sqrt{12}}=\frac1{2\sqrt3} +=0.2886751345948129\ldots.} \tag{2.4} +\] + +Since `g'` is negative below `r_*` and positive above it, the constrained minimizer is + +\[ +\boxed{r_{opt}(A)=\min\{A,r_*\}.} \tag{2.5} +\] + +Thus the near-critical isotropic mechanism predicts a **universal saturation bulge**, independent of `A` once `A>r_*`. + +## 3. Explicit piecewise rate function + +For `A<=r_*`, the constraint is active and `r=A`. Therefore + +\[ +\boxed{ +J(A)=\sqrt{1+4A^2}-1, +\qquad +0\le A\le\frac1{2\sqrt3}.} \tag{3.1} +\] + +For `A>=r_*`, insert the stationary point. Since + +\[ +\sqrt{1+4r_*^2}=\frac2{\sqrt3}, \tag{3.2} +\] + +and + +\[ +\frac2{\sqrt3}-\frac1{2\sqrt3}=\frac{\sqrt3}{2}, \tag{3.3} +\] + +we obtain + +\[ +\boxed{ +J(A)=A+\frac{\sqrt3}{2}-1, +\qquad +A\ge\frac1{2\sqrt3}.} \tag{3.4} +\] + +The derivative matches at the transition because + +\[ +\left.\frac{4A}{\sqrt{1+4A^2}}\right|_{A=r_*}=1. \tag{3.5} +\] + +So `J` is `C^1`, strictly convex before `r_*`, and exactly linear afterwards. + +## 4. Small-span Brownian overlap + +Expand (3.1): + +\[ +J(A)=2A^2-2A^4+O(A^6). \tag{4.1} +\] + +Hence + +\[ +\boxed{I_p(A)\sim2\kappa_p A^2} \tag{4.2} +\] + +under the isotropic hypothesis. + +The Brownian-bridge range moderate-deviation exponent used on this branch is + +\[ +2A^2/D_p. \tag{4.3} +\] + +Matching (4.2) therefore gives the particularly sharp near-critical prediction + +\[ +\boxed{D_p\kappa_p\to1} \tag{4.4} +\] + +in the physical normalization used by the rate ansatz. + +The 2026 bond-FK renewal theorem rigorously supports only the weaker scale relation + +\[ +D_p\kappa_p\asymp1, \tag{4.5} +\] + +so (4.4) is a genuine normalization/universality test rather than a consequence of that literature. + +## 5. No-parameter targets for #762 + +The #762 morphology plan singled out conditioned macroscopic span thresholds `A=1` and `A=2`. + +Both satisfy + +\[ +A>r_* \tag{5.1} +\] + +by a wide margin, so the isotropic loop/branch mechanism predicts that both are already in the **linear branch regime**. + +The large-deviation costs are + +\[ +\boxed{ +\frac{I_p(1)}{\kappa_p}\longrightarrow\frac{\sqrt3}{2} +=0.8660254037844386\ldots,} \tag{5.2} +\] + +\[ +\boxed{ +\frac{I_p(2)}{\kappa_p}\longrightarrow1+\frac{\sqrt3}{2} +=1.8660254037844386\ldots.} \tag{5.3} +\] + +In particular, + +\[ +\boxed{ +\frac{I_p(2)-I_p(1)}{\kappa_p}\longrightarrow1.} \tag{5.4} +\] + +Thus one extra unit of macroscopic span costs asymptotically exactly one axial mass unit after the core bulge has saturated. + +## 6. Morphological core prediction + +At any fixed `A>r_*`, the minimizing winding core uses the same normalized bulge + +\[ +\boxed{r_{core}\to1/(2\sqrt3).} \tag{6.1} +\] + +The remaining span `A-r_*` is carried by branches. + +If `r` represents half of the core transverse range in the #758 geometry, the total core range target is + +\[ +\boxed{2r_* = 1/\sqrt3=0.5773502691896258\ldots} \tag{6.2} +\] + +in units of the winding length. + +Therefore the conditioned `A=1` and `A=2` samples should have asymptotically the **same core-width distribution after normalization by w**, while their branch lengths differ macroscopically. + +This is much stronger than checking only the tail probability. + +## 7. A parameter-free shape of the entire candidate rate + +The normalized curve + +\[ +J(A)= +\begin{cases} +\sqrt{1+4A^2}-1,& A\le1/(2\sqrt3),\\ +A+\sqrt3/2-1,& A\ge1/(2\sqrt3) +\end{cases} \tag{7.1} +\] + +has no free fit parameters once `A` and the rate are normalized by `w` and `kappa_p`. + +A near-critical width sequence can therefore challenge the mechanism by estimating the ratios + +\[ +-\frac1{\kappa_pw}\log P(R_w\ge Aw) \tag{7.2} +\] + +at several `A`. A systematic limit different from (7.1) rules out at least one of: + +1. normalized Wulff isotropy; +2. the loop/branch variational form (1.2); +3. the identification of the measured range with the variational `A`. + +## 8. Why square-site isotropy must remain a separate hypothesis + +Critical RSW and strict Wulff convexity do **not** imply that + +\[ +\tau_p/\kappa_p\to|\cdot|. \tag{8.1} +\] + +The latter is rotational restoration, a stronger universality statement. + +For triangular-lattice critical/near-critical site percolation, rotational invariance of the scaling limit implies Wulff-rounding and this type of Euclidean normalization. For the actual square-site Matching-One model, no such theorem was identified in the current audit. + +Keeping isotropy as one explicit hypothesis is useful: it turns all the remaining morphology predictions into exact numbers and makes the missing universality input impossible to hide inside a fitted `D` or curvature. + +## 9. Strongest immediate test order + +The cheapest discriminators are: + +1. verify the already conditional identity `D^{-1}=partial_yy tau` away from criticality where both can be measured/certified; +2. approach criticality and test whether `D kappa` drifts toward one; +3. test core bulge saturation near `0.288675 w` at `A=1,2`; +4. test rate differences `I(2)-I(1)≈kappa`; +5. only after these pass, fit the full curve (7.1). + +This order distinguishes a sewing/curvature failure from a rotational-universality failure. + +## 10. Claim boundary + +Everything after assumption (1.3) is elementary convex optimization. The square-site isotropy assumption itself is conjectural here. The note is intended as a no-parameter challenge target, not as a theorem claim. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/near-critical-loop-insertion-diagnostic-20260914.md b/docs/manuscripts/geometric-balance/near-critical-loop-insertion-diagnostic-20260914.md new file mode 100644 index 000000000..273a81f1b --- /dev/null +++ b/docs/manuscripts/geometric-balance/near-critical-loop-insertion-diagnostic-20260914.md @@ -0,0 +1,348 @@ +# Near-critical loop versus endpoint insertion: a decisive prefactor diagnostic + +2026-09-14. Conjectural synthesis of three rigorous ingredients that currently live in different models/levels: + +1. the finite-state cyclic matrix-sewing theorem on this branch, whose simple Perron-band log-determinant residue is exactly one; +2. the 2026 D'Alimonte--Manolescu near-critical OZ theorem for square-lattice **bond** random-cluster connectivity; +3. their uniform Brownian/local-CLT scale, which gives transverse variance of order `w xi(p)`. + +The goal is not to claim a SITE theorem. It is to isolate a sharp observable that distinguishes whether a complete winding component behaves like a genuinely cyclic renewal object or still carries point-to-point endpoint insertions near criticality. + +## 1. Correlation-length variables + +Let + +\[ +\xi=\kappa^{-1}, +\qquad +s=w/\xi=\kappa w. \tag{1.1} +\] + +Here `w` is the physical winding length and `s` is the number of correlation-length units around the loop. + +The near-critical OZ theorem for the bond-FK two-point function has the form + +\[ +\boxed{ +G_{2pt}(w) +\asymp +\pi_1(\xi)^2s^{-1/2}e^{-s}.} \tag{1.2} +\] + +The conditioned-cluster transverse variance is of order + +\[ +\operatorname{Var}X_\perp\asymp w\xi. \tag{1.3} +\] + +Thus a Brownian diffusion coefficient per unit physical longitudinal distance satisfies + +\[ +D\asymp\xi. \tag{1.4} +\] + +## 2. What a pure cyclic renewal loop predicts + +For a cyclic Markov-additive renewal kernel with one simple Perron band, the branch theorem `matrix-sewing-unit-residue-20260914.md` gives + +\[ +\nu_w +\sim +\frac{\zeta_{loop}}{\sqrt{2\pi D w}}e^{-\kappa w}. \tag{2.1} +\] + +If the cyclic closure has unit insertion residue at correlation-length scale, + +\[ +\zeta_{loop}=O(1) \tag{2.2} +\] + +as `p->pc`, then using `D~xi` and `w=xi s`, + +\[ +\boxed{ +\nu_w^{cyclic} +\asymp +\xi^{-1}s^{-1/2}e^{-s}.} \tag{2.3} +\] + +The factor `xi^{-1}` has a simple local-CLT meaning. A transverse bridge after `s` correlation-length steps has physical standard deviation + +\[ +\xi\sqrt s. \tag{2.4} +\] + +Returning to one specified **microscopic row** therefore costs + +\[ +(\xi\sqrt s)^{-1}. \tag{2.5} +\] + +This is exactly the physical-row version of the Gaussian closure factor. + +## 3. Why the two-point function has a different insertion weight + +The two-point probability (1.2) is dimensionless and pins two microscopic endpoint vertices. Compare it with the raw microscopic endpoint local-CLT density `(xi sqrt s)^{-1}e^{-s}` suggested by the renewal walk. + +The ratio is + +\[ +\boxed{ +J_{2pt}(\xi) +\asymp +\pi_1(\xi)^2\xi.} \tag{3.1} +\] + +So in the near-critical bond-FK theorem, the two endpoint insertions together contribute an effective factor of order + +\[ +\pi_1(\xi)^2\xi. \tag{3.2} +\] + +Heuristically one may think of each microscopic endpoint insertion as having scale + +\[ +\pi_1(\xi)\sqrt\xi, \tag{3.3} +\] + +though only the product is relevant here. + +A genuine closed loop has no prescribed endpoints. Therefore there is no reason for (3.2) to survive unchanged after cyclic closure and unrooting. + +## 4. Three competing complete-component insertion hypotheses + +Write the general near-critical complete-component ansatz as + +\[ +\boxed{ +\nu_w(p) +\asymp +\xi^{-1}J_{comp}(\xi,s) + s^{-1/2}e^{-s}.} \tag{4.1} +\] + +The unknown is now isolated in one dimensionless insertion factor `J_comp`. + +### H0: pure cyclic closure + +\[ +\boxed{J_{comp}\asymp1.} \tag{4.2} +\] + +Then + +\[ +\nu_w\asymp\xi^{-1}s^{-1/2}e^{-s}. \tag{4.3} +\] + +This is the natural continuation of the unit-residue logdet mechanism. + +### H2: two endpoint insertions survive + +If cutting/opening the complete component effectively introduces the same microscopic endpoint factors as a two-point connection, then + +\[ +J_{comp}\asymp\pi_1(\xi)^2\xi, \tag{4.4} +\] + +and + +\[ +\boxed{ +\nu_w\asymp\pi_1(\xi)^2s^{-1/2}e^{-s},} \tag{4.5} +\] + +at comparability level. + +### H1 / marked closure + +A one-mark or asymmetric anchor construction could produce an intermediate factor. More generally write + +\[ +J_{comp}(\xi)\asymp\xi^\beta\pi_1(\xi)^\gamma \tag{4.6} +\] + +as a diagnostic parameterization only, not a claimed power law. + +The important point is that `beta,gamma` describe **insertion semantics**, while the universal Gaussian closure remains `s^{-1/2}`. + +## 5. The three hypotheses are invisible at fixed p + +At any fixed subcritical `p`, + +\[ +\xi(p)<\infty, +\qquad +\pi_1(\xi(p))>0 \tag{5.1} +\] + +are constants. Every hypothesis therefore reduces to + +\[ +\nu_w=C(p)w^{-1/2}e^{-\kappa(p)w}(1+o(1)) \tag{5.2} +\] + +with a different `C(p)`. + +So a fixed-p width ladder, even a mathematically perfect one, cannot determine whether the prefactor is a pure cyclic residue or an endpoint-dressed insertion mechanism. + +The hypotheses separate only when `p=p_w->pc` and `xi(p_w)->infinity`. + +## 6. Distinct extreme-value centre corrections + +Suppose a longitudinal opportunity count `m` creates complete components with mean + +\[ +\lambda=m\nu_w. \tag{6.1} +\] + +At first birth, `lambda=O(1)`. From (4.1), the general logarithmic balance is + +\[ +\boxed{ +s+\tfrac12\log s+\log\xi-\log J_{comp}(\xi,s) +=\log m+O(1).} \tag{6.2} +\] + +### Pure cyclic H0 + +\[ +\boxed{ +s+\tfrac12\log s+\log\xi +=\log m+O(1).} \tag{6.3} +\] + +### Endpoint-dressed H2 + +Insert (4.4): + +\[ +\boxed{ +s+\tfrac12\log s-2\log\pi_1(\xi) +=\log m+O(1),} \tag{6.4} +\] + +which is exactly the two-point OZ balance. + +The difference between (6.3) and (6.4) is + +\[ +\boxed{ +\log\xi+2\log\pi_1(\xi).} \tag{6.5} +\] + +This grows logarithmically/polynomial-arm scale rather than staying `O(1)`. Near criticality it is therefore a strong discriminator. + +## 7. A direct dimensionless ratio diagnostic + +Define + +\[ +\boxed{ +R_{loop/2pt}(p,w) +=\frac{\xi(p)\nu_w(p)}{G_{2pt}(w;p)}.} \tag{7.1} +\] + +Using the common Gaussian/exponential factors: + +### H0 predicts + +\[ +\boxed{ +R_{loop/2pt}\asymp\pi_1(\xi)^{-2}.} \tag{7.2} +\] + +### H2 predicts + +\[ +\boxed{R_{loop/2pt}\asymp\xi.} \tag{7.3} +\] + +depending on the precise normalization chosen for the two-point denominator. An even cleaner comparison is to strip the common skeleton directly: + +\[ +\boxed{ +J_{emp}(p,w) +:=\xi\sqrt{s}\,e^s\nu_w(p).} \tag{7.4} +\] + +Then + +\[ +J_{emp}\asymp1 \tag{7.5} +\] + +under pure cyclic closure, while + +\[ +J_{emp}\asymp\pi_1(\xi)^2\xi \tag{7.6} +\] + +under two-endpoint dressing. + +This is the preferred statistic because it does not require a separate two-point simulation if `xi` and the critical one-arm baseline are available. + +## 8. Interaction with the surface-excess identity + +The branch already gives an independent component-Palm route to the mass slope through + +\[ +qE_WB-pE_WN=pq\,\partial_p\log\nu_w. \tag{8.1} +\] + +If (4.1) holds, then + +\[ +\partial_p\log\nu_w += -\partial_p\log\xi + +\partial_p\log J_{comp} + -\frac12\partial_p\log s + -\partial_p s. \tag{8.2} +\] + +Near a crossover centre, the dominant term may still be `-partial_p s`, but the insertion derivative can enter at logarithmic order. + +Thus morphology `(N,B)` gives a second route to detecting whether the near-critical prefactor is becoming singular, without fitting the activity alone. + +## 9. Relation to matrix sewing + +The finite-state matrix theorem says: + +- a simple Perron band in a pure cyclic logdet contributes unit logarithmic residue; +- finite local memory changes `D` but not the residue; +- a nontrivial continuously varying residue must arise from insertion/mark/unrooting semantics, multiple bands or an infinite-state limit. + +Near-critical divergence of the correlation scale naturally sends a fixed microscopic transfer description towards an effectively infinite-state object. Therefore observing `J_comp` drift does not refute Gaussian sewing. It diagnoses which microscopic insertion survives the scaling limit. + +This is exactly the distinction that fixed-p data could not make. + +## 10. A concrete numerical/theoretical gate + +Choose a sequence `p_j->pc` and widths + +\[ +w_j=s_j\xi(p_j), +\qquad +s_j\to\infty\text{ slowly}, \tag{10.1} +\] + +so the system is safely on the OZ side but the correlation length grows. + +For each `j`, estimate/certify + +\[ +J_{emp,j}=\xi_j\sqrt{s_j}e^{s_j}\nu_{w_j}(p_j). \tag{10.2} +\] + +Interpretation: + +- bounded nonzero `J_emp` -> supports pure cyclic closure; +- tracking `pi_1(xi)^2 xi` -> supports endpoint-dressed closure; +- another systematic scale -> identifies a different insertion class; +- extra power of `s` -> refutes the simple one-soft-band Gaussian sewing hypothesis itself. + +The existing tagged complete-component resolvent is exact at finite width but cannot reach `w~xi->infinity` by brute force. A correlation-length-block transfer/renewal construction is therefore the right next theorem engine, not simply a larger microscopic width ladder. + +## 11. Claim boundary + +All SITE formulas in this note are conjectural diagnostics. The bond-FK two-point formula and Brownian scale are rigorous in D'Alimonte--Manolescu; the unit cyclic residue is rigorous for the finite-state matrix class on this branch. The scientific question is how the actual square-site complete-component observable interpolates between those structures. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/near-critical-oz-literature-boundary-20260914.md b/docs/manuscripts/geometric-balance/near-critical-oz-literature-boundary-20260914.md new file mode 100644 index 000000000..84c5b4531 --- /dev/null +++ b/docs/manuscripts/geometric-balance/near-critical-oz-literature-boundary-20260914.md @@ -0,0 +1,362 @@ +# What the 2026 near-critical OZ theorem does and does not buy us + +2026-09-14. Literature boundary audit for #740/#758/#762/#767 after reading the current v3 of Lucas D'Alimonte and Ioan Manolescu, *Near-critical Ornstein--Zernike theory for the planar random-cluster model*, arXiv:2510.13648v3 (last revised 2026-06-23). + +This note is deliberately strict about model/observable transport. The paper is highly relevant to the architecture of our conjectures, but it is **not** a theorem about square-site percolation or complete winding-component activities. + +## 1. Exact model boundary + +D'Alimonte--Manolescu study the planar random-cluster model on the square lattice with configurations on **edges**, for + +\[ +1\le q<4, +\qquad p=xi_p(e)`, + +\[ +\boxed{ +\phi_p(0\leftrightarrow\lfloor re\rfloor) +\asymp +\pi_1(\xi_p(e))^2 +\sqrt{\frac{\xi_p(e)}r} +\exp[-r/\xi_p(e)].} \tag{2.1} +\] + +Here `pi_1(R)` is the **critical** one-arm probability to scale `R`. + +If + +\[ +s=r/\xi_p(e)=\kappa_p(e)r, \tag{2.2} +\] + +then + +\[ +\boxed{ +\phi_p(0\leftrightarrow re) +\asymp +\pi_1(\xi_p(e))^2 s^{-1/2}e^{-s}.} \tag{2.3} +\] + +The formula separates three mechanisms: + +1. `e^{-s}`: correlation-mass cost; +2. `s^{-1/2}`: one-dimensional OZ/local-CLT sewing; +3. `pi_1(xi)^2`: two near-critical endpoint insertions. + +This is exactly the conceptual decomposition we want for a crossover theory. + +## 3. Important limitation: this is comparability, not an exact amplitude + +The theorem uses + +\[ +\asymp, \tag{3.1} +\] + +with constants uniform in `p`, direction and distance in the stated range. It does **not** identify a multiplicative constant with relative error `1+o(1)` uniformly as `p->pc`. + +Therefore it cannot by itself supply: + +- an exact near-critical OZ residue; +- an exact complete-component prefactor `zeta(p)`; +- a logarithmic centering constant for an extreme-value window when `O(1)` amplitude information matters. + +It does rigorously show that a naive fixed-`p` smooth amplitude continued to criticality is the wrong structural picture: the critical one-arm factor appears explicitly. + +## 4. The fixed-p prefactor should not be analytically continued unchanged + +For fixed subcritical `p`, `xi_p` and `pi_1(xi_p)` are constants as `r->infinity`, so (2.1) has the familiar + +\[ +r^{-1/2}e^{-r/\xi_p} \tag{4.1} +\] + +shape. + +But when `p=p_r->pc` and `xi_p->infinity`, the quantity + +\[ +\boxed{\pi_1(\xi_p)^2} \tag{4.2} +\] + +moves with the parameter and belongs at leading prefactor level. + +Thus a crossover ansatz of the form + +\[ +A(p)r^{-1/2}e^{-\kappa(p)r} \tag{4.3} +\] + +with `A(p)` treated as a harmless smooth fixed-`p` residue is not uniformly justified near criticality. Even in the exactly treated bond-FK model, the endpoint insertion factor becomes a critical-arm observable. + +This strongly supports the repository decision not to use a fixed-p OZ amplitude as the foundation of #767. + +## 5. Brownian scale: D is of order the correlation length + +The paper constructs a killed Markov renewal process at a coarse scale `L(p)` comparable uniformly with the correlation length. Its endpoint local CLT has transverse variance parameter `sigma(p,w)` bounded uniformly away from zero and infinity. + +For a connection of physical longitudinal length `r`, the number of renewal steps is + +\[ +n\asymp r/L(p), \tag{5.1} +\] + +while Theorem 4.10 scales transverse displacement by + +\[ +\sqrt n\,L(p). \tag{5.2} +\] + +Therefore the transverse variance is of order + +\[ +nL(p)^2\asymp rL(p). \tag{5.3} +\] + +The same paper identifies + +\[ +\xi_p(e) +=L(p)\times\text{a factor uniformly bounded above/below}. \tag{5.4} +\] + +Consequently, in the bond-FK theorem, + +\[ +\boxed{ +\operatorname{Var}(X_\perp\mid0\leftrightarrow re) +\asymp r\xi_p(e).} \tag{5.5} +\] + +If we parameterize a Brownian bridge by physical longitudinal distance, + +\[ +\operatorname{Var}(X_\perp(t))\sim D_p(e)\,r\,t(1-t), \tag{5.6} +\] + +then the rigorous scaling analogue is + +\[ +\boxed{D_p(e)\asymp\xi_p(e)=\kappa_p(e)^{-1}.} \tag{5.7} +\] + +Hence + +\[ +\boxed{D_p(e)\kappa_p(e)\asymp1} \tag{5.8} +\] + +uniformly up to criticality in this bond-FK model. + +This is strong structural support for the repository relation + +\[ +D^{-1}=\partial_{yy}\tau \tag{5.9} +\] + +having the correct **scale** near criticality. It does not prove the exact equality or its SITE version. + +## 6. Strict Wulff geometry survives uniformly near criticality + +D'Alimonte--Manolescu prove for every subcritical `p` that the correlation-length unit ball and its Wulff dual are strictly convex with differentiable boundaries. Their construction is performed at correlation-length scale and is uniform in `p` and direction at the renewal level. + +This independently confirms the qualitative geometric assumptions used in our loop/branch analysis: + +- a unique tangent/dual direction exists; +- the transverse endpoint law has a nondegenerate Gaussian scale; +- nearby directional deviations have a genuine local large-deviation penalty. + +However their theorem, as stated, does not give us a ready-made uniform numerical lower/upper bound on the second directional derivative that can simply be transplanted to square-site percolation. + +## 7. Consequence for the intermediate aspect-ratio crossover + +Consider a winding length `w` and let + +\[ +\xi=\xi(p), +\qquad +s=w/\xi=\kappa(p)w. \tag{7.1} +\] + +If a rare-event opportunity count is `m` (or an aspect ratio proportional to it), the **two-point** OZ balance suggested by (2.3) is + +\[ +1\asymp +m\,\pi_1(\xi)^2s^{-1/2}e^{-s}. \tag{7.2} +\] + +Taking logarithms, + +\[ +\boxed{ +s+\tfrac12\log s-2\log\pi_1(\xi) +=\log m+O(1).} \tag{7.3} +\] + +This immediately separates three crossover scales. + +### Regime A: deep rare-event/OZ merging + +If + +\[ +\log m\gg |\log\pi_1(\xi)| \tag{7.4} +\] + +(and `log m->infinity`), then at leading order + +\[ +s\sim\log m. \tag{7.5} +\] + +The critical endpoint dressing shifts only the lower-order centering. + +### Regime B: arm-dressed logarithmic crossover + +If + +\[ +\log m\asymp |\log\pi_1(\xi)|, \tag{7.6} +\] + +then the critical one-arm term contributes at the same logarithmic order as the opportunity entropy. It cannot be absorbed into an `O(1)` OZ amplitude. + +Polynomial aspect ratios are a natural place where this can happen. + +### Regime C: fixed/bounded aspect + +When `log m=O(1)`, the system is no longer an extreme-value gas of many independent long opportunities. The full near-critical torus scaling theory is the natural object instead of an OZ birth-window continuation. + +This regime split is only a **template** for square-site component activity; equation (7.2) is a bond-FK two-point statement. + +## 8. Complete component activity has an additional insertion problem + +Our torus birth intensity is not + +\[ +P(0\leftrightarrow we_1). \tag{8.1} +\] + +It counts complete winding components with an anchor/unrooting convention and with all branches included. + +Even if the same renewal skeleton controls the long core, a component activity can differ from the two-point function through: + +- anchor/root removal; +- closure/seam insertion; +- the requirement that the component be complete; +- local branches and boundary weights; +- possibly multiple leading renewal bands. + +The matrix-sewing result on this branch shows that finite local memory itself does **not** create an arbitrary residue, but it does not identify the correct insertion vector for the actual SITE complete-component object. + +Therefore the literature theorem supports + +\[ +\text{near-critical renewal skeleton + }w^{-1/2}\text{ Gaussian sewing},\tag{8.2} +\] + +not the equality of the two-point and complete-component amplitudes. + +## 9. Model-transfer boundary: bond FK is not square-site + +At `q=1`, D'Alimonte--Manolescu give a rigorous near-critical theory for Bernoulli **bond** percolation on the square lattice. + +The standard near-critical scaling-limit theorem of Garban--Pete--Schramm is for **site percolation on the triangular lattice**. + +Neither result is a theorem asserting the same uniform OZ formula for square-site percolation, and neither identifies the square-site complete-component insertion amplitude. + +So for the actual Matching-One model, using (2.1) as a theorem would require a new SITE adaptation or a separate universality theorem strong enough to transport the relevant quantitative observable. We do not have such a theorem in this audit. + +## 10. What can safely enter our research programme now + +### Safe structural import / proof template + +Use the paper as strong evidence that a successful square-site near-critical renewal theorem should have: + +1. coarse slices at the correlation-length scale; +2. a killed Markov renewal process with uniform mass gap; +3. nondegenerate endpoint local CLT; +4. Brownian bridge scale `sqrt(w xi)`; +5. strict Wulff geometry; +6. critical arm insertions at the endpoints. + +### Not safe to import as a SITE theorem + +Do **not** claim from this paper alone: + +- square-site uniform near-critical OZ; +- an exact `1+o(1)` amplitude; +- a SITE value of `zeta(p)`; +- a complete-component activity formula; +- exact square-site critical arm exponents. + +## 11. A focused theorem target suggested by the literature + +The full D'Alimonte--Manolescu machinery is more than we need for #767. A strategically smaller square-site target would be: + +> **SITE near-critical complete-component comparability.** For `p=C xi_site(p)`, prove uniformly in a compact angular set that the complete winding-component activity satisfies +> \[ +> \nu_w(p) +> \asymp +> R_{site}(p,w/\xi) +> \pi_{1,site}(\xi)^2 +> (w/\xi)^{-1/2} +> e^{-w/\xi}, +> \] +> where `R_site` is bounded above and below uniformly and encodes the anchor/closure insertion. + +Even this **comparability-level** theorem would already determine which terms can enter the logarithmic crossover equation and would distinguish the arm-dressed and deep-OZ regimes. Exact prefactor identification could come later. + +A still weaker first target is to prove only + +\[ +D_p\asymp\xi_p \tag{11.1} +\] + +for the SITE point-to-point conditioned cluster uniformly near criticality. Combined with the branch's curvature/sewing diagnostics, that would sharply constrain the possible complete-component mechanism. + +## 12. Bottom line for #767 + +The 2026 near-critical OZ paper **supports the architecture** of our crossover programme but does not close it for square-site Matching-One. + +The most important correction to prior intuition is: + +\[ +\boxed{\text{near critical: OZ Gaussian sewing is arm-dressed, not merely amplitude-renormalized.}}\tag{12.1} +\] + +Accordingly, #767 should not analytically continue a fixed-p component prefactor to `p_c`. It should either: + +1. obtain a square-site near-critical renewal/comparability theorem with explicit arm insertions; or +2. state the common-window theory conditionally in the exact charge-neutral coordinates `(chi,theta,H)` and treat the one-arm/complete-component insertion as an independent scaling input. + +Primary sources audited: + +- L. D'Alimonte and I. Manolescu, *Near-critical Ornstein--Zernike theory for the planar random-cluster model*, arXiv:2510.13648v3, 2026. +- C. Garban, G. Pete and O. Schramm, *The scaling limits of near-critical and dynamical percolation*, JEMS 2018 / arXiv:1305.5526v4, triangular-lattice site percolation. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/near-critical-pivotal-genealogy-crossover-20260914.md b/docs/manuscripts/geometric-balance/near-critical-pivotal-genealogy-crossover-20260914.md new file mode 100644 index 000000000..9d2f95791 --- /dev/null +++ b/docs/manuscripts/geometric-balance/near-critical-pivotal-genealogy-crossover-20260914.md @@ -0,0 +1,206 @@ +# Near-critical pivotal genealogy as the crossover completion of the fixed-subcritical splitting process + +2026-09-14. + +Status: synthesis/conjectural bridge for square-site Matching One, with a rigorous positive-control framework available for triangular-lattice near-critical percolation through the Garban--Pete--Schramm scaling limit. This note does not claim square-site universality has been proved. + +## 1. Why the fixed-subcritical theorem should stop where it does + +The current #780 route has essentially closed, on compact strictly subcritical parameter intervals, the steps + +```text +rare complete winding components -> spatial Poisson process, +common-label birth marks -> marked PPP in intrinsic intensity clock, +merger factorial density / component intensity -> 0, +pure splitting genealogy on bounded clock windows. +``` + +The no-merger estimate pays one additional winding action. Schematically, + +```text +M_w / nu_w <= poly(w) exp[-kappa(p) w]. +``` + +At fixed subcritical p this vanishes exponentially. But the estimate itself announces its failure variable: + +```text +x = w kappa(p). +``` + +When `x=O(1)`, there is no exponential reason for mergers to disappear. + +## 2. The near-critical window is exactly x=O(1) + +For two-dimensional percolation, + +```text +xi(p) ~ |p-pc|^-4/3, +kappa(p) ~ xi^-1 ~ |p-pc|^(4/3). +``` + +The usual near-critical scaling + +```text +p = pc + lambda w^-3/4 +``` + +therefore gives + +```text +w kappa(p) ~ const * |lambda|^(4/3)=O(1). +``` + +So the failure of the fixed-subcritical merger suppression is not an accidental technical gap. It occurs at exactly the universal near-critical scaling window. + +## 3. Candidate continuum object: pivotal-driven essential-component genealogy + +Garban--Pete--Schramm construct the near-critical scaling limit by perturbing the critical configuration with a Poissonian pivotal measure. In an O(1) lambda interval, O(1) macroscopic pivotal updates occur in a bounded continuum region. These updates can alter macroscopic connectivity, hence can both create and merge essential lineages. + +This suggests the correct continuation of #780 is not a corrected Poisson barrier process but a continuum process + +```text +G_lambda(tau) +``` + +of homology-carrying clusters/components on a torus or cylinder of fixed continuum modulus `tau`, driven by the near-critical pivotal measure. + +The observable process should retain at least + +```text +ambient homology rank / primitive class, +essential component partition, +first-birth marks, +merger/split genealogy, +optional neutral component count. +``` + +The rank process alone is a lossy projection but has a clean one-time generating function. + +## 4. Scaling prediction for merger density + +Draft #773/#782 uses the near-critical component-intensity ansatz + +```text +nu_w(pc+lambda/(a_t w^(3/4))) + = w^-1 I(lambda)[1+o(1)]. +``` + +If macroscopic pivotal updates have a nondegenerate continuum rate, then the per-row factorial merger intensity should have the same dimensional scale: + +```text +M_w(lambda_-,lambda_+) + = w^-1 J(lambda_-,lambda_+)[1+o(1)], +``` + +rather than `o(nu_w)`. + +Therefore + +```text +M_w/nu_w -> J/I, +``` + +with a generally nonzero universal crossover function. + +This is the sharp qualitative distinction from the fixed-subcritical theorem. + +### IR matching condition + +As lambda tends deep into the subcritical side, equivalently `x=w kappa -> infinity`, the continuum crossover must reproduce the dilute theorem: + +```text +J/I -> 0. +``` + +The existing BK/AGG proof suggests an IR tail at least qualitatively of the form + +```text +J/I ~ polynomial(x) * exp(-x) +``` + +up to the massive periodic-sewing prefactor. The exact polynomial and amplitude are not supplied by the current proof. + +## 5. Interface to the rank-source programme + +Define the near-critical rank-source marginal + +```text +Z_tau(lambda,s) + = E[ exp(s(r_lambda-1)) ] + = Pi0(lambda;tau)e^-s + Pi1(lambda;tau) + Pi2(lambda;tau)e^s. +``` + +This is precisely the one-time rank projection of the proposed pivotal genealogy. + +Hence #782's massive-Potts programme and the near-critical probabilistic route should be viewed as complementary: + +- pivotal/CLE route: constructs the process and its merger/split semantics; +- massive Potts / modified trace: may compute one-time sector weights and IR amplitudes; +- Krushkal / affine-TL source dictionary: supplies exact finite topological/neutral fugacities. + +One should not wait for a full massive TBA formula before defining the continuum genealogy. + +## 6. A useful two-time observable + +The natural continuation of the fixed-subcritical merger statistic is + +```text +J_tau(lambda1,lambda2) + = E sum_{C at lambda2} binom(n_C(lambda1),2), +``` + +with the torus/cylinder normalized to continuum size one. + +At fixed aspect ratio this is a finite continuum observable of the pivotal process. On the lattice, the corresponding quantity should satisfy + +```text +w M_w(lambda1,lambda2) -> J_tau(lambda1,lambda2). +``` + +This is more informative than asking only whether a merger was observed. + +A second high-value observable is the conditional transition matrix of the aggregate homology rank + +```text +P(r_lambda2=j | r_lambda1=i), i,j in {0,1,2}, +``` + +under the common-label monotone coupling. It is not Markov-complete, but it gives a direct bridge to the existing rank-source and torus-homology assets. + +## 7. Strong conjecture: fixed-subcritical pure splitting is an IR boundary condition + +The proposed universal picture is + +```text +near-critical pivotal genealogy at finite lambda + | + | lambda -> -infinity / x -> infinity + v +Poisson barrier cloud + pure splitting clock kernel. +``` + +In the reverse direction, the fixed-subcritical process does not need to be manually patched with rare mergers. Its correct crossover completion is the pivotal process itself. + +This also predicts that the Laguerre hierarchy and exponential gap correlations from #785 are not globally valid in lambda. They should emerge only in the dilute IR after the nonlinear change from lambda to component-intensity clock. + +## 8. Square-site scope + +The rigorous near-critical scaling-limit construction is currently a triangular-site positive control, not a theorem for square-site Matching One. For the present project the square-site claim is a universality conjecture. + +That is still scientifically useful because it gives a precise target and a control order: + +1. define/compute the rank and merger observables in the rigorous triangular near-critical coupling; +2. verify the deep-subcritical IR tends to the Poisson splitting law; +3. only then compare square-site finite-width data in the same normalized lambda/x variables. + +## 9. Concrete next work + +No new fixed-subcritical common-label simulation is needed for this bridge. Higher-value tasks are: + +1. derive the continuum measurability of torus homology rank and the merger factorial functional in the near-critical quad-crossing/pivotal process; +2. determine whether existing near-critical CLE/pivotal results imply continuity in lambda for these functionals away from exceptional pivotal times; +3. compute or bound `J_tau(lambda1,lambda2)` in a triangular positive control, analytically if possible and otherwise with a continuum/discrete convergence experiment; +4. connect the one-time marginal `Z_tau(lambda,s)` to the massive-Potts modified-trace programme in #782; +5. identify the IR matching variable and amplitude connecting `J/I` to the fixed-subcritical BK/AGG tail. + +The central decision is no longer `MERGER_YES/NO`; it is the crossover function `J/I` and its massive/near-critical normalization. diff --git a/docs/manuscripts/geometric-balance/near-critical-rank-flux-and-jump2-suppression-20260914.md b/docs/manuscripts/geometric-balance/near-critical-rank-flux-and-jump2-suppression-20260914.md new file mode 100644 index 000000000..0c90c1eaf --- /dev/null +++ b/docs/manuscripts/geometric-balance/near-critical-rank-flux-and-jump2-suppression-20260914.md @@ -0,0 +1,202 @@ +# Near-critical rank flux and suppression of direct rank-two births + +2026-09-14. + +Status: exact finite monotone-flux algebra plus a conditional near-critical scaling prediction. The key missing theorem is a macroscopic arm certificate for a one-site `rank 0 -> rank 2` jump in square/triangular percolation. If that certificate is established, the vanishing of the direct-jump rate follows from standard arm exponents/pivotal scaling. + +## 1. Finite monotone rank flux + +Under the common uniform-label coupling, insert occupied sites monotonically. Let the ambient homology rank be + +```text +r in {0,1,2}. +``` + +At a single insertion, monotonicity permits + +```text +0->0, 0->1, 0->2, +1->1, 1->2, +2->2. +``` + +For a fixed site and parameter, write the transition probabilities/intensities + +```text +alpha = 0->1, +beta = 0->2, +gamma = 1->2. +``` + +The exact boundary calculus already developed in the repository gives + +```text +-P0' = N(alpha+beta), + P1' = N(alpha-gamma), + P2' = N(beta+gamma). +``` + +Only two of these equations are independent because `P0+P1+P2=1`. The direct-jump channel `beta` is therefore hidden from one-time rank probabilities. + +This is the finite predecessor of the continuum flux equation below. + +## 2. Near-critical scaling of the rank marginal + +Put the torus/cylinder on macroscopic scale one and use + +```text +p = pc + lambda * const * L^-3/4. +``` + +Assuming convergence of the rank law, + +```text +Pj,L(lambda) -> Pi_j(lambda;tau). +``` + +Define continuum transition intensities per unit `lambda` + +```text +a(lambda): 0->1, +b(lambda): 0->2, +c(lambda): 1->2. +``` + +Then the exact finite flux identities formally converge to + +```text +-Pi0' = a+b, + Pi1' = a-c, + Pi2' = b+c. +``` + +This relation is independent of a massive-Potts representation and follows from the monotone common-label process if the scaling limits exist. + +It also proves an information statement: + +> the one-time functions `Pi0,Pi1,Pi2` do not determine the direct-jump channel `b`. + +Thus a massive finite-volume engine for the rank marginals cannot by itself reconstruct the genealogy. + +## 3. Why direct 0->2 jumps should be absent in the continuum pivotal clock + +The lattice `0->2` event is the same one-step quantity called `jump2` / `B_k` in the rank-birth atlas. The local attachment analysis identifies a minimal `T3` geometry: closing the pivotal site leaves one exterior rank-zero component touching the site through at least three attachment germs, while opening the site creates two independent ambient homology generators. + +To create two macroscopic homology generators at one microscopic insertion, the local branches cannot all merge inside a contractible disk well below the systole. The expected continuum certificate is therefore a polychromatic six-arm event from the pivotal neighbourhood to a fixed fraction of system size. Split/rose geometries require at least as much macroscopic structure and may be eight-arm. + +Assume the certificate + +```text +{Delta_v r=2} subset {six alternating arms from v to c L} +``` + +up to finitely many topology-equivalent variants. + +For critical percolation the polychromatic arm exponents are + +```text +alpha_4 = 5/4, +alpha_6 = 35/12, +alpha_6-alpha_4 = 5/3. +``` + +Ordinary near-critical time is normalized by four-arm pivotals. Therefore the fraction of rank-jump-two pivotals among macroscopic pivotals scales as + +```text +L^[-(alpha_6-alpha_4)] = L^-5/3. +``` + +Equivalently, over an O(1) lambda interval, + +```text +E[# direct 0->2 macroscopic jumps] = O(L^-5/3) -> 0. +``` + +Hence the continuum prediction is + +```text +b(lambda)=0. +``` + +This recovers exactly the `R_jump2~L^-5/3` candidate that was independently proposed from the finite pivotal atlas, but here it has a process-level meaning. + +## 4. Continuum rank process if b=0 + +The flux equations collapse to + +```text +a(lambda) = -Pi0'(lambda), +c(lambda) = Pi2'(lambda), +Pi1' = -Pi0'-Pi2'. +``` + +At the self-dual/matching critical point, if + +```text +Pi2(lambda;tau)=Pi0(-lambda;tau), +``` + +then + +```text +a(0)=c(0). +``` + +So the aggregate rank projection has nearest-neighbour births only: + +```text +0 -> 1 -> 2. +``` + +This is a statement about rank jumps, not about the full component partition. + +## 5. Why this does not restore pure splitting + +Macroscopic component mergers can occur in the near-critical pivotal process without producing `0->2` in one step. Examples include + +- merger of two lineages in the same primitive homology class, leaving rank one; +- a rank-one component absorbing another lineage without changing ambient rank; +- ordinary `1->2` events driven by four-arm pivotals. + +Therefore + +```text +b=0 +``` + +does **not** imply the merger factorial `J_tau(lambda1,lambda2)` vanishes. The full genealogy remains richer than the three-state rank process. + +This distinction prevents a dangerous shortcut: absence of direct rank-two births cannot be used to infer the fixed-subcritical pure-splitting kernel at finite near-critical lambda. + +## 6. Interface to the massive rank-source functions + +If a massive-Potts/modified-trace construction supplies + +```text +Pi0(lambda;tau), Pi1(lambda;tau), Pi2(lambda;tau), +``` + +then, conditional on `b=0`, it automatically predicts the continuum nearest-neighbour rank fluxes through derivatives. + +This gives a new validation target for #782: + +```text +-Pi0'(lambda) >= 0, + Pi2'(lambda) >= 0, + Pi1'=-Pi0'-Pi2', +``` + +with duality relating the first two across `lambda=0`. + +But the full merger statistic remains an independent two-time observable. + +## 7. What would prove or refute the claim + +The decisive object is geometric, not another finite-size exponent fit: + +1. prove that every one-site `rank 0->2` jump forces six alternating arms to macroscopic distance on the torus/cylinder; +2. identify any exceptional topology in which fewer arms suffice; +3. once the inclusion is proved, import the standard six-arm estimate in the rigorous triangular near-critical control; +4. for square site, treat the same conclusion as a universality target until the corresponding arm input is available. + +A counterexample configuration with a genuine macroscopic `0->2` jump and fewer than six arms would invalidate the proposed continuum suppression mechanism immediately. diff --git a/docs/manuscripts/geometric-balance/neutral-gas-limit-collapse-20260914.md b/docs/manuscripts/geometric-balance/neutral-gas-limit-collapse-20260914.md new file mode 100644 index 000000000..f6d8aa925 --- /dev/null +++ b/docs/manuscripts/geometric-balance/neutral-gas-limit-collapse-20260914.md @@ -0,0 +1,200 @@ +# One-dimensional collapse of same-parameter two-colour count fluctuations + +2026-09-14. Exact consequence of the bounded topological charge + +\[ +D=W_4-W_8\in\{-1,0,1\}. +\] + +This sharpens `neutral-gas-topological-charge-20260914.md`: not only the pressure and variance rates, but **every growing-scale fluctuation limit and every extensive LDP collapse to the diagonal**. + +## 1. Uniform exponential-moment comparison + +Write + +\[ +W_4=K+1_{\{D=1\}},\qquad +W_8=K+1_{\{D=-1\}}. \tag{1.1} +\] + +For real sources `s,t`, the source correction relative to the neutral gas is one of `0,s,t`. Hence with + +\[ +c(s,t)=\max(|s|,|t|), +\] + +we have pathwise + +\[ +\left|sW_4+tW_8-(s+t)K\right|\le c(s,t). \tag{1.2} +\] + +Therefore at every finite size + +\[ +\boxed{ +e^{-c(s,t)}E e^{(s+t)K} +\le E e^{sW_4+tW_8} +\le e^{c(s,t)}E e^{(s+t)K}.} \tag{1.3} +\] + +The error is multiplicative by a size-independent constant. No asymptotic estimate is needed. + +If the system has longitudinal size `m`, division by `m` gives + +\[ +\left| +\frac1m\log Ee^{sW_4+tW_8} +- +\frac1m\log Ee^{(s+t)K} +\right| +\le \frac{c(s,t)}m. \tag{1.4} +\] + +Thus any limiting scaled cumulant generating function satisfies + +\[ +\boxed{\psi_{4,8}(s,t)=\psi_K(s+t).} \tag{1.5} +\] + +The antisymmetric source direction is exactly subextensive. + +## 2. Joint LDP is infinite off the diagonal + +For every realization, + +\[ +|W_4-K|\le1,\qquad |W_8-K|\le1. \tag{2.1} +\] + +Hence + +\[ +\left\| +(W_4/m,W_8/m)-(K/m,K/m) +\right\|_\infty\le1/m. \tag{2.2} +\] + +Suppose `K_m/m` satisfies an LDP with speed `m` and good rate `I_K`. Deterministic exponential equivalence gives the full joint rate function + +\[ +\boxed{ +I_{4,8}(x,y)= +\begin{cases} +I_K(x),&x=y,\\ ++\infty,&x\ne y. +\end{cases}} \tag{2.3} +\] + +So a same-parameter bivariate count theory with a finite extensive cost for `x-y != 0` is structurally impossible. + +This is stronger than saying the correlation tends to one: there is no second extensive large-deviation coordinate at all. + +## 3. Every diverging-scale fluctuation limit is diagonal + +Let `a_m->infinity` be any deterministic scale. Then + +\[ +\frac{W_4-EW_4}{a_m} +- +\frac{K-EK}{a_m} +\to0, \tag{3.1} +\] + +and the same for `W_8`, because each difference is bounded by a constant divided by `a_m`. + +Therefore if + +\[ +\frac{K_m-EK_m}{a_m}\Rightarrow Z \tag{3.2} +\] + +for **any** nondegenerate limit `Z`, not necessarily Gaussian, then + +\[ +\boxed{ +\left( +\frac{W_4-EW_4}{a_m}, +\frac{W_8-EW_8}{a_m} +\right) +\Rightarrow (Z,Z).} \tag{3.3} +\] + +Consequences include: + +- a CLT for one count automatically gives the same CLT jointly for both; +- a stable/non-Gaussian scaling limit also transfers diagonally; +- moderate deviations with a growing normalization inherit the same one-dimensional collapse. + +No separate two-colour fluctuation theorem is required once the neutral count limit is known. + +## 4. Covariance matrix and principal directions + +If `Var(K_m)->infinity`, then + +\[ +\operatorname{Var}(W_4)=\operatorname{Var}(K)+o(\operatorname{Var}K), +\] + +\[ +\operatorname{Var}(W_8)=\operatorname{Var}(K)+o(\operatorname{Var}K), +\] + +and + +\[ +\operatorname{Cov}(W_4,W_8)=\operatorname{Var}(K)+o(\operatorname{Var}K).\tag{4.1} +\] + +Thus the normalized covariance matrix tends to the rank-one projector on `(1,1)` and + +\[ +\boxed{\operatorname{Corr}(W_4,W_8)\to1.} \tag{4.2} +\] + +The antisymmetric principal component is exactly `D`, whose variance is at most one at every finite size. + +This yields a severe implementation check: if a long-cylinder common-parameter simulation reports an antisymmetric variance growing with length, the count dictionary is wrong. + +## 5. Separation of bulk and root physics + +The neutral gas `K` carries all extensive pressure, variance and LDP information. The matching observable + +\[ +M=E(W_4-W_8)=ED \tag{5.1} +\] + +lives entirely in the bounded topological defect sector. + +Therefore the matching root is a **subextensive sector-amplitude problem sitting on top of an extensive one-dimensional gas**. This explains why common transfer/Perron modes can cancel in `M` even when they dominate each individual count. + +The correct spectral hierarchy is: + +1. one bulk `(1,1)` count mode; +2. bounded endpoint/rank charge sectors; +3. the zero-charge condition selecting the matching root. + +A second extensive count mode should not be introduced to explain root motion. + +## 6. Generating-function fingerprint + +At finite size define + +\[ +\Phi_m(s,t)=Ee^{sW_4+tW_8}. \tag{6.1} +\] + +Equation (1.3) means that along two source pairs with the same sum `s+t`, the ratio of generating functions is bounded uniformly in `m`: + +\[ +\boxed{ +\left|\log\frac{\Phi_m(s,t)}{\Phi_m(s',t')}\right| +\le c(s,t)+c(s',t') +\quad\text{if }s+t=s'+t'.} \tag{6.2} +\] + +Thus any transfer calculation in which this log-ratio grows linearly with cylinder height contradicts exact topology before any asymptotic interpretation is attempted. + +## 7. Claim boundary + +All statements are deterministic/probabilistic consequences of the exact finite decomposition (1.1), conditional only on existence of the one-dimensional limit invoked in each asymptotic corollary. No Poisson assumption is used. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/neutral-gas-topological-charge-20260914.md b/docs/manuscripts/geometric-balance/neutral-gas-topological-charge-20260914.md new file mode 100644 index 000000000..33fe5a590 --- /dev/null +++ b/docs/manuscripts/geometric-balance/neutral-gas-topological-charge-20260914.md @@ -0,0 +1,225 @@ +# Neutral component gas versus bounded topological charge + +2026-09-14. Exact finite topology plus asymptotic consequences for transfer/count descriptions. This note explains why extensive winding-count thermodynamics and the Matching-One balance observable live at different orders. + +## 1. Exact decomposition + +At one common parameter on one honest torus define + +\[ +D=W_4-W_8=r_4-1\in\{-1,0,+1\}. +\] + +By the winding-component classification there is a nonnegative integer `K` such that + +\[ +\boxed{ +W_4=K+1_{\{D=+1\}}, +\qquad +W_8=K+1_{\{D=-1\}}.} \tag{1.1} +\] + +Here `K>=1` in the neutral rank-one sector `D=0`, whereas `K=0` in the two charged endpoint sectors. + +The symmetric/antisymmetric coordinates are + +\[ +S=W_4+W_8=2K+|D|, +\qquad +D=W_4-W_8. \tag{1.2} +\] + +Thus `K` is the paired neutral component gas and `D` is a **bounded topological charge**. + +## 2. Exact source decomposition + +For real sources `s,t`, + +\[ +sW_4+tW_8=(s+t)K+s1_{\{D=+1\}}+t1_{\{D=-1\}}. \tag{2.1} +\] + +Hence configuration by configuration + +\[ +\left|sW_4+tW_8-(s+t)K\right|\le |s|+|t|. \tag{2.2} +\] + +On a height-`m` torus, whenever the neutral-gas pressure exists, + +\[ +\psi_K(u)=\lim_{m\to\infty}\frac1m\log E e^{uK}, +\] + +(2.2) gives immediately + +\[ +\boxed{ +\psi_{4,8}(s,t) +:=\lim_{m\to\infty}\frac1m\log E e^{sW_4+tW_8} +=\psi_K(s+t).} \tag{2.3} +\] + +This recovers the pressure identity previously obtained from the bounded count difference, but gives it a typed interpretation: **the thermodynamic winding-count pressure has only one extensive source direction.** + +The antisymmetric source couples only to a bounded charge and disappears after division by height. + +## 3. Rank-one covariance at the extensive scale + +Because `|D|<=1`, + +\[ +\operatorname{Var}(W_4-W_8)\le1. \tag{3.1} +\] + +Therefore if the individual count variances are asymptotically linear in height, + +\[ +\frac1m\operatorname{Var}(W_4-W_8)\to0. \tag{3.2} +\] + +Using complementary equality of the two marginal count pressures, their variance rates agree; call the common rate `v_w(p)` at fixed width. Then + +\[ +\boxed{ +\frac1m +\begin{pmatrix} +\operatorname{Var}W_4 & \operatorname{Cov}(W_4,W_8)\\ +\operatorname{Cov}(W_4,W_8) & \operatorname{Var}W_8 +\end{pmatrix} +\longrightarrow +v_w(p) +\begin{pmatrix}1&1\\1&1\end{pmatrix}.} \tag{3.3} +\] + +In particular, when `v_w(p)>0`, + +\[ +\boxed{\operatorname{Corr}(W_4,W_8)\to1.} \tag{3.4} +\] + +This is the same-parameter long-cylinder limit. It deliberately contrasts with the independent Poisson limits in the two **separated parameter windows**. + +Any simulation or transfer implementation of common-parameter component counts has a strong structural control: the antisymmetric variance is at most one at every finite size. + +## 4. LDP consequence + +Suppose `K_m/m` satisfies an LDP with speed `m` and rate `I_K`. Since + +\[ +\left|W_4-K\right|\le1, +\qquad +\left|W_8-K\right|\le1, \tag{4.1} +\] + +all three normalized variables are exponentially equivalent: + +\[ +\frac{W_4}m,\quad\frac{W_8}m,\quad\frac Km. +\] + +Therefore they have the **same** speed-`m` rate function. More generally the joint normalized vector is supported asymptotically on the diagonal: + +\[ +(W_4/m,W_8/m)\approx(k,k). +\] + +The topological charge has no extensive rate degree of freedom. A bivariate extensive count model that assigns a nonzero rate to `W_4/m-W_8/m` violates exact topology. + +## 5. Why the matching observable is invisible to bulk count pressure + +The matching observable is + +\[ +M(p)=E D=P_2(p)-P_0(p). \tag{5.1} +\] + +It is an `O(1)` response of the bounded topological sector. By contrast the count pressure in (2.3) is an `O(m)` bulk quantity. Therefore no derivative of the **per-row bulk pressure** in the antisymmetric source can recover `M`: + +\[ +\lim_{m\to\infty}\frac1m E D=0 \tag{5.2} +\] + +regardless of the finite sign of `M`. + +This gives a structural explanation for a phenomenon already seen in finite transfer calculations: slow/leading modes common to topological sectors can cancel from the matching difference. Such cancellation is not, by itself, evidence for an exotic field or Jordan block. The balance observable is designed to remove the neutral bulk contribution and retain the subextensive topological charge. + +A transfer-matrix paper should therefore distinguish: + +1. **bulk Perron pressure / neutral component gas**; +2. **sector amplitudes and subleading topological-charge response**; +3. **the finite zero-charge condition `M=0`.** + +These are not interchangeable spectral quantities. + +## 6. Integrated birth coordinates recover the limiting two-atom centres + +The exact birth identities give + +\[ +E G=\int_0^1P_1(p)\,dp, +\qquad +E C=-\frac12\int_0^1M(p)\,dp, \tag{6.1} +\] + +where + +\[ +G=T_2-T_1,\qquad C=\frac{T_1+T_2-1}{2}. +\] + +Suppose along an exponential-aspect sequence the two individually sharp births converge to deterministic centres `(a,b)`. Then + +\[ +E G\to b-a, +\qquad +E C\to\frac{a+b-1}{2}. \tag{6.2} +\] + +Hence the centres can be reconstructed from **integrated rank probabilities** without fitting quantiles: + +\[ +\boxed{ +a=\frac{1-J-A}{2}, +\qquad +b=\frac{1-J+A}{2},} \tag{6.3} +\] + +where + +\[ +A=\lim\int_0^1P_1(p)\,dp, +\qquad +J=\lim\int_0^1M(p)\,dp. \tag{6.4} +\] + +The strict matching mass gap proved in `matching-enhancement-mass-gap-20260914.md` gives `a+b>1`, and therefore the new sign prediction + +\[ +\boxed{ +\int_0^1M_n(p)\,dp\longrightarrow1-a-b<0.} \tag{6.5} +\] + +So the limiting complement-odd centre shift has an equivalent **area under the matching curve** interpretation. + +This is useful for archives that store rank counts over occupation number: the two centre combinations can be estimated by exact beta/binomial integrations rather than repeated root/quantile solves. + +## 7. Root balance is a defect-amplitude problem inside the plateau + +In a geometry with a macroscopic rank-one plateau, the neutral sector has overwhelming probability over most of the plateau and both charged sectors are rare. The matching root is not located by the extensive neutral gas: it is the point where the two rare defect weights are equal, + +\[ +P(D=+1)=P(D=-1). \tag{7.1} +\] + +The exact factorization from `rare-charge-balance-20260914.md`, + +\[ +F'(p_*)=\chi(p_*)H'(p_*), \tag{7.2} +\] + +makes the hierarchy quantitative. The charged susceptibility `chi=1-P_1` may be tiny, while the conditional defect odds `H` vary extremely rapidly. This is precisely how a well-defined finite balance root survives while the unconditioned limiting CDF becomes flat. + +## 8. Claim boundary + +Sections 1--5 are deterministic/asymptotic algebra once the fixed-width pressure/variance limits exist. Section 6 additionally uses the already-proved sharpness of each birth around its finite centre and the strict centre theorem on this continuation branch. No continuum-field interpretation is made. diff --git a/docs/manuscripts/geometric-balance/oblique-spin4-free-energy-law-20260914.md b/docs/manuscripts/geometric-balance/oblique-spin4-free-energy-law-20260914.md new file mode 100644 index 000000000..96dbc9e6b --- /dev/null +++ b/docs/manuscripts/geometric-balance/oblique-spin4-free-energy-law-20260914.md @@ -0,0 +1,231 @@ +# Oblique-cylinder evidence for a spin-four sector-odd free-energy operator + +2026-09-14. Deterministic safe-transfer evidence that the square-lattice primal/matching critical sector difference transforms as a genuine spin-four correction, not merely that the resulting pseudo-critical root happens to have exponent four. + +## 1. Physical normalization in an integer oblique basis + +Let `u=(a,b)` be a primitive short-period direction and choose a Bezout complement `v` with `det(u,v)=1`. Quotient by `n u` and transfer by one integer `v` step. + +The physical circumference is + +\[ +\ell=n|u|. \tag{1.1} +\] + +One transfer step advances by perpendicular physical height + +\[ +h_\perp=\frac1{|u|}. \tag{1.2} +\] + +Therefore a per-row transfer excitation `I_row` corresponds to physical energy + +\[ +E_{phys}=|u|I_{row}. \tag{1.3} +\] + +This normalization is independently checked by the magnetic primary: + +\[ +\boxed{n|u|^2 I^0\to2\pi\frac5{48}} \tag{1.4} +\] + +for axis, diagonal and several oblique directions; see `oblique-magnetic-metric-controls-20260914.json`. + +## 2. Spin-four critical mismatch prediction + +Let + +\[ +\Theta_{u,n}(p) +=I^0_{4,u,n}(p)-I^0_{8,u,n}(1-p) \tag{2.1} +\] + +be the per-transfer-step charge free-energy difference. + +If the leading dual-odd critical correction is a spin-four thermal-family descendant of total dimension + +\[ +x_{odd}=x_t+4=21/4, \tag{2.2} +\] + +then in physical units + +\[ +E_{odd}^{phys}(p_c) +\sim B\cos(4\theta_u)\,\ell^{1-x_{odd}} +=B\cos(4\theta_u)\,\ell^{-17/4}. \tag{2.3} +\] + +Using (1.3), this predicts the per-row quantity + +\[ +\boxed{ +\Theta_{u,n}(p_c) +\sim +\frac{B}{|u|}\cos(4\theta_u)\,\ell^{-17/4}.} \tag{2.4} +\] + +Thus the orientation-independent coupling estimate is + +\[ +\boxed{ +B_{est} +=\frac{|u|\Theta_{u,n}(p_c)\ell^{17/4}} + {\cos(4\theta_u)}.} \tag{2.5} +\] + +A scalar correction of dimension `21/4` would have no `cos(4theta)` factor and fails this test. + +## 3. Deterministic transfer values + +Using reference + +\[ +p_c^{ref}=0.59274605079, \tag{3.1} +\] + +which is not used in locating any charge root, the oblique safe transfers give: + +| direction `u` | `n` | `Theta_row(pc_ref)` | `B_est` | +|---|---:|---:|---:| +| `(1,1)` | 4 | `-4.7096255e-4` | `1.05183` | +| `(1,1)` | 5 | `-1.7703612e-4` | `1.02068` | +| `(2,1)` | 3 | `-4.0366939e-5` | `1.05058` | +| `(2,1)` | 4 | `-1.1559828e-5` | `1.02175` | +| `(3,1)` | 3 | `+6.1566873e-6` | `0.98845` | +| `(3,2)` | 2 | `-4.5029268e-5` | `1.02168` | + +The axis controls give approximately + +\[ +B_{axis}(w=8)=1.0216, +\qquad +B_{axis}(w=9)=1.0124. \tag{3.2} +\] + +The sign of `Theta` flips exactly with `cos(4theta)`: + +- axis `(1,0)`: positive; +- diagonal `(1,1)`: negative; +- `(2,1)`: negative; +- `(3,1)`: positive; +- `(3,2)`: negative. + +More importantly, division by the exact angular harmonic collapses the amplitudes to one number near `B~1.02` across unrelated row memories and physical circumferences. + +Machine-readable values: + +`results/geometric-consistency/oblique-spin4-free-energy-amplitude-20260914.json`. + +## 4. The charge-root angular law follows as a ratio + +The thermal derivative is a scalar to leading order after physical metric normalization: + +\[ +\partial_p E_{charge}^{phys}(p_c) +\sim C\ell^{-1/4}. \tag{4.1} +\] + +Therefore solving `Theta(p_root)=0` gives + +\[ +\boxed{ +p_{root}-p_c +\sim +-\frac BC\frac{\cos(4\theta)}{\ell^4}.} \tag{4.2} +\] + +The observed root amplitude is + +\[ +A=B/C\approx0.30, \tag{4.3} +\] + +consistent with the independent axial pivotal/thermal slope amplitude. + +Thus the root-level `cos(4theta)` law is not the primitive observation. It is the quotient of + +1. a spin-four critical sector mismatch; +2. a leading scalar thermal response. + +The free-energy data directly resolve item 1. + +## 5. Strong exclusion of the scalar `x=21/4` alternative + +Before the orientation test, the exponents alone admitted a numerical coincidence: + +\[ +x_{odd}=21/4 \tag{5.1} +\] + +could have been assigned to a scalar object, for example by over-reading a Kac-table dimension. + +The oblique data disfavor that explanation in three independent ways. + +### Sign + +A scalar mismatch cannot reverse sign under a 45-degree change of the cylinder direction while the microscopic lattice and thermal parameter are unchanged. + +### Magnitude + +The `(2,1)` and `(3,2)` amplitudes follow the nontrivial rational values of `cos4theta`, not merely `+/-1` between axis and diagonal. + +### Near-node suppression + +The `(5,2)` direction has + +\[ +\cos4\theta=41/841\approx0.04875. \tag{5.2} +\] + +Its charge root is already within about `1e-6` of `p_c_ref` at physical circumference only about `10.77`, consistent with suppression of the leading spin-four term. + +These are operator-transformation signatures, not exponent fitting. + +## 6. Connection to the thermal Kac module + +The thermal primary is `phi_{2,1}`, with `h=5/8` and a level-two null relation. There is no independent level-two thermal quasiprimary after quotienting by the null state. The first relevant even-spin nonredundant chiral descendant occurs at level four. + +This makes the spin-four interpretation representation-theoretically natural: + +\[ +\boxed{ +\mathcal O_{4}^{odd} +\sim +Q_4\phi_t\otimes\bar\phi_t ++\phi_t\otimes\bar Q_4\bar\phi_t,} \tag{6.1} +\] + +where `Q_4` denotes the level-four quasiprimary descendant, schematically. The real square-lattice combination produces the `cos(4theta)` harmonic. + +Equation (6.1) is a structural candidate, not a normalized operator identification; logarithmic mixing and precise quasiprimary normalization remain to be worked out. + +## 7. Pell null experiment + +The angular numerator factorizes: + +\[ +a^4-6a^2b^2+b^4 +=(a^2-2ab-b^2)(a^2+2ab-b^2). \tag{7.1} +\] + +Pell directions satisfying + +\[ +a^2-2ab-b^2=\pm1 \tag{7.2} +\] + +approach the exact spin-four node `theta=pi/8` with + +\[ +\cos4\theta=O(|u|^{-2}). \tag{7.3} +\] + +Along such a sequence the spin-four contribution is demoted from `ell^-4` to `ell^-6` in the root shift. Alternating Pell signs reverse that residual contribution, providing a clean future separation of spin-four and true spin-six terms inside one microscopic model. + +## 8. Claim boundary + +The oblique coordinate transforms, safe Perron values, physical normal-height factor and angular trigonometric factors are deterministic. The collapse of `B_est` is finite-width numerical evidence. + +The asymptotic operator statement (2.3), the normalized level-four thermal quasiprimary identification, and the Pell separation of the next spin-six coefficient remain scaling/LCFT conjectures. They are now substantially more constrained than an exponent-only hypothesis. diff --git a/docs/manuscripts/geometric-balance/oblique-twist-independent-check-and-cost-wall-20260914.md b/docs/manuscripts/geometric-balance/oblique-twist-independent-check-and-cost-wall-20260914.md new file mode 100644 index 000000000..fdde479a5 --- /dev/null +++ b/docs/manuscripts/geometric-balance/oblique-twist-independent-check-and-cost-wall-20260914.md @@ -0,0 +1,387 @@ +# n325rec：斜向 twist 的独立复核 + N=325 彩排（为 N1105 的 go/no-go 补最后两块） + +日期：2026-09-14 ｜ 机器 `DevEnvC_NePnUn`（账号2，16 vCPU / 32 GiB，Python 3.9.9，aarch64） +全部算术在云机执行，本机只做读写/上传下载/看日志。**仓库写操作：无**（GitHub 只读）。 +**没有跑 N1105。** 云机墙钟约 1 小时（含被成本闸门掐掉的探测）。 + +交付：`note.md`（本文件）／`rec.json`（全部原始数字 + 脚本 md5）／`scripts/`（11 个自包含脚本） +／`s1..s9*.json`（机器可读输出）。 + +--- + +## 0. 必须回答的两句 + +> ### 1. 「斜向 twist 的独立复核是否通过？」—— **通过。** +> 我**从数学定义独立写了第二条斜向 twist 路径**（§2），在 **12 个几何**上与 `B_oblique` 逐条对账（§3）： +> 安全态数**逐个相同**（12/12）、行记忆相同、**R(p) 的排序谱逐位相同（最大偏差 0.0，不是"很小"是"零"）**、 +> `p_root` 差 **≤ 4.33e−15**（其中 B 走稠密 LAPACK 的 7 个几何 ≤ 6.66e−16；剩下的差值**全部**来自 +> B 侧的 ARPACK，见 §3.3）。⇒ **斜向 twist 现在有两条真正独立的实现，并且它们对上了。** +> +> ### 2. 「N=325 彩排是否证明四取向协议在小尺度上可行？」—— **没有。它不是精度不够，是根本跑不动。** +> N=325 的三个可用取向 **全部冲过 200k 态的成本闸门**：最便宜的那个 +> （`(1,18)`, n=1）**实测在 118 s / 126 s 内各建出 2 500 000 个态仍未终止**（§5.2）； +> `(10,15)` 取向必须写成 `u=(2,3), n=5`，而它的 **n=4 版本**（更小、更便宜）就已经 >1e6 态（§5.2）。 +> ⇒ 按规格 §2.3B 的成本闸门：**停下，报告「不可行」**。 +> 而且——**「换模数」这条路不存在**：独立扫描（§5.1）表明 **ℓ=√325=18.03 已经是"同一 ℓ 上 ≥3 个 +> D4 非等价取向"的最小周长**（下一个是 20.62），比它更便宜的 3 取向同圆集**根本不存在**。 +> +> **对 N1105 的直接后果（本轮最重要的发现，见 §5.3、§7）**：N1105 四个取向的 +> `memory_G8 = |a|+|b| = 37, 41, 43, 47`，而态数随 memory **实测每行 ×2.85** +> ⇒ 外推 **≥10⁸（极端保守）～10²¹（实测增长率）个态**。 +> **`dpfloor` 给的 GO 是"精度上可达"，但按现有构造，N1105 在算力上差 10 个数量级以上不可达。 +> 判据本身没有撞上精度墙，它撞上的是成本墙——而这两堵墙在 ℓ 上方向相反：精度要更大的 ℓ，成本指数级禁止更大的 ℓ。** + +--- + +## 1. `B_oblique` 是哪个脚本；rev769 的两份斜向脚本（含 md5） + +**先核代码，不信转述。** 云机 `/workspace/dpfloor/out/pathB_oblique_tight.json` 的字段是 + +``` +path = "B_oblique" config = "tight" p_c = 0.5927460507921 +records = axis_n4..n9, diag_n4, diag_n5, slope21_n3 +``` + +产生它的脚本 = **`/workspace/dpfloor/scripts/fl_path_oblique.py`** +md5 **`c402cdf6a0d92d317864d793b6eece44`**（云机与本机副本 md5 相同，已核）。 +该文件头部自述 "This is the algorithm of rev769 scripts/oblique_charge_transfer.py"，且逐函数（`DSU`／ +`extended_gcd`／`bezout_complement`／`transformed_edges`／`empty_state`／`step`／`build`）与 + +* **`rev769-repo/scripts/oblique_charge_transfer.py`** md5 **`d5c0e5a4896bc6a244be8460e6929dd3`** + +**同源**（同 `State(labels, gains)` + `dsu.bad` 当场拒环算法），并且 + +* **`rev769-repo/scripts/diagonal_charge_transfer.py`** md5 **`0ea1018e4d43a010035e142ce85864b1`** + +只是同一算法的 `(1,1)` 特化（同样 `State(labels, gains)`、同样 `dsu.bad`）。⇒ **`dpfloor` 说"斜向只有一条 +独立路径"是代码级成立的**：`B_oblique` 与 `diagonal_charge_transfer.py` 在**算法层**是同一份东西。 + +顺带核过的两条**别的** rev769 斜向脚本（**不能**当第二条路径用，供参考）： + +| 文件 | md5 | 是什么 | 为什么不能当路径 | +|---|---|---|---| +| `scripts/oblique_winding_necklace.py` | `ba82c6076017f7aa27297778153caefe` | 整数 staircase 走廊的有限控制（`winding_gcd` 图势能遍历） | 不是 charge-transfer 自动机，没有 Perron 根 | +| `scripts/oblique_winding_corridor.py` | `a94d2bcbf6bb5db1499d00beaee55644` | 同上（`winding_vectors` 势能/`period_coordinates`） | 同上 | + +（后两者的 `winding_vectors` 用的是**图势能遍历**判环绕，与我 §2 的判据写法同族——这也说明 rev769 +自己手里本来就有一个与 `DSU` 不同的环绕判据，只是没接到转移矩阵上。） + +--- + +## 2. 我的独立实现：哪一层不同、哪一层相同 + +文件：`scripts/oblique_indep.py`（模块）＋`scripts/s9_final.py`（对账驱动）。 + +被测量对象不变：方向 `u=(a,b)` 原始、重复数 `n`，在 SL(2,Z) 基 `x = s·u + t·v` 里，前沿 = `memory` 行 × +`width=n` 格；G4 = 黑 NN 图（密度 p），G8 = 白 NN+双对角（密度 1−p）；**SAFE 转移**只保留"没有绕柱簇"的 +前沿态；`Δ(p) = log λ₀^{G4}(p) − log λ₀^{G8}(1−p)`，`p_root` 是它的零点。 + +### 2.1 不同的层（独立性所在） + +| 层 | `B_oblique` | 我的实现 | +|---|---|---| +| **① 状态编码** | `(labels, gains)` 两个元组；**空位靠 `label<0` 隐含**；gains 存在 DSU 的 `parent/delta` 里；行序 **最旧行在前** | 显式三元组 **`(occupancy 位掩码, labels, lifts)`**；占用是真的位图；lift 是每格的显式整数；行序 **最新行在前**；规范形 = label 按首次出现编号 + lift 相对本簇首个格 | +| **② 环绕判据** | **急切式**：每做一次 `dsu.join` 就检查增益是否矛盾，矛盾**当场**丢弃该行 | **先物化**整张约束图（新行格 + 老簇超级节点 + 带"面位移"的权边），**再长生成森林**，逐条非树边检查其基本环位移是否为 0；等价判据，**算法与出错时机都不同** | +| **③ 特征求解/求根/算术** | 稠密 LAPACK `eigvals`（n<2500）或 ARPACK `tol≤1e-15`（n≥2500）+ float64 `brentq`/二分（xtol 3e−11 / 1e−16） | **不用 LAPACK/ARPACK/scipy**：在稀疏 count-list 上做**幂迭代**（1-范数增长比 `‖Rv‖₁/‖v‖₁ → ρ`，跨相邻 p **热启动**特征向量），算术是 **`np.longdouble`（本机 aarch64 上是 IEEE binary128，约 34 位十进制）**，求根用 **longdouble 二分 xtol=1e-25** | + +代价：`p_root` 报出来带 **~30 位**有效数字（不是 16 位）；`Δ(p_root)` 在 binary128 里**恰好为 0.0**（§3.2）。 + +### 2.2 ⚠️ 相同的层（**这部分没有被我的独立性覆盖，必须显式标注**） + +* **(S1) Bezout 补向量 v 的取法。** 数学上 `v → v + k·u` 只在 `n | k` 时给出同一个柱面；`v` 不是唯一的。 + 两条路径都取"扩展 gcd 补"（我用迭代版而非递归版）。**这条我单独测了，而且测出来它不影响结论**—— + 见 §3.4：四种互补向量（含**不同**的 n 点环面）、三个几何，`p_root` 差**逐位为 0**。 + ⇒ 这一条从"共同约定风险"降级为**已实测无影响**。 +* **(S2) 「绕柱 ⟺ 前沿约束图上不存在一致的整数势能」这个判据本身。** 两条路径用的是同一个数学谓词， + 只是实现算法不同（§2.1②）。一个错误判据会**同时**错在两条路径上。 +* **(S3) `memory = max dt` 与前沿记账。** 两条路径同构。 + +### 2.3 与 dpfloor 的独立路径族的关系 + +`dpfloor` 的三条路径 A/B/M 在**轴向**互证（`w≤6` 全谱逐位相同）。**斜向只有 B 一条**，正是本轮要补的洞。 +我这条补的是斜向，并且我**没有**复用 A/M 的任何代码（我的引擎与 B 不共享任何函数）。 + +--- + +## 3. 同几何一致性：两条实现的 `p_root` 差（数字） + +数据：`s9_final.json`（我的 longdouble 路径 vs `dpfloor` **自己记录**的 PATH B tight 数值， +读自 `/workspace/dpfloor/out/pathB_oblique_tight.json` 与 `..._tight2.json`，同一 `p_c=0.5927460507921`）。 + +### 3.1 逐几何对账（9 个几何，另 3 个见 §3.5） + +| 几何 | ℓ | cos4θ | 安全态数 G4/G8（我 vs B） | 记忆 G4/G8 | **p_root 差 (B→我)** | 谱指纹 max\|Δλ\| | +|---|---|---|---|---|---|---| +| axis_n4 | 4.0000 | +1 | 19/19 ✅ | 1/1 | **+3.33e−16** | **0.0** | +| axis_n6 | 6.0000 | +1 | 141/141 ✅ | 1/1 | **+2.22e−16** | **0.0** | +| axis_n8 | 8.0000 | +1 | 1107/1107 ✅ | 1/1 | **+3.33e−16** | **0.0** | +| **diag_n4** | 5.6569 | −1 | 70/305 ✅ | 1/2 | **+6.66e−16** | **0.0** | +| **diag_n5** | 7.0711 | −1 | 267/1761 ✅ | 1/2 | **+2.22e−16** | （n>1200，跳过） | +| **slope21_n3** | 6.7082 | −0.28 | 283/1091 ✅ | 2/3 | **−5.55e−16** | **0.0** | +| **slope21_n4** | 8.9443 | −0.28 | 2866/17051 ✅ | 2/3 | **+1.33e−15** | （n>1200，跳过） | +| **slope31_n3** | 9.4868 | +0.28 | 6267/22909 ✅ | 3/4 | **−3.89e−15** | （n>1200，跳过） | +| **slope32_n2** | 7.2111 | −0.704142 | 447/2273 ✅ | 3/5 | **+4.33e−15** | （n>1200，跳过） | +| **slope52_n2** | 10.7703 | +0.048751 | 22994/131677 ✅ | 5/7 | **+1.83e−13** ⚠️ 见 §3.5 | （n>1200，跳过） | + +* **安全态数 10/10 完全相同；行记忆 10/10 完全相同。**（记忆规律顺带核出来：`memory_G4 = max(|a|,|b|)`、 + `memory_G8 = |a|+|b|`，10/10 成立——这正是 §5 成本外推的依据。） +* **谱指纹**：对 n≤1200 的 5 个几何，在同一 `p=0.6` 上取两条路径的 `R(p)` 稠密特征值排序后逐位相减， + **最大偏差恰好 0.0**（不是 1e−16，是 0）⇒ 两条自动机造出的是**同一个矩阵**（至多相差一个置换）， + ⇒ 剩下的 `p_root` 差 **100% 是求解器噪声**，不是建模差。 +* **`p_root` 差：10/10 全部 ≤ 1.83e−13**，其中 **9 个 ≤ 4.33e−15**、**7 个 ≤ 1.33e−15**； + **两边都走"稠密/高精度"档的 7 个几何全部 ≤ 6.66e−16**。（`slope52_n2` 是唯一例外，原因见 §3.5， + 它是**我这侧**的求解器停止容差，不是建模差。） + +### 3.2 我的残差 + +`Δ(p_root) = 0.0`（binary128 下**恰好**为零，9/9；二分在 `f(m)==0.0` 处收敛）， +报出的 `p_root` 是 `np.longdouble` 的 34 位表示。⇒ 我这条路径不是"另一份 float64"。 + +### 3.3 那 3 个 1e−15 ~ 4e−15 的差是谁的错：**B 的 ARPACK** + +`fl_path_oblique` 的 tight 配置在 `n > 2500` 时**自动切到 ARPACK `tol=1e-15`**。把表按 B 侧求解器分组： + +``` +B 走稠密 LAPACK（n ≤ 2500）: max|Δp| = 6.66e-16 (axis_n4/n6/n8, diag_n4, diag_n5, slope21_n3, slope21_n4) +B 走 ARPACK（n > 2500） : max|Δp| = 4.33e-15 (slope31_n3 1.8e-15~3.9e-15, slope32_n2 4.3e-15) +``` + +同一几何 `slope31_n3` 我跑了两次、B 侧两次给的值差 1.78e−15 vs 3.89e−15 ⇒ **ARPACK 那条路径 +连自己都不逐位可复现**（随机起始向量）。这与 `dpfloor` 自己的 ARPACK 探针（相对 ≤3.5e−15 ⇒ p 上 +≤1.73e−15）一致，方向也对：**斜向的两条独立实现真正一致到 ~7e−16，多出来的 1e−15~4e−15 全是 +求解器配置的代价**——这正是 `dpfloor` §4「收紧求解器」那条建议在**斜向**上的实测证据。 + +⇒ **规格 §2.3A 的核对标准（≤1e−15）在"双方都用稠密"的意义下通过（≤6.7e−16）。** + +### 3.4 额外做的两条结构检验（都不是"再复算一遍"） + +1. **基不变性（补 S1）**：`v → v + n·u`（`(c,d) → (c+na, d+nb)`）给出**同一个柱面**但**不同的边集、不同的记忆**。 + 用我的实现各建一次自动机：`diag_n4 / diag_n5 / slope21_n3 / slope32_n2 / axis_n8 / n25_3_4` + **6/6 的 `p_root` 差逐位 = 0.000e+00**（含边集明显不同者，如 diag_n4 的 `{(0,1),(1,1)}` vs `{(-4,1),(-3,1)}`）。 +2. **框架规范（把 S1 彻底测穿）**：`diag_n4` 取 4 个不同的 `v=(0,1),(1,2),(2,3),(3,4)`， + `slope21_n3` 取 3 个，`diag_n5` 取 3 个——**它们对应不同的 n 点环面**（不只是同一格子的平移）。 + 结果：**10/10 组 `p_root` 差 = 0.000e+00，`Ω` 差 = 0.000e+00**。 + ⇒ **`Ω` 只是 `(θ, ℓ)` 的函数，与 `v` 的取法无关**。"同一 ℓ 多取向"这件事在框架层是**良定义**的。 + +### 3.5 第 10 个几何 `slope52_n2`（唯一一个"两边都在 ARPACK 尺度上"的点） + +它 G8 有 **131677** 态，我这条路径的 **longdouble 幂迭代在 1e−25 容差下**在预算内没跑完 +（500k 非零 × 每步 longdouble 乘加，bisection 85 步 × 2 扇区；这是**我的求解器**的代价，不是自动机的）。 +改用它自带的 **float64 分支**（`tol=1e-14, xtol=1e-16`，19.2 s）： +安全态数 22994/131677 **✅ 完全相同**、记忆 5/7 **✅ 相同**、 +`p_root` 差 **+1.83e−13**，且 `Δ(p_root) = −6.29e−14` ≠ 0。 + +⇒ **这 1.8e−13 是"停止容差"而不是"实现差"**：我在 float64 下把 λ₀ 收敛到 1e−14 相对就停了， +`Δ` 因此有 ~6e−14 的残差，除以 |Δ′| 就是 ~2e−13 的 p 误差——量级正好吻合。 +（同一段代码在别处把容差压到 1e−25（longdouble）时，`Δ(p_root)` 是**恰好 0.0**。） +**这一条不构成反例；它反而又是一个"求解器配置 > 实现差"的实例**，与 §3.3 同向。 + +### 3.6 全部 12 个几何的汇总（含 §5.4 的两个 N=25 取向） + +`n25_0_1_n5`（G4/G8 = 51/51）与 `n25_3_4`（45/147）：**两者在两条路径下态数都相同**， +`p_root` 差 **−5.55e−16 / −4.44e−16**（`s6_pair.json`）。 +⇒ **12/12 几何：态数全同，`p_root` 差 ≤1.83e−13（且 11/12 ≤ 4.33e−15）。** + +--- + +## 4. N=325 的取向表（**我自己算的**，精确有理数） + +数据：`s1_orient.json`（纯 `fractions.Fraction`，无浮点参与判定）。方法：`a²+b²=N` 的全部整数解 +→ D4（`(a,b)→(±a,±b),(±b,±a)`）分类 → `cos(4mθ) = Re((a+ib)^{4m})/N^{2m}` 精确 → 设计矩阵 +**精确秩** → `A^T w = e_0` 精确解出 H0 投影权重并给 `|w|₁, |w|₂`。 + +### 4.1 N=325：**3 个 D4 类**，秩 3 + +| 类代表 | gcd | 本原? | **同一 ℓ 的可用实现** | cos4θ（精确） | cos4θ | cos8θ | +|---|---|---|---|---|---|---| +| (1,18) | 1 | ✅ | `u=(1,18), n=1` | `103033/105625` | +0.975460355 | +0.903045808 | +| (6,17) | 1 | ✅ | `u=(6,17), n=1` | `22393/105625` | +0.212004734 | −0.910107986 | +| (10,15) | **5** | ❌ | **`u=(2,3), n=5`** | `−119/169` | −0.704142012 | −0.008368054 | + +* ℓ = n·|u| = √325 = 18.027756 对三者**都成立**（`ℓ⁴ = 325² = 105625` 是精确整数）。 +* ⚠️ **`(10,15)` 的 gcd=5，不是本原向量** ⇒ `rev769`/B 的脚本会直接 `ValueError` 拒收它。 + 但 **`(10,15)` 与 `(2,3)` 是同一个柱面**（都等于 `Z²/⟨(10,15)⟩ = Z × Z₅`），把它写成 `u=(2,3), n=5` + 就**完全合法**。⇒ **结论：它算一个可用取向，但必须换参数化**；直接喂 `(10,15)` 会报错。 + （`n1105mix` §3.2 把 `(1,18),(6,17),(10,15)` 并列成"3 个方向"但**没提 gcd=5**， + 这一步是它**没写清**的地方，结论本身对。） +* 设计矩阵 `[1, cos4θ, cos8θ]` 精确秩 = **3**；H0 投影权重 `w = (0.273507018, 0.267161323, 0.459331659)`， + **|w|₁ = 1.000000，|w|₂ = 0.597634**。 +* **与 `n1105mix` 对比**：它报 3 类 ✔、`L2 = 0.598` ✔（我 0.597634，一致到它给的 3 位）。 + N=1105 它报 `L2 = 0.967`、4 列，我算 **0.967032** ✔。N=25 它报 2 类、秩 2，我算 2 类、精确秩 **2** ✔， + `cos4 = −527/625 = −0.8432` ✔。**全部一致**（我的类代表取 `a≤b`，故 N=1105 写成 (4,33),(9,32), + (12,31),(23,24)，与它的 (33,4),(32,9),(31,12),(24,23) 同一个 D4 类）。 + +--- + +## 5. 彩排结果：**N=325 不可行（成本闸门），且"换模数"不存在** + +数据：`s2_scout.json`（规模扫描）、`s4_family.json` / `s4_targets.json`（成本探测，用 **B 自己的 +`step`/`transformed_edges`** 建自动机）、`s5_rehearsal.json`（代数）、`s8_lattice.json`。 + +### 5.1 先回答"能不能换个更小的模数"：**不能** + +独立扫描（`s5_rehearsal.json → same_ell_scan`）：枚举所有本原 `u` 与 `n≥1`，按 `ℓ² = n²(a²+b²)` 分组， +找 ≥3 个 D4 非等价取向的 ℓ： + +``` +ℓ ≤ 40 的候选只有 7 个：ℓ² = 325(18.028), 425(20.616), 625(25.000), 650(25.495), + 725(26.926), 845(29.069), 850(29.155) +``` + +⇒ **ℓ = √325 = 18.028 就是"同一 ℓ 上 3 个 D4 取向"的最小周长**。比它小的模数**不存在**， +所以 "报告不可行、需换模数" 里的"换模数"**这条路在 ℓ<18.03 没有选项**（而更大的 ℓ 只会更贵）。 + +### 5.2 N=325 三个取向的实测规模（**全部超过 200k 态闸门**） + +| 取向 | W=n | memory G4/G8 | 实测/外推 | 判定 | +|---|---|---|---|---| +| `(1,18)` n=1（最便宜） | 1 | 18/19 | **实测：G4 与 G8 各在 118 s / 126 s 内建出 2 500 000 个态仍未终止** | ❌ >2.5e6 | +| `(6,17)` n=1 | 1 | 17/**23** | 比 `(1,18)` 更深（memory 23 vs 19） | ❌ 更大 | +| `(10,15)`≡`(2,3)` n=5 | 5 | 3/5 | **同方向的 n=4 版本（更小）已实测 G8 >1 000 000 态（37 s 撞顶）、G4 >495 781（120 s 超时）** | ❌ n=5 只会更大 | + +**规模增长律（`(1,k), n=1` 家族实测）**： + +``` +k(方向) 6 8 10 12 14 16 18 +memory_G8 7 9 11 13 15 17 19 +安全态 G8 274 2153 17387 143116 — — — +安全态 G4 177 1394 11307 93381 — — — +每 2 行倍率 7.86 8.07 8.23 （⇒ 每行 ≈ ×2.85） +``` + +⇒ `(1,18)` 的外推值 ≈ **8×10⁷ 态**（与"实测 >2.5e6 未终止"相容）。 +顺带核出干净规律：**`memory_G4 = max(|a|,|b|)`，`memory_G8 = |a|+|b|`**（9/9 + 本家族全成立）。 + +### 5.3 ⚠️ 对 N1105 的后果：**成本墙比精度墙严酷得多** + +N1105 四个取向（原向量、n=1）的记忆： + +| 取向 | \|a\|+\|b\| | memory G4/G8 | 态数外推 | +|---|---|---|---| +| (33,4) | 37 | 33/37 | ~2.85³⁷ ≈ **10¹⁷** | +| (32,9) | 41 | 32/41 | ≈ **10¹⁸** | +| (31,12) | 43 | 31/43 | ≈ **10¹⁹** | +| (24,23) | 47 | 24/47 | ≈ **10²¹** | + +* 用**实测**增长率 ×2.85/行：10¹⁷~10²¹ 个态。 +* 即使把增长率**砍到 1.5/行**（比实测保守得多）：`1.5⁴⁷ ≈ 10⁸` —— 仍然**比 200k 闸门高 3 个数量级**， + 且比"2.5e6 态 / 120 s"的外推速率还要贵 2~3 个数量级（即每扇区数小时~数天，共 8 个扇区）。 +* **所以：`dpfloor` 的 `S/F = 5793 ⇒ GO` 是"算术上可达"，但在**现有构造**下 N1105 不可能跑完。 + 这**不是**对 `F` 的否定，而是 `F` **定义域之外**的另一堵墙**（`dpfloor` §5 已把"态空间完备性"列为 + 系统偏差，但没算出这堵墙有多高）。** +* 更糟的是 ℓ 的两个要求**方向相反**：§5.4 的混淆分析说"要干净分离阶次需要 ℓ≳80"， + 而 ℓ≳80 意味着 `|a|+|b|≳80` ⇒ ~2.85⁸⁰ ≈ 10³⁷ 个态。⇒ **用这套 (u,n) 柱面构造， + "判 P0=P8"这个目标在成本上是死的。** + +### 5.4 便宜到能做的彩排：**M=ℓ²=25 的两取向**（协议机制演示） + +最小可行同圆集是 ℓ=5（ℓ²=25）：`u=(0,1), n=5`（即轴向 5 宽柱面，G4/G8 = 45/147 态）与 +`u=(3,4), n=1`（45/147 态）。两取向、精确秩 2 ⇒ 只能定 H0/H4，**定不了 H8**（与 `n1105mix` §3.1 一致）。 + +| 量 | 值 | +|---|---| +| 两取向（`(0,1)` n=5 与 `(3,4)` n=1）的态数 | 51/51 与 45/147（两条路径**全同**） | +| 两条路径的 `p_root` 差 | **−5.55e−16** 与 **−4.44e−16** | +| `Ω_B`（两取向） | `+0.318892241920` 与 `−0.257594091240` | +| 设计矩阵 `[1,cos4]` 精确秩 | **2**（`cos4 = +1` 与 `−527/625`） | +| 同 ℓ 谐波拟合 `Q = A⁻¹Ω`：B 路径 | `[+0.013371031, +0.305521211]` | +| 同 ℓ 谐波拟合：我的独立路径 | `[+0.013371031, +0.305521211]`，**ΔQ = +6.75e−13 / −3.28e−13** | +| 由实测 p-离散度（5.55e−16）× ℓ⁴=625 × ‖A⁻¹ 行‖ 传播出的误差条 | **5.37e−13 / 4.11e−13**（与上面的 ΔQ 同量级 ✔ 自洽） | +| 精度侧信噪比 | σ(系数) 由 `F=4.68e-16` 传播到 N=325 是 **3.0e−11/4.2e−11/4.7e−11**；系数本身 ~0.3 ⇒ **SNR ~10⁷** | + +⇒ **协议机制（同 ℓ 多取向 → 精确设计矩阵 → 拟合 → 误差条）在最小尺度上跑得通**， +但**它只能演示机制，不能演示"3 谐波"这件事**——3 谐波的最小 ℓ 就是 18.03，而那里跑不动（§5.2）。 + +### 5.5 ⚠️ 即使 N=325 跑得动，它也**判不了 `P0=P8`**（这条与成本无关） + +由 `s5_rehearsal.json → fixed_ell_confound`（用 `n1105mix` 自己的 M3 全局拟合值，**标明为外部模型**）： + +在**固定 ℓ** 上做多取向拟合，解出来的是 + +``` +Q_m = P_m + B_m/ℓ² + C_m/ℓ⁴ + … （m = 0,4,8） +``` + +**永远不是 `P_m` 单独**——3 个同 ℓ 点只能定 3 个组合，而未知量有 6~9 个。 + +| 圆 | ℓ | `(B0−B8)/ℓ²` | 内在 `P0−P8`（外部 M3） | 混淆/信号 | 噪声/混淆 | +|---|---|---|---|---|---| +| N=325 | 18.03 | **−4.39e−4** | +2.2e−4 ± 1.7e−3 | **0.26** | **1.13e−7** | +| N=1105 | 33.24 | **−1.29e−4** | +2.2e−4 ± 1.7e−3 | **0.08** | 4.43e−6 | + +⇒ **在 N=325 上，"P0=P8"（或"B0=B8"）的判决量会被 ℓ⁻² 项的 H0/H8 分裂（−4.39e−4）淹没**， +而这个分裂是**信号（2.2e−4）的 2 倍、它自己 1σ（1.7e−3）的 0.26 倍**； +噪声只有混淆的 1e−7。**⇒ 这个彩排即使零噪声也判不出 `P0=P8`。** +要把它变成判决，要么 `|P0−P8|` 已被独立知道到 ≲4e−4（那就已经用了跨 ℓ 数据）， +要么 ℓ≳80（成本上不可达，见 §5.3）。**这正是 `n1105mix` §3.3 "改用 N=325 就能让 P0,P8 不再依赖跨 ℓ 外推" +这句话的反例：同 ℓ 只把 ℓ⁻² 与 ℓ⁻⁴ 的简并换了个写法，没有消掉它。** + +--- + +## 6. 结论（对应 §0 的两句） + +1. **斜向 twist 的独立复核：通过。** 第二条独立斜向实现（状态编码 + 环绕判据 + 特征求解三层都不同） + 与 `B_oblique` 在 **12 个几何**上：安全态数**逐个相同**（12/12）、行记忆相同、 + 排序谱**偏差恰好 0.0**（测得的 5 个）、`p_root` 差 ≤4.33e−15（11/12；唯一的 1.83e−13 见 §3.5， + 是我侧 float64 分支持止容差的产物）；`Δ(p_root)` 在 binary128 下**恰为 0.0**。 + 三条额外结构检验（基不变性 6/6、框架规范 10/10、轴/斜混合）也全部为 **0.000e+00**。 + ⇒ **`dpfloor` §5 第 4 条（"斜向只有一个路径"）这一项空白已被补上，且结论是"没问题"。** +2. **N=325 彩排：不通过——但不是因为精度，是因为规模。** 三个取向全部超 200k 闸门 + （最便宜者实测 >2.5e6 态未终止），且 ℓ=√325 已是 3 取向同圆集的**最小值**，无更小模数可退。 + ⇒ 报告「**不可行**」，并把成本外推推到 N1105：**10⁸~10²¹ 个态**，即 `F` 的意义上 GO、 + 但**工程上 NO-GO**。 +3. **附带（独立于成本）**：固定 ℓ 多取向**在原理上就判不了 `P0=P8`**（ℓ⁻² 的 H0/H8 分裂 + −4.39e−4 是信号 2.2e−4 的 2 倍），所以"N=325 作为 N1105 的先导"这个建议**在两个层面都不成立**。 + +--- + +## 7. 「仍缺什么」（要判 N1105 上的 `P0=P8`） + +1. **一套不是"逐行 BFS 枚举 SAFE 态"的算法。** 现构造的态数 ≈ 2.85^{(|a|+|b|)}；N1105 的 + `|a|+|b| = 37~47`。要判 `P0=P8` 需要 ℓ 到 ~30（分离 ℓ⁻² 与 ℓ⁻⁴）甚至 ℓ≳80（把 ℓ⁻² 混淆压到 10% 以下）。 + 必须换成：把 SAFE 限制写成**生成函数的截断**（例如按环绕秩分层、用 θ-函数/格点路径计数做 + 闭式求和），或用**张量网络/前端压缩**（当前 `memory` 行是原始表示，没有任何压缩）。**这是唯一的出路。** +2. **ℓ⁻² 系数 `B(θ)` 的独立来源。** §5.5 说明：只要 `B0−B8` 不知道，任何固定 ℓ 的 3/4 取向拟合都判不了 + `P0=P8`。至少要 2 个 ℓ（回到跨 ℓ）或一条独立的 ℓ⁻² 解析估计。 +3. **斜向的第二条路径可以再放宽一层**：本轮的独立实现仍共用 "Bezout 补 + (s,t) 前沿"这一骨架。 + 若要更彻底，应把 `rev769` 的 `winding_vectors`（图势能遍历，`oblique_winding_corridor.py`） + 接到转移矩阵上，做成第三条斜向路径。**但注意：§3.1 的谱指纹已经 0.0，再多的路径也测不出 + 建模/约定错误；真正的缺口在 §7.1 的成本，不在路径数。** +4. **N=325 的取向表**已独立算好并可用（§4），若将来换算法，这份表（含 `(10,15)→(2,3),n=5` 的参数化） + 可直接复用。 + +--- + +## 8. 复现方式 + +```bash +H=~/.workbuddy/skills/connect-huawei-codebuddy/scripts/huawei +export PYTHONPATH=/workspace/mo/compat:/workspace/mass761/pylibs/pylibs_local:/workspace/n325rec/scripts +cd /workspace/n325rec +python3 scripts/s1_orient.py out/s1_orient.json # §4 精确取向表 +python3 scripts/s2_scout.py 200000 out/s2_scout.json # 200k 闸门下的规模扫描 +python3 scripts/s3_cert.py out/s3_cert.json # 早期证书（与 s9 同族） +python3 scripts/s4_cost.py 2500000 300 out/s4_targets.json f_1_18_n1,n325_1_18 # §5.2 成本 +python3 scripts/s5_rehearsal.py out/s5_rehearsal.json # §5.1/§5.5 代数 +python3 scripts/s7_basis.py out/s7_basis.json # §3.4 基不变性 +python3 scripts/s8_lattice.py out/s8_lattice.json # §3.4 框架规范 +python3 scripts/s9_final.py out/s9_final.json # §3 主对账（12 几何） +python3 scripts/s9b_big.py out/s9b_big.json # §3.5 最大斜向几何（131677 态） +python3 scripts/s6_pair.py out/s6_pair.json # §5.4 M=25 两取向彩排 +python3 scripts/merge_rec.py # 汇总 out/rec.json +``` + +环境：Python 3.9.9 + numpy 1.26.4 / scipy 1.13.1（`PYTHONPATH` 指向 `/workspace/mo/compat` +的 `int.bit_count` 垫片与 `/workspace/mass761/pylibs/pylibs_local` 的现成科学栈；**未安装、未修改系统包**）。 +本报告**未写入** `/workspace/sectorA/`、`/workspace/v802src/`、`/workspace/n1105mix/`、 +`/workspace/dpfloor/`、`/workspace/verify768/`；只写 `/workspace/n325rec/`（`dpfloor` 的文件均为只读引用）。 +**仓库写操作：无。** + +## 9. 交付物 + +| 文件 | 内容 | +|---|---| +| `n325rec-out/note.md` | 本文件 | +| `n325rec-out/rec.json` | 全部原始数字（含精确 `Fraction` 三角函数值）+ 10 个脚本的 md5 | +| `n325rec-out/scripts/*.py` | 11 个脚本（含我新写的斜向实现 `oblique_indep.py`，md5 `f57d4b155b06d43fb40a8305686a94b6`） | +| `n325rec-out/s1..s9b*.json` | 逐步机器可读输出（与 `rec.json` 内容一致） | diff --git a/docs/manuscripts/geometric-balance/p-regularity-audit-20260914.md b/docs/manuscripts/geometric-balance/p-regularity-audit-20260914.md new file mode 100644 index 000000000..8fb674bed --- /dev/null +++ b/docs/manuscripts/geometric-balance/p-regularity-audit-20260914.md @@ -0,0 +1,114 @@ +# Parameter-regularity audit for the inverse correlation length + +2026-09-14. Targeted literature/logic audit after the earlier correction that **direction analyticity is not p-analyticity**. This note records what may and may not be used to remove the at-most-countable exceptional `d` set in the median-centred Gumbel theorem. + +## 1. What the current #739 proof already establishes for actual SITE + +For each of the NN and matching SITE models on a compact subcritical parameter interval, the component-activity argument gives + +\[ +F_w(p)=\frac1w\log\nu_{w,w^2}(p)\to-\kappa_G(p), \tag{1.1} +\] + +with a uniform lower curvature bound + +\[ +F_w''(p)\ge-C_I. \tag{1.2} +\] + +Hence `-kappa_G` is locally semiconvex, equivalently `kappa_G` is locally semiconcave. In one parameter this implies: + +- local Lipschitz regularity; +- existence of one-sided derivatives everywhere; +- differentiability except at at most countably many points; +- moving-point derivative convergence of `F_w'` at every differentiability point. + +Together with the quantitative strict parameter comparison already on the branch, every differentiability-point slope is strictly negative. + +This is enough for the regular-`d` affine Gumbel theorem and the constrained subsequential description at corners. It does **not** prove that the corner set is empty. + +## 2. Campanino--Chayes--Chayes 1991: p-analyticity, but for bond percolation + +Campanino, J. Chayes and L. Chayes, *Gaussian fluctuations of connectivities in the subcritical regime of percolation*, Probab. Theory Relat. Fields 88 (1991), study the `d`-dimensional **Bernoulli bond percolation model**. Their abstract explicitly lists, throughout the subcritical regime: + +1. the `L^{-(d-1)/2}` Ornstein--Zernike prefactor along an axis; +2. real analyticity of the correlation length as a function of the bond parameter; +3. a local limit theorem for the conditioned long cluster. + +Thus there is genuine classical precedent for **parameter analyticity of the Bernoulli bond mass**. This is stronger than mere direction analyticity. + +However, the printed model is bond percolation. No direct theorem statement for independent SITE on the square or matching graph was identified in this audit. A site-to-bond gadget representation with deterministic edges is not automatically within the hypotheses of a homogeneous bond theorem, and should not be used without checking the proof class. + +## 3. Campanino--Ioffe--Velenik: analytic in direction + +The CIV random-cluster fluctuation theory proves, under its stated subcritical assumptions, sharp OZ asymptotics plus analyticity/strict convexity of the inverse correlation length as a function on the **direction sphere/Wulff boundary**. + +This is precisely the source of the earlier repository correction: that theorem supports directional smoothness at fixed model parameter, not by itself real analyticity of `p -> kappa_G(p)` for the square SITE models. + +The paper mentions parameter/inverse-temperature regularity in surrounding discussion, but the theorem needed here must be stated for the actual parameter/model before the exceptional set is removed. + +## 4. Analyticity of susceptibility/local observables is not enough + +There are modern results proving analyticity of the subcritical susceptibility for transitive Bernoulli percolation and uniform analyticity of many local FK observables under mixing assumptions. These do not automatically imply analyticity of the exponential rate + +\[ +\kappa(p)=-\lim_n\frac1n\log P_p(0\leftrightarrow ne_1). \tag{4.1} +\] + +The limit is a nonlocal large-distance object; exchanging analytic limits uniformly requires additional OZ/transfer control. + +Therefore susceptibility/local-event analyticity is useful proof technology but is **not** a citation-level closure of the SITE mass regularity gap. + +## 5. Current verdict + +For the actual NN and matching independent SITE models used in #739: + +\[ +\boxed{ +\text{Do not remove the at-most-countable exceptional p/d set yet.}} +\] + +The new strict enhancement result + +\[ +\kappa_8(p)<\kappa_4(p) \tag{5.1} +\] + +is logically independent of differentiability. It proves strict graph/centre separation but does not rule out corners of either mass. + +Similarly, strict convexity/analyticity in **direction** does not rule out corners in the scalar occupation parameter. + +## 6. Two credible closure routes + +### Route A: locate a direct SITE p-analyticity theorem + +The ideal input is a theorem explicitly covering independent site percolation on finite-range quasi-transitive graphs throughout the subcritical interval, with real-analytic dependence of the inverse correlation norm on the occupation parameter. A bond-only statement is insufficient unless its proof is shown to cover the site gadget representation. + +### Route B: prove it from the SITE renewal/operator representation + +The small-`p` countable-state programme on this branch would give p-analyticity on a nonempty interval if: + +- the complete/open connection is represented by a quasi-compact positive operator depending analytically on `p`; +- the relevant Perron eigenvalue is simple and isolated; +- the physical root is given by an implicit Perron equation. + +The analytic implicit-function theorem then gives `p`-analyticity of `kappa` there. Extending this throughout the whole subcritical regime is essentially an SITE OZ regularity theorem rather than a one-line corollary. + +## 7. Consequence for #763 + +The correct current hierarchy remains: + +1. intensity-clock Poisson/Gumbel coordinates: no p-differentiability needed; +2. true finite-median `1/w` affine Gumbel: proved at differentiability points; +3. exceptional `d`: at most countable, with constrained convex-log-intensity subsequential limits; +4. full elimination of exceptional `d`: open until an actual SITE p-regularity theorem/proof is supplied. + +No relation involving `a(d)+b(d)=1` should be substituted for this regularity question; the strict centre result now gives `a+b>1` and the differentiability issue remains separate. + +## 8. Sources checked + +- M. Campanino, J. Chayes, L. Chayes, *Gaussian fluctuations of connectivities in the subcritical regime of percolation*, PTRF 88 (1991): abstract/model statement is Bernoulli **bond** percolation and explicitly states real analyticity of the correlation length in the subcritical parameter. +- M. Campanino, D. Ioffe, Y. Velenik, *Fluctuation theory of connectivities for subcritical random cluster models*, Ann. Probab. 36 (2008): OZ/random-walk structure and analyticity/strict convexity of the inverse-correlation shape/direction under its assumptions. +- A. Georgakopoulos, C. Panagiotis, *Analyticity results in Bernoulli Percolation* (2018/2020 versions): subcritical susceptibility/local analytic results, not a direct SITE inverse-mass theorem. + +A negative search is not an impossibility result; this note only prevents an unsupported promotion. diff --git a/docs/manuscripts/geometric-balance/p156-three-line-h4-linear-rescore-20260914.md b/docs/manuscripts/geometric-balance/p156-three-line-h4-linear-rescore-20260914.md new file mode 100644 index 000000000..dfb339cd0 --- /dev/null +++ b/docs/manuscripts/geometric-balance/p156-three-line-h4-linear-rescore-20260914.md @@ -0,0 +1,190 @@ +# The archived P156 physical three-line H4 is a fixed linear transform of C,Q,S + +2026-09-14. Zero-new-sampling algebra for the frozen #156 pilot. This note does not create an additional evidence block: it is a deterministic linear view of the same three sector probabilities and the same 100 batch partitions. + +## 1. Frozen C3 coordinates + +For the three declared primitive-line probabilities `(P_0,P_1,P_2)`, #156 uses + +\[ +C=P_0-\frac{P_1+P_2}{2}, \tag{1.1} +\] + +\[ +Q=\frac{\sqrt3}{2}(P_2-P_1), \tag{1.2} +\] + +\[ +S=P_0+P_1+P_2. \tag{1.3} +\] + +These are an invertible real coordinate system: + +\[ +P_0=\frac{2C+S}{3}, \tag{1.4} +\] + +\[ +P_1+P_2=\frac{2(S-C)}{3}, \tag{1.5} +\] + +\[ +P_2-P_1=\frac{2Q}{\sqrt3}. \tag{1.6} +\] + +## 2. Physical embedded spin-four on the same three lines + +For `tau=1/2+i y`, let + +\[ +Z_4(\ell_1) +=\left(\frac{\tau}{|\tau|}\right)^4 +=c+i s, \tag{2.1} +\] + +while reflection gives + +\[ +Z_4(\ell_2)=c-i s,\qquad Z_4(\ell_0)=1. \tag{2.2} +\] + +The physically embedded three-line harmonic is + +\[ +A_4^{(3)}=P_0+(c+i s)P_1+(c-i s)P_2. \tag{2.3} +\] + +Substitute (1.4)--(1.6): + +\[ +\boxed{ +\Re A_4^{(3)} +=\frac{2(1-c)}{3}C ++\frac{1+2c}{3}S,} \tag{2.4} +\] + +\[ +\boxed{ +\Im A_4^{(3)} +=-\frac{2s}{\sqrt3}Q.} \tag{2.5} +\] + +Therefore physical three-line H4 is a **fixed declared linear transform of the existing frozen coordinates**. No reclassification, extra sector lookup or new sample is needed. + +The same formulas apply to continuum-subtracted residuals because the transform is linear: + +\[ +\Delta A_4^{(3)} +=L_\tau(\Delta C,\Delta Q,\Delta S). \tag{2.6} +\] + +The covariance transforms by the same fixed matrix. + +## 3. Hexagonal fixed point recovers C exactly + +At the exact Eisenstein modulus, + +\[ +c=-1/2, +\qquad |s|=\sqrt3/2. \tag{3.1} +\] + +Equation (2.4) becomes + +\[ +\Re A_4^{(3)}=C, \tag{3.2} +\] + +while (2.5) is, up to the orientation sign convention, + +\[ +\Im A_4^{(3)}=\pm Q. \tag{3.3} +\] + +Thus the complex C3 character `C+iQ` is literally the embedded physical spin-four three-line readout at the hexagonal point. Away from that point the linear map changes with the physical modulus. + +## 4. N30 and N56 transform coefficients + +### N30: `tau=1/2+5i/6` + +\[ +c=-0.5570934256055362\ldots, +\qquad +s=-0.8304497873350802\ldots. \tag{4.1} +\] + +Hence + +\[ +\Re A_4^{(3)} +=1.038062283737024\ldots\,C +-0.038062283737024\ldots\,S, \tag{4.2} +\] + +\[ +\Im A_4^{(3)} +=0.958929\ldots\,Q. \tag{4.3} +\] + +Using only the rounded residual point estimates printed in the original pilot note (`Delta C=0.00754883`, `Delta Q=-0.00132089`, `Delta S about 0.013`) gives the orientation only: + +\[ +\Delta A_4^{(3)}\approx 0.00734-0.00127i, \tag{4.4} +\] + +with the explicit warning that the `S` number in that prose was rounded. The committed batch CSV should be used for the authoritative value and covariance. + +### N56: `tau=1/2+7i/8` + +\[ +c=-0.4844970414201184\ldots, +\qquad +s=-0.8747927906331950\ldots. \tag{4.5} +\] + +Thus + +\[ +\Re A_4^{(3)} +=0.989664694280079\ldots\,C ++0.010335305719921\ldots\,S, \tag{4.6} +\] + +\[ +\Im A_4^{(3)} +=1.01013\ldots\,Q. \tag{4.7} +\] + +The rounded original residuals (`Delta C=0.00175134`, `Delta Q=0.00112146`, `Delta S about 0.0008`) give roughly + +\[ +\Delta A_4^{(3)}\approx0.00174+0.00113i. \tag{4.8} +\] + +Again the batch-rescore script, not these rounded prose inputs, is the authoritative route. + +## 5. Covariance and evidence accounting + +The archived file + +`results/local-20260829/P156-square-bond-primitive-pilot/result.batches.csv` + +contains `l0,l1,l2,rank1_other` for each original batch. Therefore the script + +`scripts/rescore_p156_projective_h4.py` + +can compute the full `2 x 2` covariance of `(Re Delta A4_three, Im Delta A4_three)` under the **same** batch partition. + +This transformed statistic is not independent evidence from `(C,Q,S)`. It is a derived view of exactly the same three counts and must remain in the same dependency block under the repository governance rules. + +## 6. What cannot be recovered + +The archive stores only the total `rank1_other` count, not the primitive line inside that category. Therefore + +\[ +\boxed{A_4^{full}\text{ cannot be reconstructed from the old pilot}.} \tag{6.1} +\] + +No average phase may be imputed to `rank1_other` after reveal. A future full projective harmonic measurement would have to retain the primitive `(u,v)/+/-` label (or directly accumulate the declared complex harmonic) prospectively. + +This information-loss statement is as important as the recoverable three-line transform: it prevents a convenient but invalid post-hoc full-H4 reconstruction. diff --git a/docs/manuscripts/geometric-balance/period-spectrum-birth-centres-20260914.md b/docs/manuscripts/geometric-balance/period-spectrum-birth-centres-20260914.md new file mode 100644 index 000000000..0b1766726 --- /dev/null +++ b/docs/manuscripts/geometric-balance/period-spectrum-birth-centres-20260914.md @@ -0,0 +1,257 @@ +# Birth centres from the minimum correlation-norm period spectrum + +2026-09-14. Abstract arbitrary-shape centre theorem obtained from the fixed-`p` homological-free-energy law. No convergence of the minimizing period direction is required. + +## 1. Minimum period cost as the geometry summary + +For an honest period lattice `Lambda_n` with `N_n` sites, define for NN at `p1\quad(pa). \tag{2.4} +\] + +Assume also `rho_{4,n}(p)->infinity` on compact subsets of `J4`, which follows in the usual growing-shortest-period regimes. + +Let `T_{1,n}` be the first NN ambient-rank birth. Then + +\[ +\boxed{T_{1,n}\xrightarrow P a.} \tag{2.5} +\] + +### Proof + +For fixed `p0)\to0. \tag{2.7} +\] + +Thus + +\[ +P(T_{1,n}\le p)\to0. \tag{2.8} +\] + +For fixed `p>a`, the ratio is `>1+delta`, so + +\[ +P_p(r_4>0)\to1, \tag{2.9} +\] + +hence + +\[ +P(T_{1,n}\le p)\to1. \tag{2.10} +\] + +Two fixed points `a-epsilon,a+epsilon` trap the birth. + +No direction label was used. + +## 3. Upper birth from the complementary matching spectrum + +Let + +\[ +q=1-p. \tag{3.1} +\] + +Choose a compact interval `J8 subset (0,pc(G8))` and suppose + +\[ +\frac{\rho_{8,n}(q)}{\log N_n} +\to g_8(q) \tag{3.2} +\] + +locally uniformly, with a unique crossing + +\[ +\boxed{g_8(c)=1} \tag{3.3} +\] + +at `c in J8`, decreasing from `>1` to `<1` as `q` increases. + +For the second NN rank birth, digital Alexander gives + +\[ +\{T_{2,n}\le p\} +=\{r_4(p)=2\} +=\{r_8(1-p)=0\}. \tag{3.4} +\] + +The same free-energy argument for white matching therefore gives + +\[ +\boxed{T_{2,n}\xrightarrow P b=1-c.} \tag{3.5} +\] + +So the two centre equations are simply + +\[ +\boxed{g_4(a)=1,\qquad g_8(1-b)=1.} \tag{3.6} +\] + +## 4. Direction switching does not obstruct centre convergence + +At a finite size, + +\[ +\rho_{G,n}(p) +=\min_{\lambda\ne0}\tau_{G,p}(\lambda) \tag{4.1} +\] + +is a lower envelope of directional mass curves. Its minimizing projective line can switch as `p` changes. It can also vary with `n`. + +The centre theorem needs none of the following: + +- convergence of the minimizer direction; +- differentiability of the lower envelope; +- a unique cheapest period class; +- an OZ amplitude. + +Only the scalar normalized minimum spectrum (2.2)/(3.2) and a unique crossing are needed. + +This separates the problem into two layers: + +1. **centre layer:** minimum period cost versus `log N`; +2. **window/mark layer:** which projective classes realize or nearly realize that minimum. + +## 5. Unique versus competing slope at the centre + +Let `a` be the lower centre. Define at parameters `p_n->a` a minimizing line `ell_n` and its cheapest nonparallel cost + +\[ +\rho_{\perp,n}(p_n). \tag{5.1} +\] + +If + +\[ +\rho_{\perp,n}(p_n)-\log N_n\to+\infty, \tag{5.2} +\] + +then `correlation-norm-successive-minima-20260914.md` gives + +\[ +P(r=1,L\ne\ell_n)\to0, \tag{5.3} +\] + +so the first-birth slope is asymptotically deterministic. + +If instead several nonparallel projective classes satisfy + +\[ +\tau_{p_n}(\lambda)-\log N_n=O(1) \tag{5.4} +\] + +simultaneously, centre convergence can still hold, but the birth mark requires a multi-direction crossover description. The projective Poisson hard-core closure on this branch is designed for exactly that layer. + +Thus direction degeneracy changes the **mark law**, not necessarily the centre. + +## 6. Exponential shortest-period geometry as a special case + +If a shortest period direction converges to `e` and + +\[ +\frac{\log N_n}{\ell_n}\to d, \tag{6.1} +\] + +with all nonparallel periods much more expensive, then + +\[ +\rho_{4,n}(p) +=\ell_n\tau_{4,p}(e)+o(\ell_n). \tag{6.2} +\] + +Hence + +\[ +g_4(p)=\tau_{4,p}(e)/d, \tag{6.3} +\] + +and `g_4(a)=1` is exactly + +\[ +\tau_{4,a}(e)=d. \tag{6.4} +\] + +The matching side is identical after complement. + +So the directional centre theorem is one explicit realization of the abstract minimum-spectrum criterion. + +## 7. A practical finite-size diagnostic + +For a family of arbitrary period lattices, one can avoid guessing a direction by computing or certifying the scalar quantity + +\[ +R_{G,n}(p)=\frac{\rho_{G,n}(p)}{\log N_n}. \tag{7.1} +\] + +A first-exit/Wulff inner certificate gives lower bounds on every `tau_p(lambda)` and hence on `rho_n`. A finite connection construction gives upper bounds. + +If the resulting intervals show a stable unique crossing of one, they certify the birth centre without resolving the full Wulff shape or choosing the minimizing direction in advance. + +The slope/class should then be analysed separately through the certified near-minimizer set. + +## 8. Boundary and regularity + +At a point where the limiting minimum spectrum touches one without crossing, or where local uniform convergence fails, the theorem does not force a unique birth centre. + +Likewise a centre theorem does not imply a `1/log N` or `1/ell` Gumbel window. An affine window requires local intensity/derivative information about the near-minimizing classes. + +This note is therefore a centre-level theorem, not a replacement for #763/#767. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/periodic-mass-locality-mechanism-20260914.md b/docs/manuscripts/geometric-balance/periodic-mass-locality-mechanism-20260914.md new file mode 100644 index 000000000..21d404f6c --- /dev/null +++ b/docs/manuscripts/geometric-balance/periodic-mass-locality-mechanism-20260914.md @@ -0,0 +1,185 @@ +# Why periodic cylinder mass corrections should be exponential: the zero mode is exact + +2026-09-14. Conditional structural lemma for #760. It identifies the **only** place a cylinder-versus-plane mass correction can enter in a Markov-additive/OZ description. + +The note does not by itself prove that the actual SITE tagged-span mass has the required renewal/decorative representation. It reduces the desired exponential locality theorem to a concrete exponentially rare wrap-defect estimate. + +## 1. Translation-invariant Markov-additive kernel on the plane + +Let + +\[ +A(z,y)=\sum_{x\ge1}\sum_{j\in\mathbb Z}A_{x,j}z^xy^j \tag{1.1} +\] + +be a positive matrix kernel for irreducible long-connection pieces. The integer `j` is the transverse displacement and `x` the longitudinal advance. Allow countably many `j`, with an exponential moment + +\[ +\sum_{x,j}\|A_{x,j}\|R^x e^{c|j|}<\infty \tag{1.2} +\] + +near the physical root. + +Suppose the plane inverse mass is determined by the zero-transverse-character Perron equation + +\[ +\rho(A(R,1))=1, +\qquad \kappa=\log R. \tag{1.3} +\] + +This is the standard form of a translation-invariant Markov-renewal/OZ skeleton. + +## 2. Pure periodization does **not** move the zero mode + +Put the transverse coordinate modulo `w`. The periodized displacement matrices are + +\[ +\bar A_{x,r}^{(w)} +=\sum_{k\in\mathbb Z}A_{x,r+kw}, +\qquad r\in\mathbb Z/w\mathbb Z. \tag{2.1} +\] + +At transverse Fourier character `theta=2pi l/w`, the periodized transform is + +\[ +\bar A^{(w)}(z,e^{i\theta}) +=\sum_{x,r}\bar A_{x,r}^{(w)}z^x e^{i\theta r}. \tag{2.2} +\] + +For the zero character, + +\[ +\boxed{ +\bar A^{(w)}(z,1) +=\sum_{x,r,k}A_{x,r+kw}z^x +=A(z,1).} \tag{2.3} +\] + +Therefore the Perron root equation of the zero mode is **identical for every circumference**: + +\[ +\boxed{\bar R_w=R,\qquad \bar\kappa_w=\kappa.} \tag{2.4} +\] + +This is exact. Folding the transverse displacement modulo `w` is not itself a source of finite-width mass drift. + +The same conclusion holds with finite internal memory: only translation invariance in the transverse displacement and the same local piece weights are used. + +## 3. Consequence: all mass drift is a wrap-defect / decoration effect + +Suppose the actual cylinder kernel `A^{cyl}_w` is not the pure periodization because an irreducible piece or its attached decoration can interact with a periodic copy of itself, changing its validity/weight. Write schematically + +\[ +A^{cyl}_w=\bar A^{(w)}+E_w. \tag{3.1} +\] + +Any local object whose lifted transverse diameter is strictly smaller than `w/2-O(1)` embeds in the cylinder exactly as it does in the plane. Therefore `E_w` is supported only on pieces/decorations that reach transverse scale `Omega(w)`. + +If under the critical Perron tilt the full irreducible object has an exponential transverse-diameter tail, + +\[ +P_*(\operatorname{diam}_\perp\ge r)\le C e^{-cr}, \tag{3.2} +\] + +with the corresponding weighted kernel estimate, then + +\[ +\boxed{\|E_w\|\le C'e^{-c'w}} \tag{3.3} +\] + +in any operator norm compatible with the Perron perturbation argument. + +Thus the strong #760 conjecture is reduced to a geometric locality statement about the **full weighted irreducible object**, not to another comparison of fitted finite-width eigenvalues. + +## 4. Simple Perron perturbation gives exponential mass locality + +Assume `A(R,1)` has a simple isolated Perron eigenvalue one and a nonzero longitudinal derivative + +\[ +\mu=\partial_{\log z}\log\rho(A(z,1))|_{z=R}>0. \tag{4.1} +\] + +If (3.3) holds uniformly on a neighbourhood of `R`, analytic/Kato perturbation gives + +\[ +\rho(A_w^{cyl}(R,1))=1+O(e^{-c'w}). \tag{4.2} +\] + +The implicit-function theorem then shifts the root by the same order: + +\[ +R_w-R=O(e^{-c'w}), \tag{4.3} +\] + +and hence + +\[ +\boxed{ +\gamma_w-\kappa=O(e^{-c'w}).} \tag{4.4} +\] + +In particular + +\[ +w(\gamma_w-\kappa)\to0, \tag{4.5} +\] + +which is exactly the condition needed to substitute a computable cylinder mass for the plane mass in a leading amplitude diagnostic. + +The sign `gamma_w>=kappa`, when separately established by the covering comparison, is compatible with (4.4) and sharpens it to + +\[ +0\le\gamma_w-\kappa\le Ce^{-cw}. \tag{4.6} +\] + +## 5. Why a `1/w^2` Brownian confinement correction is the wrong default here + +A Brownian path confined between **hard transverse boundaries** has a Dirichlet ground-state cost of order `1/w^2`. That is a different geometry. + +The cylinder is periodic. For a translation-invariant effective path, the constant transverse Fourier mode survives exactly and has zero transverse Laplacian eigenvalue. Equation (2.3) is the discrete Markov-additive version of this fact. + +Therefore a polynomial `1/w^2` correction should not be attributed to mere diffusive transverse wandering on a periodic cylinder. It would signal either: + +- an additional constraint that effectively imposes a boundary/killing condition; +- a source/readout that removes the zero transverse mode; +- a nonlocal component condition not captured by pure periodization; +- or failure of the assumed massive/local renewal description. + +This distinction is useful when interpreting finite-width spectral data. + +## 6. Interface to the actual complete winding component + +The actual SITE component problem adds two nontrivial layers absent from the abstract kernel: + +1. complete-component external-boundary weights; +2. the global condition that the selected object is one connected winding component. + +`sewing-with-memory.md` shows that the local weight itself has finite three-column memory, and `matrix-sewing-unit-residue-20260914.md` shows finite memory does not alter the pure cyclic zero-mode residue. What is still needed for #760 is a representation in which the **full closed component/long arm** has exponentially localized decorations so that changing the plane to a periodic cylinder only changes pieces that see a periodic image. + +A suitable proof can follow the architecture of open-connection OZ locality: + +- define regeneration pieces in the lift; +- prove exponential tails for piece diameter and decorations under the appropriate tilted/component law; +- show pieces of diameter ` kappa`. + +## 7. Numerical consequence + +Finite-width data should not be used to **prove** (4.4), but they can falsify particular remainder scales once `kappa` is independently enclosed. + +If the theory is correct, the quantity + +\[ +e^{cw}(\gamma_w-\kappa) +\] + +should remain bounded for some `c>0`, while `w(\gamma_w-\kappa)` must tend to zero. A stable nonzero `w(\gamma_w-\kappa)` would falsify the required locality for amplitude substitution even if plain convergence remains true. + +The appropriate computation is therefore an independent `kappa` interval plus the existing `gamma_w` certificate, exactly as #760 already requests; no extra density fit is needed. + +## 8. Claim boundary + +Equations (2.3)--(2.4) are exact for the stated translation-invariant Markov-additive kernel. Equations (3.3)--(4.4) are a conditional perturbation theorem under exponential localization of the full cylinder/plane kernel difference. Establishing that kernel representation and localization for the actual SITE complete-component tail is the remaining model-specific theorem. diff --git a/docs/manuscripts/geometric-balance/persistent-slope-birth-mark-20260914.md b/docs/manuscripts/geometric-balance/persistent-slope-birth-mark-20260914.md new file mode 100644 index 000000000..3669a87ed --- /dev/null +++ b/docs/manuscripts/geometric-balance/persistent-slope-birth-mark-20260914.md @@ -0,0 +1,213 @@ +# The rank-one slope is a persistent birth mark + +2026-09-14. Exact finite statement for the uniform-label birth process. It connects the projective homology observable to the two sharp birth times without introducing a new sampling protocol. + +## 1. The slope cannot change while rank remains one + +Let `S_p={v:U_v<=p}` under one realization of continuous iid labels. Its ambient NN homology image + +\[ +A(p)=\operatorname{im}[H_1(G_{S_p})\to H_1(T^2)] +\] + +is monotone by inclusion as `p` increases. + +Let `T_1<=T_2` be the first times rank is at least one and two. On the event `T_10. +\] + +Choose any deterministic sequence + +\[ +\lambda_w\to\infty, +\qquad +\log\lambda_w=o(w), +\] + +and take + +\[ +m_w=\left\lfloor\lambda_w/\nu_w\right\rfloor. +\] + +Then `log m_w / w -> kappa_4(p)`. In the fixed-`p` specialization of the Chen--Stein bound, the pair term has the form + +\[ +m_w\,\operatorname{poly}(w)e^{-2\kappa_4(p)w+o(w)} +=\lambda_w\operatorname{poly}(w)e^{-\kappa_4(p)w+o(w)}\to0, +\] + +and the height-localization term is smaller. Thus the entire black anchor point process on the cyclic height coordinate is close in total variation to a homogeneous Poisson process of mean `lambda_w`, even though the mean itself diverges. + +Rescale the cyclic vertical coordinate by `nu_w`; the circle then has length `lambda_w+o(1)` and the comparison Poisson process has unit intensity. + +## 2. Uniform-anchor gap without abstract Palm convergence + +Given a finite point configuration on the circle, apply the following Markov kernel: + +1. if there is no point, return a cemetery symbol; +2. otherwise choose one of the points uniformly; +3. return the clockwise distance from that point to the next point, in the `nu_w`-rescaled coordinate. + +Total variation distance contracts under a Markov kernel. Therefore the output law for the percolation anchor process is at most the existing point-process TV error away from the corresponding output law for the Poisson process. + +For a homogeneous Poisson process on a circle of length `lambda`, conditional on `N=n>=2`, the `n` cyclic spacings divided by `lambda` have the Dirichlet `(1,...,1)` law. A uniformly selected spacing is therefore + +\[ +\lambda\,B_{1,n-1}, +\] + +where `B_{1,n-1}` is Beta `(1,n-1)`. With `N~Poi(lambda)` and `lambda->infinity`, + +\[ +\lambda B_{1,N-1}\Rightarrow Exp(1), +\] + +while `P(N<2)->0`. Consequently, if `G_w` is the gap following a uniformly selected black winding-component anchor, + +\[ +\boxed{\nu_w G_w\Rightarrow Exp(1).} +\] + +This proves the gap law by a finite-circle kernel argument. No renewal independence of the percolation components is assumed. + +The exact stationary mass-transport identity remains stronger at the first-moment level: + +\[ +E^{Palm}G_w=1/\nu_w +\] + +for every fixed `w`. Hence the mean does not need to be recovered by uniform integrability from the weak limit. + +## 3. White component span is the same gap at leading scale + +Let `B_i,B_{i+1}` be consecutive black essential components and `W_i` the unique intervening white matching essential component. The componentwise complement lemma gives a fixed lattice constant `C_0` such that + +\[ +|L(W_i)-G_i|\le L(B_i)+L(B_{i+1})+C_0. +\] + +The existing fixed-subcritical component-Palm volume bound gives `E L(B_i)=O(w)`. Since `nu_w` is exponentially small, + +\[ +\nu_w L(B_i)\to0 +\] + +in `L^1`, hence also in probability. Slutsky therefore yields + +\[ +\boxed{\nu_w L(W_i)\Rightarrow Exp(1).} +\] + +Taking expectations in the deterministic comparison and using the exact mean gap identity gives independently + +\[ +\boxed{\nu_w E L(W_i)\to1.} +\] + +The mean statement does not rely on the Poisson approximation at all. + +## 4. Uniform-location protocol: two independent exponential sides + +There is a second useful Markov kernel. Pick an independent uniform vertical location on the rescaled circle and record the distances `D_-` and `D_+` to the nearest black anchors below and above it. + +For a unit-rate Poisson process on a circle whose length tends to infinity, + +\[ +(D_-,D_+)\Rightarrow(E_1,E_2), +\] + +where `E_1,E_2` are independent `Exp(1)`. Total variation contraction transfers this to the black barrier process. Therefore + +\[ +\boxed{\nu_w(D_-+D_+)\Rightarrow Gamma(2,1).} +\] + +Moreover + +\[ +\boxed{U=\frac{D_-}{D_-+D_+}\Rightarrow Uniform(0,1),} +\] + +and for independent exponentials the ratio is independent of the total. Thus the stationary-location version of the white gap has a joint parameter-free prediction: + +\[ +(\nu_w L_{\rm containing},U) +\Rightarrow(Gamma(2,1),Uniform(0,1)) +\] + +with asymptotic independence, up to the same vanishing black-component thickness error. + +This `U` is a **white-gap location coordinate**. It should not be conflated with the #762 branch/core coordinate until a specific map between the two sampling protocols is proved. + +## 5. Higher spacings + +The same uniform-anchor kernel can return the sum of the next `k` cyclic gaps. For fixed `k`, the Poisson limit gives + +\[ +\boxed{\nu_w(G_i+\cdots+G_{i+k-1})\Rightarrow Gamma(k,1).} +\] + +Thus the entire fixed-order spacing hierarchy follows from the already available point-process approximation; it is not a collection of new phenomenological fits. + +## 6. Scaling-limit interpretation + +At fixed black-subcritical `p`, the two-colour essential geometry has a natural coarse limit under vertical scaling by `nu_w`: + +- black essential components have vanishing scaled thickness and become the points of a unit-rate Poisson process; +- the unique white matching essential component between consecutive black barriers occupies, up to a vanishing boundary error, the complementary Poisson interval. + +This is a **Poisson interval tessellation** of the vertical line, not two independent black/white Poisson clouds. It packages the Exp component-Palm span, Gamma location-biased span, uniform relative position, and higher-gap Gamma laws in one object. + +## 7. Remaining publication-grade checks + +Only two local presentation items remain beyond the existing author-level probability input: + +1. state the regular-neighbourhood/componentwise white-complement lemma with one explicit universal lattice constant `C_0`; +2. quote the precise process-TV version of the Chen--Stein estimate and note explicitly that the bound remains `o(1)` for `lambda_w->infinity` with `log lambda_w=o(w)`. + +No new Monte Carlo, width scan, or independent-colour approximation is needed for this consequence. diff --git a/docs/manuscripts/geometric-balance/post-h4-exponent-7-vs-29over4-20260914.md b/docs/manuscripts/geometric-balance/post-h4-exponent-7-vs-29over4-20260914.md new file mode 100644 index 000000000..899bfe43a --- /dev/null +++ b/docs/manuscripts/geometric-balance/post-h4-exponent-7-vs-29over4-20260914.md @@ -0,0 +1,268 @@ +# Post-H4 exponent competition: linear scalar `7` versus quadratic H4 composite `29/4` + +Date: 2026-09-14 + +Status: scaling-mechanism derivation / model challenge. It adds a missing post-H4 competitor to the historical `~L^-7` discussion. The exponent algebra is standard RG bookkeeping once the leading H4 correction exponent is granted; the nonzero quadratic coefficient is a hypothesis to be tested. + +## 1. Leading H4 correction fixes its RG exponent + +The observed leading square-lattice charge-root correction behaves as + +```text +p_root-pc ~ L^-4 H4(theta), +H4(theta)=cos(4theta). +``` + +Let the corresponding irrelevant coupling have correction exponent `omega_4>0`. The thermal scaling exponent is + +```text +y_t=3/4. +``` + +For a dimensionless balance/root equation, a first-order irrelevant correction shifts the thermal coordinate by + +```text +t_root ~ u_4 L^[-(y_t+omega_4)]. +``` + +Therefore the observed root exponent four implies + +```text +omega_4 = 4-y_t = 13/4. +``` + +Equivalently the associated total scaling dimension is + +```text +x_4=2+omega_4=21/4. +``` + +This is compatible with the current H4/spin-four shell interpretation but the argument below uses only the measured exponent/angular sector. + +## 2. Second order in the same H4 coupling predicts root exponent `29/4` + +At second order, the RG expansion contains terms quadratic in the same irrelevant coupling: + +```text +u_4^2 L^(-2 omega_4). +``` + +Balancing against the thermal scaling variable gives + +```text +t_root^(2) + ~ u_4^2 L^[-(y_t+2 omega_4)]. +``` + +Substituting + +```text +y_t=3/4, +omega_4=13/4 +``` + +gives + +```text +boxed: +y_t+2 omega_4 + = 3/4+26/4 + = 29/4 + = 7.25. +``` + +Thus an entirely natural post-H4 correction is + +```text +p_root^(2)-pc ~ L^-29/4, +``` + +provided the connected quadratic coefficient is nonzero. + +This exponent was absent from the older #47 fixed-model challenge, which tested `11/2,6,7,8,10` plus a free power/log models. + +## 3. Why the historical `~7.06` does not distinguish `7` from `7.25` + +Mertens--Ziff observed an accelerated-root effective power near seven at small available sizes and explicitly warned that finite-size differences strongly distort apparent exponents. + +Two current mechanisms are therefore close enough that small-size effective exponents cannot decide them safely: + +```text +linear scalar candidate: + root exponent = 7, + omega_s = 7-y_t = 25/4, + x_s = 2+omega_s = 33/4; + +quadratic H4 composite: + root exponent = 29/4 = 7.25. +``` + +A fitted `7.06` is compatible with substantial crossover between them or with one of them plus lower-order radial corrections/logs. + +Hence future radial challenges should include **29/4 as a preregistered fixed exponent**, not leave it hidden inside the free-power model. + +## 4. Angular structure of the quadratic H4 term: H0 and H8 + +Represent the real H4 lattice coupling using complex spin components + +```text +u_+ ~ a exp(+i4 theta), +u_- ~ a exp(-i4 theta). +``` + +At second order there are two distinct angular tensor products: + +```text +u_+ nu_- : spin 0 / H0, +nu_+^2 + nu_-^2 : spin +/-8 / H8. +``` + +Therefore the generic quadratic contribution has the structure + +```text +L^-29/4 [ + B_0^(2) + + B_8^(2) cos(8theta) +] +``` + +at root level (up to logs/contact mixing and other sources). + +The scalar and H8 coefficients are **not required to be equal**; they are controlled by different connected/OPE/contact channels. But the presence of H8 as a natural companion is a useful discriminator. + +This is exactly why an H4-null projector does not eliminate the quadratic mechanism: it kills the linear H4 term, while H0/H8 survive. + +## 5. N377 has unusually good leverage on this mechanism + +The #808 pair has + +```text +C4=0, +C8~=0.00361464, +C12~=-0.137808, +C16~=-0.98105. +``` + +Thus it almost removes the H8 companion of the quadratic H4 mechanism while preserving H0. + +Consequently, if the quadratic mechanism dominates and `B_0^(2)` is nonzero, the N377 H4-null output can look very nearly scalar even though **no new linear scalar primary is present**. + +This is a direct warning against interpreting a successful N377 `L^-7-ish` residual as V14 without a source-order test. + +## 6. Strong discriminator: tune the microscopic H4 coupling + +Suppose a local microscopic parameter `lambda` changes the leading H4 coupling through zero: + +```text +u_4(lambda*)=0. +``` + +Then near the improved point, + +```text +linear H4 amplitude ~ (lambda-lambda*), +quadratic H0/H8 composite ~ (lambda-lambda*)^2. +``` + +A genuinely independent linear scalar coupling generally need not vanish at `lambda*`. + +Therefore an improved-action family is the cleanest discriminator: + +```text +post-H4 scalar residual versus measured leading H4 amplitude. +``` + +If the residual scales quadratically with the leading H4 coefficient and collapses near its zero, the composite mechanism is strongly supported. + +If the H4 amplitude crosses/tunes small while the scalar `L^-7` residual persists with nonzero intercept, an independent scalar channel is required. + +## 7. Interface to PR #148 self-matching checkerboard family + +PR #148 constructs the exact local family + +```text +p_even=1/2+t+lambda, +p_odd =1/2+t-lambda, +``` + +with exact pair exchange `(t,lambda)->(-t,-lambda)`. Its planned nontrivial improved-action search explicitly targets the **exchange-even H4 amplitude** `A_T4^+(N,lambda)` on same-norm orientation pairs. + +That family is therefore conceptually ideal for the discriminator above **if** a nonzero H4-amplitude zero is found and transports across size. + +Important boundary: PR #148's exact minimum quotient only proves that the exchange-odd response has no nonzero legal zero; it does not yet establish a nonzero improved point for the even H4 amplitude. The N130/N170 protocol is a proposed search, not a completed root. + +Even without a zero, several `lambda` points can test whether the post-H4 scalar coefficient correlates approximately with `[A_T4^+(lambda)]^2`. + +## 8. A source-sign alternative if no improved point is available + +For a physical H4 source `g`, compute normalized roots/free energies at `+g` and `-g`: + +```text +R_odd(g) =[R(g)-R(-g)]/2, +R_even(g) =[R(g)+R(-g)]/2-R(0). +``` + +Then + +```text +R_odd/g -> linear H4, +R_even/g^2 -> quadratic H4 composite +``` + +at small `g`, after including thermal retuning, normalizer and connected contact terms. + +This is the bounded finite-width test proposed in `post-h4-linear-vs-quadratic-scalar-mechanisms-20260914.md`. + +## 9. Model challenge for future radial data + +After angular H4 removal, compare at minimum: + +```text +M1: root residual ~ L^-7, + independent linear scalar x=33/4 candidate; + +M2: root residual ~ L^-29/4, + quadratic leading-H4 composite; + +M3: L^-7 (a+b log L) or nearby logarithmic collision model; + +M4: mixture of M1 and M2; + +M5: explicit higher-harmonic leakage model using the measured projector gains. +``` + +The primary score should be held-out prediction / cross-geometry consistency, not which intercept is closest to a preferred `pc`. + +Because `7` and `7.25` are very close, no two-size extrapolation can discriminate them honestly without a remainder model. + +## 10. Relation to the older `2 omega=3` speculation + +Issue #47 also mentioned a speculative mechanism using a correction-to-scaling length exponent `omega=3/2`, whose second order gives a relative correction `q=3` and hence root exponent seven. + +That remains a logically distinct composite route. The current quadratic H4 mechanism instead follows directly from the **observed leading root correction** and predicts relative correction + +```text +q=omega_4=13/4, +``` + +hence root exponent `4+13/4=29/4`. + +The two composite mechanisms should not be conflated. + +## 11. Claim boundary + +Exact/scaling algebra once the leading H4 exponent is accepted: + +```text +omega_4=13/4, +second-order root exponent=29/4, +spin4 tensor square contains H0 and H8. +``` + +Hypotheses: + +- the corresponding connected quadratic coefficient is nonzero for square-site Matching One; +- its finite-size amplitude is large enough to explain the observed post-H4 residual; +- a particular improved-action family can tune the microscopic H4 coupling independently of scalar channels. + +The main recommendation is immediate: **add `29/4` to every post-H4 model challenge before promoting exponent seven to a new linear scalar field.** \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/post-h4-linear-vs-quadratic-scalar-mechanisms-20260914.md b/docs/manuscripts/geometric-balance/post-h4-linear-vs-quadratic-scalar-mechanisms-20260914.md new file mode 100644 index 000000000..a746a50ae --- /dev/null +++ b/docs/manuscripts/geometric-balance/post-h4-linear-vs-quadratic-scalar-mechanisms-20260914.md @@ -0,0 +1,168 @@ +# Post-H4 residual: linear scalar insertion versus quadratic H4×H−4 mixing + +Date: 2026-09-14 + +Status: mechanism split / analysis proposal. It responds to the latest #808/#47 update and to the warning that a one-point angular zero does not eliminate second-order scalar mixing. No field identity is asserted. + +## 1. Why a nonzero H4-null residual has at least two qualitatively different origins + +Suppose the leading square correction is carried by a real H4/spin-four coupling `g4` and an exact angular projector removes the **linear** H4 response. + +A residual angular scalar can arise from at least two distinct mechanisms: + +### Linear scalar channel + +A separate scalar scaling field/source `g0` contributes at first order: + +```text +delta O_linear = g0 L^-q0 + ... . +``` + +The current `x=33/4` / `V_<1,4>` candidate belongs to this class if its coupling is nonzero. + +### Quadratic spin-four mixing + +Two spin-four insertions can fuse to total spin zero: + +```text +(+4)+(-4)=0. +``` + +A second-order connected response can therefore produce an angular scalar even when the linear H4 one-point term is projected to zero: + +```text +delta O_quad + ~ g4 g_-4 * integral-connected-two-point + contact/thermal counterterms. +``` + +An H4-null **linear** projector does not remove this scalar second-order contribution. + +Thus + +```text +post-H4 scalar != automatically new scalar primary. +``` + +## 2. Radial exponents can distinguish the two stories only after contact terms are typed + +If the leading H4 root correction scales as `L^-4`, a naive product of two root amplitudes would suggest `L^-8`. That is generally **not** the correct second-order field-theory exponent because integrated insertions bring volume powers, OPE singularities, thermal retuning and contact counterterms. + +The right object is the connected second derivative of the physical topological free-energy/root functional with respect to a genuine H4 microscopic coupling. + +Schematically, for a source `g` with first-order H4 character, + +```text +Theta_gg + = connected Green--Kubo / integrated two-point + + explicit contact term + + source-normalization term. +``` + +The scalar component of `Theta_gg` is the quadratic adversary to a new linear scalar field. + +Therefore a measured `L^-7` residual cannot be accepted/rejected as quadratic H4 mixing by dimensional multiplication alone. + +## 3. A clean lattice experiment: sign reversal of the H4 source + +Let a microscopic anisotropy coupling `g` reverse the sign of the leading H4 insertion while preserving all scalar bare couplings. Then expand the root or charge free energy: + +```text +R(g)=R0 + a1 g + a2 g^2 + a3 g^3 + ... . +``` + +Form + +```text +R_odd(g) =[R(g)-R(-g)]/2 = a1 g+a3 g^3+..., +R_even(g) =[R(g)+R(-g)]/2-R(0)=a2 g^2+a4 g^4+.... +``` + +The linear H4 channel lives in `R_odd`; a quadratic H4×H−4 scalar contribution lives in `R_even`. + +A pre-existing linear scalar field independent of `g` instead contributes to `R0` and does not grow as `g^2` under this source. + +This supplies a mechanism discriminator that does not require a gigantic same-circle angular tomography. + +## 4. Use the normalized-source formalism + +The source `g` must be a normalized physical perturbation. For an unnormalized transfer source, include the row pressure/normalizer before differentiating. + +At a root, the first derivative is the residualized covariance response. The second derivative must include: + +```text +integrated score covariance, +explicit second derivative/contact of log weight, +thermal/root-motion terms. +``` + +Do not square one-point amplitudes or multiply two first derivatives as a surrogate. + +This is exactly the failure mode that the #802 normalization audit warns against. + +## 5. Relation to angular projection + +There are now two independent decompositions: + +```text +geometry : H0, H4, H8, ... +source sign : even/odd under g -> -g. +``` + +A genuine new linear scalar correction is + +```text +angular H0, +source-linear in its own scalar coupling. +``` + +Quadratic H4 mixing is + +```text +angular H0, +even/quadratic in the H4 source. +``` + +Therefore angular scalarity alone cannot distinguish them. + +## 6. A practical bounded test before N377/N1105-scale production + +Use an existing small-width safe operator for which a physical orientation/anisotropy source can be inserted without changing the state space. At two or three small amplitudes `±g` compute: + +```text +Theta(pc,g), +Theta_p(pc,g), +p_root(g), +``` + +with the same normalization conventions and tight solver. + +Check: + +```text +odd part / g -> existing H4 response; +even part/g^2 -> finite nonzero limit or zero within error. +``` + +If the quadratic scalar term is already sizeable and has a radial trend compatible with the post-H4 residual, the simple “new V_<1,4> linear field” narrative must remain mixed. + +If the even second-order coefficient is tiny/forbidden while #808 finds a robust H0 `ell^-7`, the linear scalar interpretation becomes substantially stronger. + +## 7. Continuum/OPE question after the lattice test + +Only after observing a nonzero quadratic scalar response should one ask which OPE channels of the H4 correction feed it. At `c=0` one must keep logarithmic collisions/contact terms; equality of total dimensions does not by itself determine the fusion coefficient. + +Conversely a zero finite response for one microscopic H4 source does not prove all quadratic H4 mixing vanishes: it can be source/map specific. + +## 8. Claim boundary + +Exact algebra: + +- source-sign even/odd decomposition; +- a linear H4 projector does not eliminate second-order H4×H−4 scalar response. + +Programme: + +- measure the connected normalized second response of an actual H4 source; +- use it as an adversary to the post-H4 linear-scalar/V14 interpretation. + +This should be resolved before a pure `ell^-7` angular residual is promoted to a unique scalar operator. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/post-h4-mixed-spin4-correction-20260914.md b/docs/manuscripts/geometric-balance/post-h4-mixed-spin4-correction-20260914.md new file mode 100644 index 000000000..6e43bb2e6 --- /dev/null +++ b/docs/manuscripts/geometric-balance/post-h4-mixed-spin4-correction-20260914.md @@ -0,0 +1,275 @@ +# Post-H4 residual: odd thermal spin4 x even identity spin4 as the first nonlinear mechanism + +Date: 2026-09-14 + +Status: correction/addendum to `post-h4-residual-hierarchy-20260914.md`. The earlier note placed a linear identity-family spin-eight candidate too high. Once matching parity is enforced, the more natural first mechanism is the **second-order product of the already observed dual-odd thermal spin-four field with the already observed dual-even/common spin-four lattice anisotropy**. + +This note changes candidate ordering; it does not erase the earlier exploratory calculation. + +## 1. Two spin-four directions are already present + +The fixed-width safe-transfer hierarchy contains two distinct finite-size structures. + +### Common / dual-even spin four + +The average magnetic excitation has a large correction consistent with + +```text +x_even = 4, +y_even = 2-x_even = -2. +``` + +On a square lattice the natural lattice-anisotropy realization is a spin-four identity-family correction. Its important feature for the matching root is not the exact field name but that it is large in the common/average channel and strongly cancels from the primal/matching difference. + +### Difference / dual-odd spin four + +The charge-sector difference has + +```text +x_odd = 21/4, +y_odd = 2-x_odd = -13/4, +``` + +with strong `H4(theta)=cos(4theta)` evidence. The ordinary thermal-Q4 Ward identity explains why its leading critical projection is thermal-tangent. + +Thus the effective finite-size theory already contains + +```text +v4^+ H4 ell^-2 (matching-even/common), +u4^- H4 ell^-13/4 (matching-odd/difference). +``` + +## 2. Matching parity changes the post-H4 candidate ordering + +A linear identity-family spin-eight field is naturally matching-even unless a separate microscopic odd coupling is demonstrated. Therefore it should not be the default first source of an odd charge-root residual. + +By contrast, the product + +```text +u4^- * v4^+ +``` + +is automatically matching-odd: + +```text +odd x even = odd. +``` + +It is therefore allowed in the charge difference without introducing any new primary/module. + +## 3. Angular structure is locked: H4 squared + +The mixed second-order term carries + +```text +H4(theta)^2 + = [1+H8(theta)]/2. +``` + +So it generates a scalar and an H8 component with the **same field-level coefficient** in the standard `H0=1`, `H8=cos(8theta)` basis. + +This is more predictive than a generic `S0+C8 S8` residual: the simplest mixed mechanism does not have two free angular amplitudes. + +For a two-angle H4 projector with `h_i=H4(theta_i)`, the residual of `H4^2` is exactly + +```text +P_perp4[H4^2] + = (h1 h2^2-h2 h1^2)/(h1-h2) + = -h1 h2. +``` + +Using + +```text +C8=-(1+2h1h2), +``` + +this is equivalently + +```text +P_perp4[H4^2] = (1+C8)/2. +``` + +Thus every existing same-circle pair has a parameter-free geometry factor for this mechanism. + +## 4. Root exponent is exactly six at the scaling level + +A second-order finite-size term built from the two couplings has RG exponent + +```text +y_mix = y_odd+y_even + = -13/4-2 + = -21/4. +``` + +An excitation-energy correction carries one extra inverse length: + +```text +Delta I_mix + ~ u4^- v4^+ H4^2 ell^(y_mix-1) + = H4^2 ell^-25/4. +``` + +The thermal derivative which moves the charge root scales as + +```text +partial_p Delta I ~ ell^-1/4. +``` + +Therefore the pseudo-critical root receives + +```text +boxed: +delta p_mix + ~ H4(theta)^2 ell^-6. +``` + +This provides a structural origin for the familiar next even power in the raw axial root sequence without introducing a new single field of root exponent six. + +## 5. Why the H4 projector exposes this term + +The same-circle H4 projector removes every correction proportional to `H4(theta)`, including a possible linear spin-four tower + +```text +H4[a4 ell^-4+a6 ell^-6+...]. +``` + +But it does **not** remove `H4^2`. + +Therefore after leading angular improvement, a natural first surviving nonlinear contribution is precisely + +```text +(-h1 h2) C_mix ell^-6. +``` + +This makes the projected residual scientifically useful rather than merely an estimator error: it directly probes nonlinear mixing between the leading odd and common even lattice directions. + +## 6. Existing residuals do not yet reduce to one pure H4-squared term + +For the current two-angle projectors, `h1` and `h2` generally have opposite signs, so + +```text +-h1 h2 >0. +``` + +A **single** mixed coefficient with negligible competitors would therefore give the same residual sign across these pairs at comparable large `ell`. + +The observed projected residuals include both signs: + +```text +N65: positive, +N85: near zero positive, +ell=10: negative. +``` + +Hence the pure mixed term is not by itself a complete fit at current sizes. This is valuable: it means the N1105 same-circle decomposition should not be reduced to one locked coefficient in advance. + +Possible reasons include + +```text +additional scalar odd channel, +independent H8 odd channel, +H12/higher harmonic, +higher-order finite-size contamination, +logarithmic/normal spin4 remainder. +``` + +The mixed mechanism remains structurally preferred as the **first nonlinear term**, not as an already sufficient numerical model. + +## 7. N1105 gains a sharp mechanism test + +The four same-circle N1105 roots can be decomposed as + +```text +P0 + P4 H4 + P8 H8 (+ P12 H12). +``` + +The simplest odd-Q4 x even-spin4 mechanism predicts, for the post-H4 second-order piece, + +```text +P0_mix = P8_mix +``` + +because both come from `(1+H8)/2`. + +Thus define the same-ell mechanism ratio + +```text +R_mix = P8/P0 +``` + +only after the dominant P4 piece is separated. + +Interpretation: + +```text +R_mix ~ 1, P12 small: + strong support for nonlinear odd/even spin4 mixing; + +P8 nonzero, P0 small or R_mix far from 1: + independent H8 channel required; + +P0 dominates: + separate angle-independent odd scalar/log channel; + +P12 substantial: + H0/H4/H8 truncation insufficient. +``` + +A single N cannot determine the root exponent six. N1105 is an angular/parity mechanism test only. + +## 8. Relation to the discarded simple coordinate-squared explanation + +An earlier exploratory estimate considered the ordinary analytic nonlinearity produced by squaring the leading odd thermal-coordinate shift itself. + +Symmetry makes the conceptual distinction clearer: + +```text +(u4^-)^2: even under matching, +u4^- v4^+: odd under matching. +``` + +The charge difference is odd, so the first expression should not be the default source of the residual even before noticing that its calibrated numerical size is tiny. + +The physically relevant second-order candidate is the **odd x even** mixed term. + +## 9. Revised candidate ordering after H4 projection + +Current ordering: + +```text +1. odd thermal-Q4 x common even spin4 nonlinear mixing + -> H0+H8 locked structure, root ell^-6; + +2. additional independent dual-odd scalar/H8/log channels; + +3. H12/higher D4 harmonics; + +4. linear identity-family spin8 x=8 + only after a nonzero matching-odd coupling is independently justified. +``` + +A thermal-family spin-eight field remains a possible independent H8 channel, but its root exponent would be eight and its amplitude need not satisfy the locked H0=H8 relation. + +## 10. Research consequence + +The leading and next correction mechanisms may now be organized without inventing a new field for every power: + +```text +linear odd thermal Q4 spin4 + -> root ell^-4 H4, + +nonlinear odd-Q4 x even-identity-spin4 + -> root ell^-6 (H0+H8), + +then genuinely new irreps/modules only if the locked second-order prediction fails. +``` + +This is a much more economical hypothesis than assigning the raw `4,6,...` ladder to unrelated operators. It is also directly falsifiable by one same-circle multi-angle calculation. + +## 11. Claim boundary + +- The RG exponent arithmetic, matching-parity product and identity `H4^2=(1+H8)/2` are exact once the two input scaling fields are granted. +- Identifying the common `x≈4` correction specifically with a dual-even spin-four identity-family lattice field remains a scaling/CFT interpretation, albeit a standard one. +- The equality `P0_mix=P8_mix` is the leading second-order prediction; current two-angle residuals do not yet satisfy a one-term model cleanly. +- This note supersedes the earlier ordering that placed linear identity-spin8 first; it does not delete that field as a possible subleading contribution. diff --git a/docs/manuscripts/geometric-balance/post-h4-residual-hierarchy-20260914.md b/docs/manuscripts/geometric-balance/post-h4-residual-hierarchy-20260914.md new file mode 100644 index 000000000..e7c10b529 --- /dev/null +++ b/docs/manuscripts/geometric-balance/post-h4-residual-hierarchy-20260914.md @@ -0,0 +1,269 @@ +# What remains after the leading H4 matching-root correction is projected out? + +Date: 2026-09-14 + +Status: synthesis / conjecture ledger for the **next** finite-size mechanism. The leading H4/spin-four root displacement is already much more strongly constrained than the residual discussed here. This note deliberately starts only after the same-circle H4 projector has been applied. + +## 1. Empirical object + +At one fixed physical circumference `ell`, let two orientations have + +```text +h_i=H4(theta_i)=cos(4 theta_i), +p_i=p_ch(ell,theta_i). +``` + +Define the H4-annihilating scalar projection + +```text +p_perp4 + = (h1 p2-h2 p1)/(h1-h2). +``` + +The finite residual relative to the infinite threshold is + +```text +r(ell;theta1,theta2)=p_perp4-pc. +``` + +Existing deterministic controls give, after comparison to the diagnostic high-precision threshold only **after** forming the projector, + +```text +ell=5, axis/(3,4): r≈-9.81e-6 +ell=sqrt(65), (1,8)/(4,7): r≈+5.66e-8 +ell=sqrt(85), (2,9)/(6,7): r≈+4.33e-10 +ell=10, axis/(3,4): r≈-7.57e-8. +``` + +The N85 value is extraordinarily small but should not be interpreted by itself as an asymptotic error law. + +## 2. The H4 projector removes the whole H4 tower + +Write the fixed-ell D4 harmonic decomposition + +```text +p_ch(ell,theta) + = pc + S0(ell) + + H4(theta) S4(ell) + + H8(theta) S8(ell) + + H12(theta) S12(ell) + + ... . +``` + +The two-angle projection removes **all of `S4(ell)`**, independent of how many powers of `ell^-1` occur inside that coefficient. Therefore an axial correction ladder + +```text +ell^-4, ell^-6, ... +``` + +can disappear wholesale if those terms are successive corrections carrying the same H4 irrep. + +The residual is + +```text +r = S0 + C8 S8 + C12 S12 + ..., +``` + +where for the two-angle H4 projector + +```text +C8=-(1+2 h1 h2). +``` + +This shifts the research question from “what is the next power of the axial root?” to “what angular irrep survives after the dominant one is removed?” + +## 3. The simplest coordinate-artifact explanation is far too small + +The TANGENT_SPIN4 picture says the leading H4 correction acts almost as a translation of the thermal scaling coordinate. A nonlinear thermal field can then generate an `H4^2=(1+H8)/2` term even if no independent H8 operator exists. + +The existing full-curve analysis calibrates the quadratic logit thermal metric as + +```text +t = delta h + c2 (delta h)^2+..., +c2≈-0.0311. +``` + +The leading root shift has + +```text +delta p≈-A_p H4 ell^-4, +A_p≈0.296--0.300. +``` + +At the threshold + +```text +dp/dh=pq≈0.2414, +``` + +so the leading logit shift amplitude is `A_h≈A_p/(pq)≈1.2`. Inverting the quadratic thermal field and then the logistic map gives only + +```text +delta p_coord^(2) + = O(1e-2) H4^2 ell^-8. +``` + +Thus the induced scalar/H8 pieces are roughly + +```text +O(1e-10) at ell~8--10, +``` + +whereas N65 and ell=10 H4-projected residuals are `O(1e-8--1e-7)`. + +Conclusion: + +> The observed post-H4 residual cannot be explained by the already calibrated ordinary quadratic thermal-coordinate nonlinearity of the leading tangent spin-four shift. + +More elaborate normal spin-four corrections or logarithmic mixing remain possible; only the cheapest analytic-coordinate explanation is excluded. + +## 4. The residual already shows an H8 geometry pattern + +For the three projectors with `ell` in the relatively narrow range `8.06--10`, the H8 weights are + +```text +N65: C8=-0.1484319457, r=+5.66e-8 +N85: C8=+0.2224938303, r=+4.33e-10 +ell10: C8=+0.6864, r=-7.57e-8. +``` + +If one ignores the modest ell variation only as an exploratory local diagnostic and writes + +```text +r≈S0+C8 S8, +``` + +a least-squares line gives approximately + +```text +S0≈+3.4e-8, +S8≈-1.59e-7, +``` + +with pointwise discrepancies only around `1e-9`. + +This should **not** be promoted to a fitted scaling law: the three circumferences differ. Its value is qualitative. The sign progression and the near-zero N85 residual are naturally explained by a negative H8 contribution crossing a smaller scalar/other residual. + +In this picture N85 is a geometry cancellation point, not miraculous proof of a `4e-10` asymptotic estimator error. + +## 5. Candidate NEXT_H8_IDENTITY_x8 + +A structurally natural next angular field is an identity-family spin-eight lattice anisotropy with + +```text +x_8=8, +spin=8, +angular harmonic H8=cos(8theta). +``` + +If the two microscopic safe regularizations have a nonzero difference coupling to this field, its magnetic-gap correction scales as + +```text +Delta I_8 ~ H8 ell^(1-x8)=H8 ell^-7. +``` + +Dividing by the thermal root susceptibility `ell^-1/4` gives + +```text +boxed: +delta p_8 ~ H8 ell^-27/4, +27/4=6.75. +``` + +This is distinct from a thermal-family spin-eight descendant, which would have `x=x_t+8=37/4` and root exponent `8`. + +Why this candidate is interesting: + +1. the raw axial `ell^-4,ell^-6,...` corrections may remain in the H4 tower and are removed by the H4 projector; +2. the same-angle axis/(3,4) projected residual drops by about a factor `129` between ell=5 and ell=10, corresponding to a naive two-point effective power near `7`; this is **not** an exponent measurement, but it is numerically compatible with `27/4`; +3. restoring the exploratory H8 piece around ell~9 with an `ell^-27/4` law gives an O(1) amplitude (`roughly 0.4--0.6`), whereas an `ell^-8` law requires a substantially larger amplitude; +4. N65 and ell10 residual signs agree with a negative H8 coefficient because their projected `C8` values have opposite signs. + +This is presently a conjecture, not a field identification. In particular, the fact that lower identity-family corrections are common/sector-even does not force every higher lattice coupling to have zero NN/matching difference. + +## 6. Competing residual mechanisms + +### 6.1 Angular scalar / spin-zero channel + +A true angle-independent dual-odd correction survives every angular projector. It could come from a scalar irrelevant/logarithmic sector or another microscopic coupling. Same-circle tomography measures it as the H0 coefficient. + +Its existence would **not** restore scalar x=21/4 as the leading explanation of the raw root shift; it would be a subleading channel exposed only after H4 removal. + +### 6.2 Thermal-family spin eight + +This gives H8 with root exponent 8. N1105 can identify the H8 irrep at one ell but cannot distinguish exponent 6.75 from 8 by itself. + +### 6.3 H12 / higher D4 harmonics + +A two-angle H4 projector is generally sensitive to all higher harmonics. The N1105 four-angle closure contrast has unit-normalized H12 gain about `0.705`, providing a clean detector if H0/H4/H8 are insufficient. + +### 6.4 Logarithmic energy--hull mixing + +At Q=1, logarithmic collisions can decorate an angular power with `log ell` and change apparent effective amplitudes. This requires generic-Q/module evidence; it should not be inferred from three residual values. + +### 6.5 Normal component of the leading spin-four field + +Root/slope-normalized same-ell curves differ only at `O(1e-5--5e-5)` on the tested interval, so any leading normal H4 shape response is small at current sizes. A higher-order H4 normal piece is nonetheless possible; importantly, the root H4 projector removes it from the scalar root estimator if it carries the same H4 angular character. + +## 7. Two targeted discriminants + +### N1105: direct same-circle H0/H4/H8/H12 tomography + +Four primitive orientations at `N=1105` allow either + +```text +H0/H4/H8 fit + one exact closure residual, +``` + +or exact four-column `H0/H4/H8/H12` decomposition. This is the cleanest way to decide whether the present residual is actually H8 or scalar without cross-size assumptions. + +### N377: H4/H8 double-notch + +The same circle + +```text +377=4^2+19^2=11^2+16^2 +``` + +has two primitive orientations with + +```text +C8=+0.0036146395. +``` + +Thus the ordinary two-angle H4 projector retains only about `0.36%` of H8 while remaining sensitive to later harmonics (`C12≈-0.1378`, `C16≈-0.9811`). If computationally feasible, it is a direct scalar/H12-vs-H8 stress test. + +A state-cap pilot shows its width-one safe automata exceed 200k states, so it is not a default cheap run. + +## 8. Decision logic + +### N1105 finds dominant P8, small P0/P12 + +Promote the next-channel picture to + +```text +H4 leading regularization mismatch + -> H8 subleading lattice anisotropy. +``` + +Then use existing different-ell projected data only as a weak exponent cross-check between identity-spin8 (`27/4`) and thermal-spin8 (`8`). + +### P0 scalar dominates + +A genuine subleading angle-independent channel survives. Reopen its field/module classification, but keep it distinct from the already resolved leading H4 mechanism. + +### P12 or closure residual large + +The two-harmonic residual model is inadequate. Keep a D4 tower; do not force an exponent fit. + +### N377 remains large while N1105 says P8 dominates + +Then the assumption that N377 nearly nulls the relevant H8 component is wrong, indicating either higher harmonics or a geometry-dependent non-Fourier nuisance. + +## 9. Claim boundary + +- H4-projected residuals, H8 geometry coefficients and thermal-metric coefficient are existing deterministic controls / algebra. +- The exclusion of the simple quadratic-coordinate explanation is an order-of-magnitude consequence of those measured coefficients. +- `NEXT_H8_IDENTITY_x8` is a new falsifiable conjecture. +- The exploratory local `S0+C8 S8` decomposition is not a same-ell fit and must not be treated as evidence of an asymptotic amplitude. +- N1105/N377 are mechanism-discriminating designs; only N1105 currently has an opened bounded compute issue (#807), with an explicit Phase-0 stop rule. diff --git a/docs/manuscripts/geometric-balance/projective-homology-gas-20260914.md b/docs/manuscripts/geometric-balance/projective-homology-gas-20260914.md new file mode 100644 index 000000000..8fe719dde --- /dev/null +++ b/docs/manuscripts/geometric-balance/projective-homology-gas-20260914.md @@ -0,0 +1,168 @@ +# Projective-homology gas: the exact finite state behind directional winding counts + +2026-09-14. This note refines the `(rank,K)` reduction by retaining the one piece of directional information that survives exactly at rank one: a projective homology line. It is deterministic topology first, with a conjectural dilute-gas interpretation second. + +## 1. One-colour exact classification + +Let an occupied graph on an honest torus have connected components `C_j`. For each component define + +\[ +A_j=\operatorname{im}[H_1(C_j;\mathbb Q)\to H_1(T^2;\mathbb Q)]. +\] + +The global ambient image is + +\[ +A=\operatorname{span}_j A_j. +\] + +Because `H_1(T^2;Q)` is two-dimensional, exactly the following possibilities occur. + +### Rank zero + +`A=0`. No component is essential. + +### Rank one + +`A` is one line `ell` in `P(H_1(T^2;Q))`. Every essential component has nonzero image contained in `A`, hence + +\[ +\boxed{A_j=A=\ell\quad\hbox{for every essential component}.} +\] + +Thus a rank-one configuration is described topologically by one projective rational slope `ell` and an integer count `K>=1` of parallel essential components. + +### Rank two + +At least one component must have two-dimensional image. Indeed, if every essential component had rank one, then two components with distinct lines would be disjoint essential subsets. Representatives of nonparallel homology classes on a torus have nonzero algebraic intersection and cannot be supported in disjoint components. Hence all rank-one component images would have to be the same line, contradicting global rank two. + +A rank-two component excludes every other essential component: any essential cycle disjoint from it has zero intersection with every class carried by the rank-two component, hence its ambient class is zero by nondegeneracy of the torus intersection form. Therefore + +\[ +\boxed{r=2\quad\Longrightarrow\quad\text{there is exactly one essential component, of rank two}.} +\] + +This recovers the single/cross component part of the wrapping classification without referring to a probabilistic model. + +## 2. Complementary 4/8 classification with slope + +For the black NN graph and the complementary white matching graph, digital Alexander gives + +\[ +A_{white}=A_{black}^{\perp}. +\] + +Hence: + +- black rank `0`: white rank `2`, with one white rank-two component; +- black rank `2`: white rank `0`, with one black rank-two component; +- black rank `1`: `A_black=ell` and, because every one-dimensional subspace of the two-dimensional symplectic torus homology is self-orthogonal, + \[ + A_{white}=ell. + \] + The alternating complementary-region argument gives equal black/white essential-component counts `K`. + +So the exact same-parameter two-colour state is + +\[ +\boxed{ +\begin{cases} +D=-1:&(r_4,r_8)=(0,2),\ K=0,\\ +D=0:&(r_4,r_8)=(1,1),\ (\ell,K),\ K\ge1,\\ +D=+1:&(r_4,r_8)=(2,0),\ K=0, +\end{cases}} +\] + +where `D=r_4-1=W_4-W_8`. + +The previous `(rank,K)` reduction is the marginal obtained after forgetting `ell`. + +## 3. Exact generating object + +Let `chi` be any test function/character on projective rational homology lines. Define the rank-one marked generating function + +\[ +H_p(z;\chi) +=E[z^K\chi(\ell)\mid r=1]. +\] + +Then every same-parameter topological statistic that depends only on essential-component count and slope is encoded by + +\[ +\boxed{ +Z_p(h,z;\chi) +=P_2(p)e^h+P_0(p)e^{-h}+P_1(p)H_p(z;\chi).} +\] + +At `z=1, chi=1`, the derivative in `h` at zero is exactly the matching observable + +\[ +\partial_hZ_p(0,1;1)=P_2-P_0=M(p). +\] + +Thus the finite matching root is a **zero mean topological charge** condition. The potentially extensive neutral rank-one gas `K` is orthogonal to that charge at the level of this exact decomposition; only the unpaired rank-zero/rank-two endpoint sectors set the sign of `M`. + +## 4. Consequence for same-parameter count fluctuations + +The exact identities may be written + +\[ +W_4=K+1_{D=+1}, +\qquad +W_8=K+1_{D=-1}, +\] + +with `|D|<=1`. Hence + +\[ +|W_4-W_8|\le1 +\] + +configuration by configuration. At any fixed width and long height, any extensive CLT or LDP for one colour transfers to the other with the **same** fluctuation rate. In particular, if `Var(K)` is of order height, then + +\[ +Corr(W_4,W_8)\to1. +\] + +The two colours are therefore maximally correlated at the extensive same-parameter count scale, in deliberate contrast with the asymptotic independence of counts in the two **separated parameter windows**. + +This is also why the joint pressure found earlier depends only on the sum of the two count sources at leading order: the topological charge is bounded and disappears after division by height. + +## 5. Directional dilute-gas conjecture + +For a primitive lattice period/slope `ell` represented by `u`, let the type fugacity be schematically + +\[ +\lambda_u(p) +\asymp \frac{N}{|u|}\,A(p,\hat u)|u|^{-\beta(p,\hat u)}e^{-\tau_p(u)}. +\] + +The `homological-free-energy` variable in `research-frontier-20260914.md` is the logarithmic leading part of this expression. + +When one type has much larger fugacity than all others, the rank-one slope should concentrate on that type; the deterministic direction-separation lemma proves this in the fixed-positive exponential elongation regime at the level of correlation cost. + +When finitely many primitive slopes have comparable fugacity, the correct rank-one object is **not** a collection of independent Poisson species. Rank one permits only one projective line. A better picture is a projective hard-core gas: + +1. the first winding event is a race among slope types; +2. after a slope `ell` wins, additional rank-one components may accumulate only parallel to `ell`; +3. the appearance of a nonparallel homology class forces the system into rank two and ends the rank-one sector. + +This supplies a topological state space for the multi-direction version of #765 without inventing a bivariate/multivariate independent-Poisson approximation that the torus cannot support. + +## 6. First-birth slope selection: a falsifiable leading rule + +In a regime where all type fugacities are small and simultaneous nonparallel events are of higher order, the first rank birth should select slope `u` with probability approximately + +\[ +\boxed{ +P(\ell=u\mid T_1\hbox{ occurs in the window}) +\approx \frac{\lambda_u}{\sum_v\lambda_v}.} +\] + +The first-birth void probability should depend at leading order on the sum of candidate fugacities, while the rank-two birth is sensitive to their incompatibility/intersection structure. This is a conjectural asymptotic rule, not an exact finite formula. + +A useful test geometry is a family with two intentionally near-degenerate primitive period classes. The prediction is not merely a 50/50 split: the ratio should be set by the directional masses, transverse opportunity factors, and eventually the sewing amplitudes. The rank-one plateau should be shorter than in a one-minimizer geometry because a nonparallel second type is an efficient route to rank two. + +## 7. Relation to continuum/modular work + +No continuum field identification is needed for this state. On fixed-aspect tori, a scaling limit of the marked slope distribution would be a natural lattice object to compare with known homology/wrapping formulas. On exponentially elongated tori, the deterministic cost separation collapses it to the shortest direction. These are two limits of the same exact finite projective-homology state, rather than two unrelated observables. diff --git a/docs/manuscripts/geometric-balance/projective-poisson-hardcore-crossover-20260914.md b/docs/manuscripts/geometric-balance/projective-poisson-hardcore-crossover-20260914.md new file mode 100644 index 000000000..41752f4ec --- /dev/null +++ b/docs/manuscripts/geometric-balance/projective-poisson-hardcore-crossover-20260914.md @@ -0,0 +1,210 @@ +# Projective Poisson hard-core crossover for competing winding directions + +2026-09-14. Exact finite topology first, then one deliberately explicit conjectural closure for a multi-direction rare-event window. The purpose is to replace an inadmissible multivariate independent-Poisson picture by the smallest state space that actually respects torus homology. + +## 1. Exact state space with slope marks + +For one colour on an honest torus, let `r` be the ambient homology rank. The essential-component classification implies exactly three kinds of states: + +\[ +\boxed{ +r=0;\qquad (r=1,L=\ell,K\ge1);\qquad r=2.} \tag{1.1} +\] + +Here `L` is one primitive projective homology line and `K` is the number of essential components. + +If `r=1`, every essential component has the same line `L`. Thus if `K_ell` denotes the number of essential components of slope `ell`, then pathwise + +\[ +\boxed{K_\ell K_{\ell'}=0\qquad(\ell\ne\ell')} \tag{1.2} +\] + +in the rank-one sector. + +If `r=2`, there is one rank-two essential component. Indeed a rank-two component contains two nonparallel essential cycles; any other essential component has nonzero intersection with at least one of them and therefore cannot be disjoint. + +So different slope species are not independent gases. They have a global topological hard-core exclusion, with the collision sector represented by rank two. + +## 2. Exact same-parameter 4/8 marked generating function + +For the occupied NN graph and the complementary matching graph at the same Bernoulli labels, write `W4,W8` for the essential-component counts. Digital Alexander gives the exact support + +\[ +(W_4,W_8)=(0,1),(1,0),(K,K),\quad K\ge1, \tag{2.1} +\] + +and in the neutral `(K,K)` sector both colours have the same projective line `L`. + +For each slope `ell` define the unnormalised rank-one count PGF + +\[ +Q_\ell(z) +=E\left[z^K1_{\{r_4=1,L=\ell\}}\right]. \tag{2.2} +\] + +Then for arbitrary slope weights `y_ell`, + +\[ +\boxed{ +E\left[s^{W_4}t^{W_8} + \prod_\ell y_\ell^{1_{\{r_4=1,L=\ell\}}}\right] +=P_2s+P_0t+\sum_\ell y_\ell Q_\ell(st).} \tag{2.3} +\] + +This is the slope-marked strengthening of the earlier `P2 s + P0 t + P1 H(st)` reduction. A general bivariate copula has no place here: every rank-one contribution depends on `s,t` only through `st`, and only one slope mark can be active. + +## 3. Why independently marked Poisson components are impossible + +Suppose one tried to assign independent slope marks to a Poisson cloud of essential components. If two nonparallel marks occur with positive probability, the resulting configuration would contain two disjoint nonparallel essential components. That is topologically impossible on a torus. + +Therefore the usual marked-Poisson theorem can be used only for **local morphology marks that do not change the projective homology line**, such as span, boundary size or core shape. Projective slope is a global hard-core mark and must be handled separately. + +This distinction is important for #762/#765: morphology can be marked independently on the anchor process, direction cannot. + +## 4. Conjectural pre-topological Poisson candidate closure + +A natural multi-direction rare-event model is to place Poisson statistics one layer earlier, on **candidate directional births before topological closure**. + +Let a finite competitive set of primitive slopes be `Lambda`, and suppose + +\[ +N_\ell\sim\operatorname{Poi}(\lambda_\ell),\qquad \ell\in\Lambda,\tag{4.1} +\] + +independently. Interpret `N_ell` as the number of candidate essential objects of slope `ell` generated by local rare-event mechanisms before enforcing the global torus intersection constraint. + +Apply the exact topological closure rule: + +- if all `N_ell=0`, output rank zero; +- if exactly one slope species is nonempty, output rank one with that slope and `K=N_ell`; +- if at least two distinct slope species are nonempty, output rank two and one connected essential component. + +Call this the **projective Poisson hard-core closure**. + +It is a conjectural asymptotic mechanism, but every output state satisfies the exact topology by construction. + +## 5. Closed-form law of the projective hard-core model + +Put + +\[ +\Lambda_*=\sum_{\ell\in\Lambda}\lambda_\ell. \tag{5.1} +\] + +Then + +\[ +\boxed{P(r=0)=e^{-\Lambda_*}.} \tag{5.2} +\] + +For every slope `ell` and integer `k>=1`, + +\[ +\boxed{ +P(r=1,L=\ell,K=k) +=e^{-\Lambda_*}\frac{\lambda_\ell^k}{k!}.} \tag{5.3} +\] + +Hence + +\[ +\boxed{ +P(r=1,L=\ell) +=e^{-\Lambda_*}(e^{\lambda_\ell}-1).} \tag{5.4} +\] + +The collision/rank-two probability is the remainder + +\[ +\boxed{ +P(r=2) +=1-e^{-\Lambda_*} + \left[1+\sum_\ell(e^{\lambda_\ell}-1)\right].} \tag{5.5} +\] + +For exactly two competing slopes, + +\[ +\boxed{P(r=2)=(1-e^{-\lambda_1})(1-e^{-\lambda_2}).} \tag{5.6} +\] + +Thus at low activity, + +\[ +P(r=2) +=\sum_{\ell<\ell'}\lambda_\ell\lambda_{\ell'} ++O(\lambda^3). \tag{5.7} +\] + +Rank two is therefore a direct second-order collision observable for competing direction gases. + +Conditional on `(r=1,L=ell)`, the component count is exactly zero-truncated Poisson with parameter `lambda_ell`. + +## 6. Parameter-free internal diagnostics + +The closure has strong algebraic checks that do not require fitting a full likelihood. + +Let + +\[ +p_0=P(r=0),\qquad +p_{\ell,k}=P(r=1,L=\ell,K=k). \tag{6.1} +\] + +Equation (5.3) implies + +\[ +\boxed{\lambda_\ell=\frac{p_{\ell,1}}{p_0}.} \tag{6.2} +\] + +and for every `k>=2`, + +\[ +\boxed{ +k!\,p_0^{k-1}\frac{p_{\ell,k}}{p_{\ell,1}^k}=1.} \tag{6.3} +\] + +In particular, + +\[ +\boxed{ +2p_0p_{\ell,2}=p_{\ell,1}^2.} \tag{6.4} +\] + +There is also the global zero-state identity + +\[ +\boxed{ +-\log p_0 +=\sum_\ell\frac{p_{\ell,1}}{p_0}.} \tag{6.5} +\] + +If only a finite slope subset is measured, the right side is a lower bound and the deficit estimates unobserved directional activity. + +These identities provide cheap falsification tests for #765 directional-window data before introducing any continuum model. + +## 7. Relation to the one-direction Poisson theorem + +If only one slope has non-negligible activity, (5.5) gives `P(r=2)=0` at this lower-window scale and the model reduces to an ordinary Poisson count of parallel components: + +\[ +P(r=0)=e^{-\lambda},\qquad +P(r=1,K=k)=e^{-\lambda}\lambda^k/k!. \tag{7.1} +\] + +That is exactly the topology-compatible structure already used in the one-sided birth window. The projective hard-core closure is therefore a genuine extension of the existing Poisson picture, not a competing replacement for it. + +## 8. What should happen near a directional degeneracy + +Suppose two primitive directions have comparable homological free energies. The correct qualitative prediction is not coexistence of two independent families of disjoint winding components. Instead: + +- one species alone produces a rank-one plateau carrying that persistent slope; +- simultaneous activation of both species is converted into rank two; +- the rank-two probability is the collision channel and should rise quadratically when both activities are small; +- the first-birth slope probabilities are controlled by the relative candidate activities. + +This gives a concrete state-space answer to the warning in #765 that nonparallel cheap classes cannot simply be added as independent Poisson processes. + +## 9. Claim boundary + +Sections 1--3 are exact finite topology. Sections 4--8 define and analyse one conjectural asymptotic closure. Independence is asserted only for pre-topological candidate processes, not for actual essential components. The diagnostic identities (6.2)--(6.5) make the conjecture directly falsifiable. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/projective-slope-harmonic-control-20260914.md b/docs/manuscripts/geometric-balance/projective-slope-harmonic-control-20260914.md new file mode 100644 index 000000000..a9de8d2d7 --- /dev/null +++ b/docs/manuscripts/geometric-balance/projective-slope-harmonic-control-20260914.md @@ -0,0 +1,264 @@ +# Projective homology-slope harmonics as an exact lattice/continuum control + +2026-09-14. This note does **not** introduce a new homology-flux programme. Issue #156 already proposed primitive homology-sector characters, and merged PR #213 supplies a general-`tau` Pinson--Arguin primitive-sector continuum evaluator. The new contribution here is the exact interface supplied by the 2026-09-14 persistent 4/8 slope theorem. + +The resulting observable is a useful positive control for angular/modular pipelines because it has all three ingredients simultaneously: + +1. an unambiguous finite-lattice definition; +2. an exact primal/matching complement transport; +3. an explicit critical continuum baseline from already-merged primitive-sector probabilities. + +It is **not** a replacement for original-U and must not be scored as though it were the #275 observable. + +## 1. Rank-one projective slope + +On an honest torus, a rank-one occupied configuration has one projective rational homology line + +\[ +\ell=[u:v]\in\mathbb P(H_1(T^2;\mathbb Q)), +\] + +represented by a primitive integer pair `(u,v)`, modulo overall sign. The exact classification in `projective-homology-gas-20260914.md` says every essential component in that configuration has the same line. + +For a declared physical period basis + +\[ +\omega_1=1,\qquad \omega_2=\tau,\qquad \Im\tau>0, +\] + +the embedded physical vector is + +\[ +z_{u,v}=u+v\tau. \tag{1.1} +\] + +Because the line is unoriented, any even spin gives a well-defined projective harmonic. In particular + +\[ +\boxed{ +Z_4^{(\tau)}(u,v) +=\frac{(u+v\tau)^4}{|u+v\tau|^4}.} \tag{1.2} +\] + +Changing `(u,v)` to `(-u,-v)` leaves (1.2) unchanged. + +This is an **embedded spin-4 geometric readout**, not a claim that the rank-one sector is a particular CFT spin-4 field. A modular basis change transports both `tau` and `(u,v)` according to the already-frozen period-basis convention; the phase of (1.2) is referred to the declared physical `omega_1` axis. + +## 2. Finite-lattice observable + +For graph `G` define + +\[ +A_{4,G}(p;\tau) +=E_p^G\left[1_{\{r=1\}}Z_4^{(\tau)}(\ell)\right]. \tag{2.1} +\] + +When `P_G(r=1)>0`, define the conditional harmonic + +\[ +H_{4,G}(p;\tau) +=E_p^G[Z_4^{(\tau)}(\ell)\mid r=1] +=\frac{A_{4,G}}{P_G(r=1)}. \tag{2.2} +\] + +More generally every even projective harmonic + +\[ +Z_{2k}^{(\tau)}(u,v) +=\frac{(u+v\tau)^{2k}}{|u+v\tau|^{2k}} \tag{2.3} +\] + +is legitimate. Spin four is singled out here because it is the first nontrivial harmonic compatible with the square `C4` geometry and it directly separates axis-like from diagonal-like rank-one sectors. + +On the square torus `tau=i`, + +\[ +Z_4(u,v) +=\frac{(u+iv)^4}{(u^2+v^2)^2}. \tag{2.4} +\] + +Thus + +- axis slopes `(1,0),(0,1)` have `Z_4=+1`; +- diagonal slopes `(1,1),(1,-1)` have `Z_4=-1`. + +Reflection symmetry makes the expectation real on the square torus; the imaginary part is an exact zero control. + +## 3. Exact 4/8 complement transport + +The persistent digital-Alexander theorem gives, configuration by configuration in the rank-one sector, + +\[ +\ell_8(\omega^c)=\ell_4(\omega). \tag{3.1} +\] + +If black NN sites have density `p`, the complementary white matching sites have density `1-p`. Therefore for **every** even projective harmonic and every honest torus, + +\[ +\boxed{ +A_{2k,4}(p;\tau)=A_{2k,8}(1-p;\tau).} \tag{3.2} +\] + +Since rank-one probabilities also match under complement, + +\[ +\boxed{ +H_{2k,4}(p;\tau)=H_{2k,8}(1-p;\tau).} \tag{3.3} +\] + +This is exact finite-lattice transport, not universality and not an asymptotic approximation. + +It is stronger than equality of a coarse directional flag: the complete projective line is preserved. + +## 4. Continuum baseline from the already-merged Pinson--Arguin evaluator + +Merged PR #213 fixes the convention + +\[ +\text{engine }(u,v)\longmapsto\text{paper sector }\{u,-v\} \tag{4.1} +\] + +and supplies the critical `Q=1` continuum probability + +\[ +\pi_\tau(\{a,b\}) \tag{4.2} +\] + +for every primitive unoriented homology sector on an arbitrary complex torus. + +Therefore the continuum topological spin-4 baseline is not a fitted modular function. It is the absolutely convergent primitive-sector sum + +\[ +\boxed{ +A_4^{\rm cont}(\tau) +=\sum_{(u,v)\in\mathbb Z^2_{\rm prim}/\pm} +\pi_\tau(\{u,-v\}) +Z_4^{(\tau)}(u,v).} \tag{4.3} +\] + +The total rank-one continuum probability is + +\[ +P_1^{\rm cont}(\tau) +=\sum_{(u,v)\in\mathbb Z^2_{\rm prim}/\pm} +\pi_\tau(\{u,-v\}), \tag{4.4} +\] + +and the corresponding conditional harmonic is + +\[ +\boxed{ +H_4^{\rm cont}(\tau)=A_4^{\rm cont}(\tau)/P_1^{\rm cont}(\tau).}\tag{4.5} +\] + +No field identification enters (4.3)--(4.5). They are a direct function of the published primitive wrapping-sector probabilities. + +## 5. Square-torus exact finite controls and continuum target + +The new standard-library census `scripts/rank1_slope_harmonic_control.py` gives at `p=1/2`: + +### `L=3` + +There are 162 rank-one NN configurations out of 512, with primitive slope counts + +\[ +(1,0):78,\quad(0,1):78,\quad(1,1):3,\quad(1,-1):3. +\] + +Hence + +\[ +A_4^{(L=3)}=\frac{150}{512}=\frac{75}{256}, \tag{5.1} +\] + +and + +\[ +\boxed{H_4^{(L=3)}=\frac{25}{27}=0.9259259259\ldots}. \tag{5.2} +\] + +### `L=4` + +There are 19,932 rank-one NN configurations out of 65,536, with slope counts + +\[ +(1,0):9406,\quad(0,1):9406,\quad(1,1):560,\quad(1,-1):560. +\] + +Thus + +\[ +A_4^{(L=4)}=\frac{17692}{65536}=\frac{4423}{16384}, \tag{5.3} +\] + +and + +\[ +\boxed{H_4^{(L=4)}=\frac{4423}{4983} +=0.8876179009\ldots}. \tag{5.4} +\] + +In both systems the imaginary harmonic is exactly zero. Every observed rank-one slope is axis or diagonal, so `Z_8=1` identically at these tiny sizes. + +Using the merged PR #213 Pinson--Arguin formula at `tau=i` and summing primitive sectors gives + +\[ +P_1^{\rm cont}(i) +=0.38094744914033735446061273329420245\ldots, \tag{5.5} +\] + +\[ +A_4^{\rm cont}(i) +=0.29682713399631153163484531698494418\ldots, \tag{5.6} +\] + +and therefore + +\[ +\boxed{ +H_4^{\rm cont}(i) +=0.77918131402676301260435917980210484\ldots.} \tag{5.7} +\] + +For comparison the continuum conditional spin-eight harmonic is + +\[ +H_8^{\rm cont}(i) +=0.99924170963860929291015300312572463\ldots. \tag{5.8} +\] + +The finite values are controls, not a claimed monotone convergence theorem. The fact that `H_8=1` at `L=3,4` simply reflects the absence of longer primitive slopes in those tiny honest square quotients; the continuum sum includes them. + +## 6. Why this is a useful positive control for modular/angular analysis + +This observable cleanly separates four questions that are often conflated. + +### Lattice semantics + +The input is the exact ambient homology line of a rank-one configuration. No source normalization, derivative, rank-one denominator or continuum operator naming is needed to define it. + +### Angular calibration + +The phase `Z_4=e^{i4\theta_\ell}` is a literal geometric spin-four readout of the wrapped slope. If an angular pipeline cannot reproduce its exact lattice symmetries and the known continuum primitive-sector baseline, that pipeline should not be trusted to identify a more complicated spin-four response. + +### Modular/shape dependence + +At general `tau`, PR #213 already supplies the sector probabilities with the correct period-basis transport. Equation (4.3) therefore gives a parameter-free shape prediction for the slope harmonic once the physical embedding is declared. + +### Primal/matching transport + +Equation (3.2) supplies an exact finite complement relation unavailable for generic microscopic angular observables. This makes the channel particularly useful for debugging orientation conventions and matching-sector maps. + +## 7. Interface to #585 and #589 + +The map-resolved torus programme in #585 requires every candidate continuum sector to have a defensible lattice-to-map dictionary. The rank-one primitive homology sector has exactly such a dictionary through PR #213. The harmonic (4.3) is therefore a natural **positive control** for the projective/subspace scoring machinery before applying it to a source whose map semantics remain uncertain. + +Likewise, an angular-channel design such as #589 should first demonstrate that its orientation weighting and phase convention recover the known `Z_4` transformation of this topological observable. This does not solve the original H4/H8 mechanism question; it calibrates the geometry of the measurement. + +## 8. Strict boundary relative to original-U + +The topological slope harmonic is **not** original-U and should not be inserted as a surrogate forward column in #275. + +A candidate mechanism for original-U still owes the declared physical source-to-observable map, normalizer and moving-root counterterm. Success of the present positive control proves only that the homology/angular/modular plumbing is correct for an observable whose semantics are already known. + +That separation is precisely why this control is useful. diff --git a/docs/manuscripts/geometric-balance/projective-slope-modular-covariance-20260914.md b/docs/manuscripts/geometric-balance/projective-slope-modular-covariance-20260914.md new file mode 100644 index 000000000..aece26033 --- /dev/null +++ b/docs/manuscripts/geometric-balance/projective-slope-modular-covariance-20260914.md @@ -0,0 +1,209 @@ +# A map-resolved modular-covariant spin-4 ray from primitive homology sectors + +2026-09-14. This note turns the projective slope harmonic into a genuine modular-covariant continuum control. It consumes, rather than replaces, the already-merged Pinson--Arguin primitive-sector evaluator from PR #213. + +No continuum field is named. The covariance follows directly from two facts: + +1. modular transformations permute primitive homology sectors with their published probabilities; +2. the embedded angle of the same physical cycle rotates by the coordinate rescaling used to renormalize the torus basis. + +This gives #585 a concrete map-resolved modular-covariant ray whose lattice semantics are explicit before any fit. + +## 1. Continuum slope harmonic + +For + +\[ +T_\tau=\mathbb C/(\mathbb Z+\tau\mathbb Z),\qquad \Im\tau>0, +\] + +let the engine primitive line `(u,v)` represent the physical cycle + +\[ +z_{u,v}=u+v\tau. \tag{1.1} +\] + +The critical `Q=1` primitive-sector probability from PR #213 is + +\[ +\pi_\tau(\{u,-v\}). \tag{1.2} +\] + +For even spin `s`, define + +\[ +Z_s^{(\tau)}(u,v) +=\left(\frac{u+v\tau}{|u+v\tau|}\right)^s, \tag{1.3} +\] + +and the unnormalized continuum harmonic + +\[ +\boxed{ +A_s(\tau)= +\sum_{(u,v)\in\mathbb Z^2_{prim}/\pm} +\pi_\tau(\{u,-v\})Z_s^{(\tau)}(u,v).} \tag{1.4} +\] + +The rank-one probability + +\[ +P_1(\tau)=\sum\pi_\tau(\{u,-v\}) \tag{1.5} +\] + +is a modular scalar. Hence the conditional harmonic + +\[ +H_s(\tau)=A_s(\tau)/P_1(\tau) \tag{1.6} +\] + +has the same spin covariance as `A_s`. + +## 2. Modular covariance from physical coordinate transport + +Let + +\[ +\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL(2,\mathbb Z), +\qquad +\tau'=\gamma\tau=\frac{a\tau+b}{c\tau+d}. \tag{2.1} +\] + +The standard normalized complex coordinate on the same physical torus changes by + +\[ +z'= rac{z}{c\tau+d}. \tag{2.2} +\] + +Therefore the unit direction of every physical cycle changes by the **same phase** + +\[ +\frac{z'}{|z'|} +=\frac{|c\tau+d|}{c\tau+d}\frac{z}{|z|}. \tag{2.3} +\] + +Meanwhile the modular transformation only permutes the primitive sector labels, with probabilities transported exactly by the Pinson--Arguin laws already tested in PR #213. Summing over all primitive unoriented lines therefore gives + +\[ +\boxed{ +A_s(\gamma\tau) +=\left(\frac{|c\tau+d|}{c\tau+d}\right)^s A_s(\tau).} \tag{2.4} +\] + +Likewise + +\[ +\boxed{ +H_s(\gamma\tau) +=\left(\frac{|c\tau+d|}{c\tau+d}\right)^s H_s(\tau).} \tag{2.5} +\] + +Thus `A_s` and `H_s` are weight-zero **spin-`s` modular-covariant functions**: their magnitude is modular invariant, while their phase follows the physical frame rotation. + +### Generator checks + +For `T: tau -> tau+1`, `c=0,d=1`, so the phase is one. The sector relabelling alone leaves the harmonic unchanged. + +For `S: tau -> -1/tau`, + +\[ +A_s(-1/\tau) +=\left(\frac{|\tau|}{\tau}\right)^sA_s(\tau). \tag{2.6} +\] + +At `tau=i`, the spin-four phase equals one, consistent with the square fixed point. + +## 3. Automorphism selection rules + +Equation (2.4) immediately gives zeroes at moduli with a nontrivial torus automorphism whose frame phase is not unity in spin `s`. + +### Hexagonal fixed point + +At + +\[ +\tau_\hexagon=e^{i\pi/3}=\frac12+i\frac{\sqrt3}{2}, \tag{3.1} +\] + +the torus has a 60-degree automorphism. A spin-four harmonic acquires + +\[ +e^{i4\pi/3}\ne1. \tag{3.2} +\] + +Since the modulus is fixed, covariance forces + +\[ +\boxed{A_4(\tau_\hexagon)=H_4(\tau_\hexagon)=0.} \tag{3.3} +\] + +This is an exact continuum selection rule, not a numerical cancellation. + +Similarly spin eight also fails the 60-degree invariance condition and vanishes at the hexagonal fixed point. + +### Square fixed point + +At `tau=i`, the nontrivial automorphism is a 90-degree rotation. Spin four is invariant, so `A_4(i)` need not vanish. This is exactly what the primitive-sector sum gives. + +## 4. Numerical shape controls from the existing primitive-sector formula + +Summing the PR #213 primitive probabilities with the phase (1.3) gives the following high-precision controls. The quoted values were stable under expanding the primitive-pair box from coordinate cutoff 8 to 12; each individual sector probability itself uses the certified PR #213 evaluator. The global weighted-sector tail was not separately re-certified in this continuation, so the non-fixed-point decimals should be treated as high-precision numerical controls rather than new rigorous enclosures. + +| modulus | `P1` | `H4=A4/P1` | +|---|---:|---:| +| `i` | `0.38094744914033735446...` | `+0.77918131402676301260...` | +| `1/2+i` | `0.37162305143377145961...` | `+0.29680403442759164454...` | +| `1/2+i*sqrt(3)/2` | `0.36789317459719662787...` | `0` exactly by automorphism | +| `1/2+5i/6` | `0.36810613542505968165...` | `-0.08280183604383594154...` | +| `1/2+7i/8` | `0.36790953505947126272...` | `+0.02211526529074907955...` | + +The corresponding unnormalized square value is + +\[ +A_4(i)=0.29682713399631153163484531698494418\ldots. \tag{4.1} +\] + +At `tau=i`, the continuum conditional spin-eight value is + +\[ +H_8(i)=0.99924170963860929291015300312572463\ldots, \tag{4.2} +\] + +whereas at the hexagonal fixed point spin eight vanishes exactly by the same automorphism argument. + +The sign change of `H4` across the nearby `Re tau=1/2` shapes is therefore not an arbitrary fitted curvature: it is a prediction of the primitive-sector map with an exact zero anchored at the hexagonal automorphism point. + +## 5. Why this is a stronger #585 positive control than a generic modular function + +The map-resolved torus programme asks for candidate solution spaces with explicit connectivity semantics. Here every ingredient is typed: + +```text +lattice event = ambient rank one, +map label = primitive projective homology line, +finite readout = e^{i4 theta_line}, +continuum weights = Pinson--Arguin primitive sector probabilities, +modular law = exact sector permutation + frame rotation, +matching transport = exact digital-Alexander slope equality. +``` + +There is no free amplitude after conditioning on rank one and no post-hoc basis expansion. + +Therefore an analysis pipeline intended to compare modular-covariant spin-four subspaces should first recover this ray/shape law from a control observable with known semantics. Failure localizes an orientation, modulus, period-basis, covariance or sector-dictionary problem before a more ambiguous continuum candidate enters. + +## 6. Exact finite primal/matching transport remains separate from universality + +For every finite honest torus and every occupation probability `p`, the lattice slope harmonic satisfies + +\[ +A_{s,4}(p;\tau)=A_{s,8}(1-p;\tau) \tag{6.1} +\] + +by configurationwise slope preservation under complement. Equation (6.1) is exact and off criticality. + +By contrast, (1.4) is the **critical continuum** percolation prediction. Agreement of a finite lattice sequence with (1.4) is an ordinary scaling-limit/universality question and should retain finite-size errors. + +The two statements should not be conflated: exact complement transport is lattice topology; modular covariance is continuum geometry. + +## 7. Boundary relative to original-U + +This modular-covariant spin-four ray is deliberately a positive control, not a candidate replacement for original-U. It has no declared mapping from the #275 physical source, normalizer and moving-root counterterm. It can validate the modular/angular machinery that a future original-U candidate would use, but cannot satisfy that candidate's missing source-to-forward-column contract by itself. diff --git a/docs/manuscripts/geometric-balance/query-cavity-nondegeneracy-20260914.md b/docs/manuscripts/geometric-balance/query-cavity-nondegeneracy-20260914.md new file mode 100644 index 000000000..38602c2cf --- /dev/null +++ b/docs/manuscripts/geometric-balance/query-cavity-nondegeneracy-20260914.md @@ -0,0 +1,260 @@ +# 查询孔洞是一个真实的第二模式：正方差、低密度展开与空间谱 + +2026-09-14。续接 #777 / #772 / #764。读取的 #771 基线是 +`80ed49265ee6ae8abd88426321ad2a6017caa48f`。本记录证明剩余 bulk 方差的严格正性，并给出独立的低黑密度展开；不依赖周期簇 OZ、单位缝合振幅或巨大簇联合 CLT。 + +## 1. 对象与结论 + +平面白色是独立 matching（八邻接）site 渗流，概率 q，黑色 NN 概率 p=1-q。考虑 θ(q)>0 且 p0。一个显式下界是 + +\[ +\boxed{J(q)\ge \frac{q^{16}\theta(q)}{25}\,j_{\rm cell}(q)>0,} +\tag{1} +\] + +其中 + +\[ +j_{\rm cell}(q)=\frac{p^8q^4P(p)}{9-p^8}, +\] + +\[ +P(p)=9+18p+27p^2+36p^3+45p^4+54p^5+63p^6+8p^7 ++7p^8+6p^9+5p^{10}+4p^{11}+3p^{12}+2p^{13}+p^{14}. +\] + +这个常数很保守，不是 J 的近似值。它关闭的是“第二方向是否真实存在”，不是实际幅度的高精度计算。 + +**结论 B（低黑密度）**： + +\[ +\boxed{J(1-p)=p^8-2p^9+9p^{10}-20p^{11}+O(p^{12}).} +\tag{2} +\] + +特别地，原来的 p^8 首项猜想由下面的局部轮廓展开得到。正交化对这一展开的首次可能影响在 p^15 阶，而不是 p^8 阶。 + +**结论 C（空间谱）**：令 Γ_X(k) 表示协方差的 Fourier 级数，并以热评分场在每个波数分别投影： + +\[ +J(q;k)=\Gamma_\eta(k)-|H_{\eta\xi}(k)|^2/c. +\] + +对 k∈[-π,π]^2，一致地有 + +\[ +\boxed{\begin{aligned} +J(1-p;k)={}&p^8-2p^9 + +(5+2\cos k_x+2\cos k_y)p^{10}\\ + &-(12+4\cos k_x+4\cos k_y)p^{11}+O(p^{12}). +\end{aligned}} +\tag{3} +\] + +因此第二模式在最低阶近似白噪声，第一个非局部修正来自两个轴向相邻的查询孔洞。没有声称所有 q、所有 k 都已有同样的严格正谱下界。 + +## 2. 一个可以完全求解的白色围栏 + +取 5×5 方框，外圈 16 个站点全白并接到无限白簇。内部 3×3 的九个标签仍独立。设 T 是内圈八点的白色数量，Z 是中心标签，则 + +\[ +T\sim\operatorname{Bin}(8,q),\quad Z\sim\operatorname{Ber}(q),\quad T\perp Z. +\] + +八个内圈点都邻接外圈，必被查询；中心被查询当且仅当 T>0。内部的无限白成员数 K、黑边界数 B、查询数 U 与评分 S 恰为 + +\[ +K=T+Z1_{T>0},\quad B=8-T+(1-Z)1_{T>0},\quad +U=9-1_{T=0},\quad S=K/q-B/p. +\tag{4} +\] + +外圈的固定白点不计入随机评分。令 r=p^8，则 + +\[ +ES=0,\quad \operatorname{Var}S=(9-r)/(pq), +\] +\[ +\operatorname{Var}K=pq(9-r)+q^2r(17-r),\quad +\operatorname{Cov}(K,S)=9-r+8q p^7. +\tag{5} +\] + +最优地去掉评分后仍有 + +\[ +\min_a\operatorname{Var}(K-aS) +=q^2\left[r(1-r)-\frac{64p^{14}}{(9-r)/(pq)}\right] +=j_{\rm cell}(q). +\tag{6} +\] + +正性也可不做消元就看出：内圈全黑时 (K,B)=(0,8)；打开一个内圈“门”且中心黑时变成 (1,8)；门和中心都白时变成 (2,7)。两种增量 (1,0)、(2,-1) 不共线，且所有三个事件在 00`：**前提清单**（逐条标「已证 / 引用 / 需假设」）+ 本征值量级 + +结论 (3.5)：`J_q(k) ≥ q¹⁶ θ(q)² / (25 D_q(k)) > 0`，`D_q(k)=(T(k)−q−1)²/q⁹+(T(k)−q)²/(pq⁸)+p^(−8)`。 + +| # | 前提 | 强度 | +|---|---|---| +| P1 | 精确二阶探索恒等式 `Eξ=0`、`Cov(ξ_0,ξ_z)=c 1{z=0}`、`c=θ/(pq²)`（引 #771） | **引用（外部输入）**，我未独立重算；与上游 giant-white §5(5.1)/(6.1) 的同一 `c` 一致 | +| P2 | [AV] Antunović–Veselić arXiv:0707.1089v3 Thm 2–3（site 版）：黑色亚临界**体积指数尾** | **引用（外部）**；用于 `η,β` 的有限半径可截断与协方差和存在 | +| P3 | 正关联（FKG/Harris，site 渗流单调）→ `π(q)=P(Q↔∞|Q全白) ≥ θ(q)/q` | **标准定理**（适用；前提是匹配图渗流满足 FKG，成立） | +| P4 | 不重叠 5×5 方框**标签独立**（按构造） | **已证/构造正确**；原文明确**不**假设各框「接无穷」事件独立（只经 `π(q)` 进入） | +| P5 | `Q` 全白时改动内部 `I` 不改变任何**外部**无穷连通量 | **已证**（论证：经 `I` 的新外部联系已可经全白 `Q` 实现） | +| P6 | Grothendieck/Bessel 常数：`det V(k)=1`、`minim=q²/D_q(k)>0` | **我独立验证**（3 个 q × 5 个 T 全部 `minim==q²/den` 且 `>0`） | +| P7 | 题设 `p0`（白 matching 超临界） | **假设**（§1 明写）；这也是「严格超临界」的真正含义 | + +**「strictly supercritical 的边界：`q<1` 但多远？」（任务 D 问题）——裁定：** +`D_q(k)>0` 对**一切** `00`），所以**正性结论不需要 `q` 接近 1**， +只要白相**严格**超临界（`θ(q)>0`，即 `q>q_c(matching)`，等价地黑相 `pH1(T_L)] = 0. +``` + +Open `v` and all NN edges from `v` to occupied neighbours. Suppose the new occupied graph `A+v` has ambient rank two. + +We prove that the local rank-two birth cannot be confined to a microscopic neighbourhood. In the universal cover it forces at least three disjoint occupied arms to a fixed fraction of the systole; the split case forces at least four. Standard planar separation then supplies the corresponding matching-vacant arms. + +## 2. Rank-zero components have disjoint deck translates + +Let `C` be one occupied component of `A`. Choose one connected lift + +```text +C_tilde subset Z^2. +``` + +For a deck vector `g in Z^2`, write + +```text +C_tilde+L g. +``` + +If two distinct translates intersect, then there is a lifted occupied path from a vertex `x` to `x+L g`. Its projection is a loop in `C` with nonzero ambient homology `g`, contradicting rank zero. + +Hence + +```text +(C_tilde+L g) cap (C_tilde+L h) = empty for g != h. +``` + +This is the lifted-component form of the gain-union-find invariant already used by the safe transfer. + +## 3. Attachment germs carry deck labels + +Fix one lift `v_0` of `v`. Each occupied neighbour/germ of `v_0` belongs to one translate of one lifted base component. + +For a fixed base component `C`, collect the deck labels of the translates which touch `v_0`: + +```text +G_C = {g_1,...,g_k} subset Z^2. +``` + +When `v_0` is opened, it joins all these translates at one new vertex. Every pair `g_i,g_j` creates a lifted cycle whose deck gain is + +```text +g_i-g_j. +``` + +Therefore the homology subgroup created through component `C` is generated by + +```text +. +``` + +With several base components touching `v`, the total rank gain is the span of these difference subgroups over the different components. + +This gives the deterministic alternatives for a rank-two jump: + +### T3 / theta type + +For some base component `C`, + +```text +rank = 2. +``` + +Then `G_C` contains at least three labels whose pairwise differences span rank two. + +### split / rose type + +No one component supplies rank two, but at least two components `C,D` each supply a nonzero rank-one difference and the two deck vectors are independent. Each of those components has at least two attachment labels. + +These are precisely the lifted versions of the T3 versus Rsplit attachment classes used in the finite pivotal atlas. + +## 4. A nonzero deck difference forces macroscopic diameter + +Suppose two distinct translates of the same lifted component touch the unit neighbourhood of `v_0`: + +```text +(C_tilde+L g_i) cap B_1(v_0) != empty, +(C_tilde+L g_j) cap B_1(v_0) != empty. +``` + +Translate the second contact by `L(g_i-g_j)`. It follows that `C_tilde+L g_i` contains + +- a vertex within O(1) of `v_0`, and +- another vertex within O(1) of `v_0+L(g_i-g_j)`. + +Thus its diameter is at least + +```text +L |g_i-g_j| - O(1). +``` + +In particular, for all sufficiently large `L`, it contains a connected occupied path from the inner neighbourhood of `v_0` to radius `cL` for any fixed `c<1/3` (after choosing a lattice norm and reducing `c` by an immaterial constant). + +## 5. T3 gives three disjoint black arms + +In the T3 case choose labels `g_1,g_2,g_3` with rank-two differences. The three translates + +```text +C_tilde+L g_1, +C_tilde+L g_2, +C_tilde+L g_3 +``` + +are pairwise disjoint by the rank-zero lemma. Each touches the inner neighbourhood of `v_0`, and each has macroscopic diameter because each label differs from at least one other label. + +Therefore there are at least three mutually vertex-disjoint occupied NN arms from the neighbourhood of `v_0` to distance `cL`. + +Importantly, the fact that the three germs belong to the **same base component on the torus** does not collapse them into one arm: before `v` is opened they are different deck translates in the universal cover and hence cannot connect there. + +## 6. Split type gives at least four black arms + +If two base components `C,D` each supply an independent rank-one deck difference, then + +```text +G_C contains g_1 != g_2, +G_D contains h_1 != h_2. +``` + +The four corresponding lifted components are pairwise disjoint: distinct translates within one base component are disjoint by rank zero, while lifts of distinct base components are disjoint trivially. + +Each has macroscopic diameter by the deck-difference argument. Thus the split case supplies at least four disjoint occupied arms to `cL`. + +## 7. From distinct black arms to alternating matching-vacant arms + +Inside the planar universal-cover annulus + +```text +B_{cL}(v_0) \ B_O(1)(v_0), +``` + +the black arms above belong to distinct occupied components before `v` opens. Between consecutive black crossing components in cyclic order there must be a vacant separator; for square-site percolation the correct planar separator uses the matching NN+diagonal adjacency. + +The expected site/matching separator statement is therefore: + +```text +3 disjoint black NN arm-components + => 3 interlaced white matching arms, + +4 disjoint black NN arm-components + => 4 interlaced white matching arms. +``` + +If this standard digital-planar separator lemma is invoked with the same arm convention used in the percolation exponent literature, we obtain + +```text +T3 jump2 => six alternating arms to cL, +Rsplit => eight alternating arms to cL. +``` + +This is the only step in the present note that should be independently checked for boundary/germ conventions before calling the whole result a theorem. + +## 8. Consequence for the near-critical pivotal clock + +Assuming the six-arm inclusion, standard critical/near-critical arm estimates give + +```text +P(jump2 pivotal at a given site) + <= const * pi_6(1,L). +``` + +Ordinary macroscopic pivotal intensity is governed by the four-arm probability `pi_4(1,L)`. Therefore the fraction of jump-two pivotals is bounded by + +```text +pi_6(1,L)/pi_4(1,L). +``` + +For triangular critical percolation, the exact polychromatic exponents give + +```text +alpha_4=5/4, +alpha_6=35/12, +alpha_6-alpha_4=5/3, +``` + +hence + +```text +jump2 / ordinary pivotal = O(L^-5/3+o(1)). +``` + +In an O(1) near-critical lambda interval the expected number of direct `rank 0->2` jumps therefore tends to zero. + +For square-site Matching One, the deterministic universal-cover certificate remains valid, while importing the exact exponent is a universality step unless the required square-site arm theorem is supplied separately. + +## 9. What this does and does not prove + +If the matching-vacant separator step is certified, this note gives the missing geometric input for the conjecture + +```text +continuum direct rank jump rate b(lambda)=0. +``` + +It does **not** imply that component lineages never merge in the near-critical process. Rank-preserving mergers and ordinary `1->2` events remain possible. The full merger factorial process is strictly richer than the rank projection. + +## 10. Direct finite regression + +The finite pivotal atlas already stores the exact `jump2` histogram and T3/T4+/Rsplit classes at small `L`. The most targeted numerical regression after the proof audit is not another free exponent fit, but the ratio + +```text +R_jump2 = 2 P(Delta_v r=2) / E[Delta_v r] +``` + +which should scale like the six-arm/four-arm ratio in the T3-dominated regime. This is a secondary check; the topology/arm inclusion is the primary result. diff --git a/docs/manuscripts/geometric-balance/rank-sector-exact-tables-L3to6-20260914.md b/docs/manuscripts/geometric-balance/rank-sector-exact-tables-L3to6-20260914.md new file mode 100644 index 000000000..3e9bfac28 --- /dev/null +++ b/docs/manuscripts/geometric-balance/rank-sector-exact-tables-L3to6-20260914.md @@ -0,0 +1,394 @@ +# Note — torus rank-sector occupation polynomials (Matching-One issue #775) + +**Author:** rank775 (cloud VM `DevEnvC_TV2N0X`, `/workspace/rank775/`) +**Machine:** huawei cloud EulerOS aarch64, 16 vCPU, gcc 10.3.1. +**Status of controls:** L=3 and L=4 reproduced **digit-for-digit**; all derived +quantities reproduce the round-6 controls to ~1e-11 (see §3). +**Honesty flags are in §8.** Everything here is *computed by me* unless marked +"引用". + +--- + +## 1. Problem statement (exact) + +On the honest `L×L` torus (both directions periodic) with the **black = NN** +adjacency `{(1,0),(0,1)}`, and for each occupied subset `ω ⊆ L²`: + +- `r_black(ω)` = rank of the image of `H_1(black) → H_1(T²) ≅ Z²`, + i.e. the rank (over `Z`, equivalently over `Q`) of the subgroup of `Z²` + generated by the winding vectors of all black components. Values **0 / 1 / 2**. +- `k = |ω|`. + +Exact count to produce: + +``` +C[L,j,k] = #{ ω : r_black(ω) = j, |ω| = k }, j = 0,1,2, k = 0..L² +``` + +Required: for each `k`, `Σ_j C[L,j,k] = binom(L²,k)` (full enumeration +consistency). + +Reference algorithm: `round3b-out/scaling_and_W4W8_audit.py::percomp_ranks` +(union-find + `Z²` offsets). Cross-check algorithm (this work): universal-cover +BFS, see §2. + +--- + +## 2. Algorithm & state definition (reproducible) + +### 2.1 Rank by union-find with `Z²` offsets `ranksec_brute.c` + +For a mask `ω` (`uint64`, sites indexed `i = y*L + x`): + +1. For every black site, `par[i]=i`, `off[i]=(0,0)`, `sz=1`. +2. **Union pass** — for each black site `i` and the two forward edges + `(+x)` and `(+y)` (torus-mod `L`), if the neighbour `j` is black, union + with the offset-compensated rule + `off[rv] = off[u] + (dx,dy) − off[v]` (using union-by-size). +3. **Winding pass** — re-read `off[i], off[j]` *after* both `find` calls: + for each forward edge, `d0 = off[u]+dx−off[v]`, `d1 = off[u]+dy−off[v]`. + Consistency requires `d0 % L == 0` and `d1 % L == 0`; the winding vector is + `(d0/L, d1/L)`. +4. **Rank** — collect the non-zero winding vectors; `rank = 2` if any two are + `Z`-linearly independent, else `rank = 1` if any non-zero vector, else `0`. + +Why `off` must be re-read after `find`: a `find(u)` may compress `u` and rewrite +`off[u]`; the winding delta must use the *current* offsets of `u` and `v`, both +measured to their (now equal) roots. + +### 2.2 Independent cross-check: universal-cover BFS `rank_bis.c` + +A structurally different algorithm, used for verification only: + +- Build the component in the **universal cover `Z²`**: BFS from `(v0,(0,0))` + stepping by `(dx,dy) ∈ {(±1,0),(0,±1)}` (the **symmetric** NN neighbourhood). +- Restrict the BFS to the box `|ℓ_∞| ≤ 2L`; this is enough because every + fundamental-cycle winding has `|w|_∞ ≤ L` (a tree path inside a component has + length `< L²` edges ⇒ displacement `< L` per coordinate), so a subgroup + generator is found inside the box. +- The winding subgroup `W_c` of a component is exactly + `{ ℓ : (base_vertex, ℓ) is reachable within the box }`. Combine the subgroups + of all components; report rank 2 / 1 / 0. + +**The bug I found and fixed** (must be recorded — it produced silent wrong +results): my first cover-BFS used the *one-directional* generator set +`{(+1,0),(0,+1)}` instead of the symmetric `{(±1,0),(0,±1)}`. With +one-directional generators the BFS can only traverse each undirected edge in one +orientation, so it cannot reach the base vertex again with a non-zero lift and +returns `rank=0` for many essential configs. + +**Minimal reproduction** (L=3, mask `60 = 0b000111100`, sites +`{(2,0),(0,1),(1,1),(2,1)}` — a full middle row + one cell, which contains a +full +x winding): + +``` +directed generators {(+1,0),(0,+1)} : rank_cov(60) = 0 (WRONG) +symmetric generators {(±1,0),(0,±1)} : rank_cov(60) = 1 (correct, matches UF) +``` + +Verifier `./rank_bis cross L 0 100000` reports **0 mismatches** between the +union-find and the cover-BFS on 100k random configs at L=3 and L=4. This is the +same class of "adjacency convention" trap already documented for the white +matching graph (`w48-out/correction-note.md`); here it is the *cover-BFS +direction convention* and it is caught by the cross-check, not by the controls. + +### 2.3 Digital-Alexander-duality independent check `rank_bis.c` (mode `duality`) + +The brief's identity `r4(S) + r8(S^c) = 2` (black NN rank + white **matching** +rank of the complement) was checked exhaustively: + +| configs | violations of `r4 + r8 = 2` | +|---|---| +| L=3 (512) | **0** | +| L=4 (65536) | **0** | +| L=5 (33554432) | **0** | + +The `duality` pass also independently reproduces every `C[L,j,k]` table +(§4), confirming the union-find counting twice. + +### 2.4 How L=5 was done (full enumeration) + +Brute-force bitmask enumeration `2^25 = 33 554 432` configurations, one +union-find per mask (`ranksec_brute.c`, `-O2 -march=native`, 8 pthreads). +`33.5M × ~0.29 µs/config-thread / 8 threads ≈ 1.35 s wall`, **C_L5.json**. +The per-`k` sum equals `binom(25,k)` for every `k` (§3), so the enumeration is +exact and complete. Peak RSS ≈ tens of MB. + +--- + +## 3. Controls comparison + +### 3.1 L=3 (exact controls, given in spec) — MATCH + +``` +rank0: [1, 9, 36, 78, 90, 45, 0, 0, 0, 0] ✓ +rank1: [0, 0, 0, 6, 36, 72, 48, 0, 0, 0] ✓ +rank2: [0, 0, 0, 0, 0, 9, 36, 36, 9, 1] ✓ +``` + +### 3.2 L=4 (exact controls, given in spec) — MATCH + +``` +rank0: [1, 16, 120, 560, 1812, 4272, 7448, 9440, 8082, 3984, 792, 32, 0, 0, 0, 0, 0] ✓ +rank1: [0, 0, 0, 0, 8, 96, 560, 1984, 4580, 6368, 4704, 1472, 160, 0, 0, 0, 0] ✓ +rank2: [0, 0, 0, 0, 0, 0, 0, 16, 208, 1088, 2512, 2864, 1660, 560, 120, 16, 1] ✓ +``` + +### 3.3 `Σ_j C[L,j,k] = binom(L²,k)` — verified every `k`, no failures (L=3,4,5) + +L=5 rank totals `[20218851, 9649920, 3685661]`, sum `= 33554432 = 2^25` ✓. + +### 3.4 round-6 derived controls — reproduced to ~1e-11 (exact-rational analysis) + +| diagnostic | L=3 (mine / round-6) | L=4 (mine / round-6) | +|---|---|---| +| `sqrt(L)·ρ_KX²` | 1.1326299214 / 1.1326299214 | 1.1607768254 / 1.1607768254 | +| `L^{3/4}·(−2/g1)` | −1.5480444436 / −1.5480444436 | −1.5165645063 / −1.5165645063 | +| `g2/g1` | −0.0658659540 / −0.0658659540 | −0.0669677666 / −0.0669677666 | +| `g3/g1³` | 0.0257008455 / 0.0257008455 | 0.0275610401 / 0.0275610401 | +| `c_slope` | −0.0918877291 / −0.0918877291 | −0.0630255510 / −0.0630255510 | +| `d log c / db` | 0.0624022146 / 0.0624022146 | 0.0337934511 / 0.0337934511 | +| `d² log c / db²` | −0.1256525679 / −0.1256525679 | −0.1122527722 / −0.1122527722 | + +This pins down the definitions (which were not in the local assets — reverse +engineered and confirmed): `κ_n(K|j)` = cumulant of `K` under the measure +tilted at the exact balance root `p_L^*` and conditioned on rank `j`; +`g_n = κ_n(K|2) − κ_n(K|0)`; `h_n = κ_n(K|1) − ½(κ_n(K|0)+κ_n(K|2))`; +`c_slope = h1`; `d log c / db = −2 h1 / g1`; `d² log c / db² = 4(g1 h2 − g2 h1)/g1³`. +The root `p_L^*` solves `P2(p) = P0(p)` and is found by exact rational bisection +(`Fraction`, enclosure `2^{-67}`); `a_L = P0(p^*) = P2(p^*)`. + +> ⚠️ One definition I could NOT recover from the local assets and did NOT +> fabricate: the *exact* meaning of `theta_L` from `cosh(s) = 1 − 1/(2 a_L)` +> and the explicit bookkeeping of `(z, b, c, s)`. The numerical diagnostics +> above are all reproduced, so the operative definitions of the `c_slope` +> sieve are confirmed; the `theta_L` symbol is reported but its intended use +> is not independently re-derived here. + +--- + +## 4. `C[L,j,k]` tables + +- **L=3, L=4:** inline in §3.1–3.2 (exact). +- **L=5:** `C_L5.json` (full 26-entry array per rank). Totals + `[20218851, 9649920, 3685661]`. +- **L=6:** `C_L6.json` (full `2^36 = 68719476736` enumeration; totals + `[44162178463, 18469776586, 6087521687]`; per-`k` binomial sum verified; + cross-checked 0/50000 vs cover-BFS). See §7 for telemetry. + +--- + +## 5. Derived quantities (full table; exact-rational, float shown) + +| L | `p_L^*` | `a_L = P0=P2` | `P1` | `g1` | `g2` | `g3` | `h1 (=c_slope)` | `h2` | +|---|---|---|---|---|---|---|---|---| +| 3 | 0.586511455113 | 0.329128026631 | 0.341743946738 | 2.945015004 | −0.193976223 | 0.656462536 | −0.091887729 | −0.266397469 | +| 4 | 0.590672112331 | 0.322744098273 | 0.354511803454 | 3.730045261 | −0.249792800 | 1.430335468 | −0.063025551 | −0.386229194 | +| 5 | 0.591988256518 | 0.318613210099 | 0.362773579802 | 4.457657281 | −0.295546724 | 2.584002999 | −0.043635640 | −0.508324800 | +| 6 | 0.592395070818 | 0.316064810660 | 0.367870378680 | 5.140609643 | −0.336722929 | 4.072369808 | −0.031551490 | −0.640801795 | + +L=6 derived scalar diagnostics: +`sqrt(L)·ρ_KX² = 1.1767864644`, `L^{3/4}·(−2/g1) = −1.4915190577`, +`g2/g1 = −0.0655025283`, `g3/g1³ = 0.0299780465`, +`self_curve_slope = 0.0122753883`, `self_curve_curvature = −0.0973089716`. +L=6 rank totals: `[44162178463, 18469776586, 6087521687]` (sum = `2^36` ✓); +independent cross-check (union-find vs cover-BFS) **0/50000 mismatches**; per-`k` +binomial sum verified **no failures**. + +Scaled diagnostics: `g1/L^{3/4}`, `g2/L^{3/4}`, `g3/L^{9/4}`, `g2/g1`, +`g3/g1³`, `sqrt(L)·ρ_KX²`, `L^{3/4}·(−2/g1)`, `self_curve_slope`, +`self_curve_curvature` are in `rank-sector-polynomials.json`. + +--- + +## 6. `c_slope` sieve — 8-arm vs 6-arm + +`c_slope = h1` = `d log c / dz` at the balance root. The two candidate +asymptotics from the spec: + +- **8-arm root `L^{-4}`** ⇒ `c_slope ~ L^{-5/2}` +- **6-arm root `L^{-5/3}`** ⇒ `c_slope ~ L^{-1/6}` + +Observed `c_slope` (L=3,4,5): `-0.0918877291, -0.0630255510, -0.0436356404`. +Consecutive ratios: `0.6859, 0.6923` (nearly constant). Pairwise effective +exponent `p_eff = ln(r)/ln(L/L')`: `1.3106, 1.6477` (increasing). + +Template fit (`rms_log` in the log of `|c_slope|`, finite-width check — **not** +a critical-exponent measurement): + +| template | `rms_log` (L=3,4,5) | +|---|---| +| free `A L^{-p}` (p≈1.45, 2-param) | 0.0199 | +| `A L^{-5/2}` [8-arm], 1-param | 0.2203 | +| `A L^{-5/2}` [8-arm] with `1/L` correction (2-param) | **0.0100** | +| `A L^{-1/6}` [6-arm], 1-param | 0.2693 | +| `A L^{-1/6}` [6-arm] with `1/L` correction (2-param) | 0.1297 | + +### 6.1 Verdict (4 points: L = 3,4,5,6) + +`c_slope` (L=3,4,5,6): +`-0.0918877291, -0.0630255510, -0.0436356404, -0.0315514899`. +Consecutive ratios `0.6859, 0.6923, 0.7231` (monotonically **increasing**). +Pairwise effective exponent +`p_eff = ln(r)/ln(L/L')`: `1.3106 → 1.6477 → 1.7785` (monotonically +**rising** — the signature of a correction term, not a constant). + +Template fit, `rms_log` in `log|c_slope|` (finite-width check — **not** a +critical-exponent measurement). With 4 points the 2-param fits have **2 dof**, +so the residual is now a real goodness-of-fit signal: + +| template | `rms_log` | +|---|---| +| free `A L^{-p}` (p≈1.54, 2-param) | 0.0287 | +| `A L^{-5/2}` [8-arm], 1-param | 0.2498 | +| `A L^{-5/2}` [8-arm] with `1/L` corr (2-param, dof=2) | **0.0184** | +| `A L^{-1/6}` [6-arm], 1-param | 0.3569 | +| `A L^{-1/6}` [6-arm] with `1/L` corr (2-param, dof=2) | 0.1258 | + +- **6-arm `L^{-1/6}` is EXCLUDED** by the sieve. `L^{-1/6}` predicts + `c_slope` almost constant (step ratio ≈ 0.95); the observed ratio is + ≈ 0.69–0.72, and `p_eff` is already 1.78 and rising. Its `rms_log` + (0.357, or 0.126 with a 1/L correction) is ~7× worse than the 8-arm + `1/L`-corrected fit (0.0184). It cannot explain an order-unity drop. +- **8-arm `L^{-5/2}` is COMPATIBLE.** The `1/L`-corrected 8-arm fit is the + best (`rms_log` 0.0184, `B ≈ −1.95 < 0`), consistent with the rising + `p_eff`. The data is *in* the 8-arm family. +- **Uncertainty / cannot claim:** the asymptotic exponent is **not yet + nailed**. `p_eff` is still rising at L=6 (1.78 < 2.5); it has not + converged to 5/2, so I cannot state the asymptotic exponent is exactly + 5/2 — only that the data lies in the 8-arm family and is incompatible with + 6-arm. The 8-arm vs 6-arm call is made; the residual of the 2-param fits + still shrinks with L=7,8. **Uncertainty magnitude:** the 8-arm+1/L fit beats + the 6-arm+1/L fit by ~7× in `rms_log`, but this is a 4-point discrimination, + not a convergence proof. +- This sieve does **not** settle the value of the percolation exponent; it + only establishes that the `c_slope` series is **incompatible with 6-arm + `L^{-1/6}` scaling** and **compatible with (a corrected) 8-arm + `L^{-5/2}` scaling**. L=7 (and L=8) are needed to confirm the exponent + converges to 5/2 (see §7). + +--- + +## 7. Status of L=6 / 7 / 8 + +- **L=6** (brute force): in progress at `/workspace/rank775/in/ranksec_brute 6` + (`2^36 = 6.87e10` configs, 8 threads). Estimated wall time ≈ 40 min + (projection from the L=5 rate `0.29 µs/config-thread`). Expected peak RSS: + tens of MB (counts are `3 × 37` 64-bit integers). Telemetry will be written + to `C_L6.json` + `/workspace/rank775/log/L6.log`. This is **feasible** on the + box; no DP needed for L=6. +- **L=7, L=8**: `2^49 / 2^64` configs — brute force is infeasible + (~236 days / beyond reach). These need a **transfer-matrix + connectivity + state DP** (percolation connectivity transfer matrix). I have a concrete state + design but did **not** implement/validate it within this round; it is the + natural next step and is described below so the next agent can pick it up. + +### 7.1 DP state design (for L≥7) — "frontier column + unwrapped heights" + +Process the torus column by column (`x = 0..L-1`), frontier = current column. + +State after processing columns `0..x` (a cylinder band, `y` periodic, `x` open): + +- `M_x` : black mask of the frontier column (L bits). +- For each black row `y ∈ M_x`: a component id `c(y)` and an **unwrapped height** + `h_y ∈ Z` with `h_y ≡ y (mod L)`, unique up to a per-component constant shift. + Canonicalise per block by subtracting the first member's height. +- For each block: its accumulated **vertical winding generator** `g ∈ Z` + (subgroup of `{0}×Z` generated by the cycle windings found so far; before the + `x`-seam all cycles have `Δx < L` so they are purely vertical). +- `A` : the accumulated winding subgroup of all **closed** components so far + (rank 0/1/2 subgroup of `Z²`). + +Transition (add column `x+1` with mask `M'`): create new frontier cells as +singletons; add vertical edges within `M'` (merge with offset ±1); add horizontal +edges `(x,y)-(x+1,y)` for `y ∈ M_x ∩ M'`. A horizontal edge never creates an +`x`-winding (Δx=±1=1` paired essential components; +- `D=+1`: black rank two / white rank zero, no neutral paired winding gas. + +Thus `D` is an `O(1)` topological charge, whereas the rank-one count `K` is the neutral component gas. + +The matching observable is simply + +\[ +M(p)=E_pD=P_2(p)-P_0(p). +\] + +The finite matching root `p_*` is therefore the zero-mean-charge condition + +\[ +\boxed{E_{p_*}D=0.} +\] + +It does not require the neutral gas to be small, concentrated, or absent. + +## 2. Charge susceptibility + +The exact charge variance is + +\[ +\chi(p):=\operatorname{Var}_p(D) +=P_0(p)+P_2(p)-M(p)^2. +\] + +At the matching root, write + +\[ +P_0(p_*)=P_2(p_*)=\varepsilon_*. +\] + +Then + +\[ +\boxed{ +\chi(p_*)=2\varepsilon_*=1-P_1(p_*).} +\] + +Hence on a long rank-one plateau the topological susceptibility can be exponentially small even though the finite zero-charge point remains unique. + +## 3. Exact factorization of the median slope + +Let + +\[ +H(p)=\frac{P_2(p)}{P_0(p)+P_2(p)} +\] + +be the endpoint-sector conditional odds, and let + +\[ +F(p)=\frac12(1+M(p)) +\] + +be the two-birth mixture CDF. + +Put + +\[ +L(p)=\log\frac{P_2(p)}{P_0(p)}. +\] + +Then `H'=H(1-H)L'`. At the matching root `H=1/2`, so `H'=L'/4`. + +Also, because `P_0=P_2=epsilon_*`, + +\[ +M'(p_*) +=P_2'(p_*)-P_0'(p_*) +=\varepsilon_*L'(p_*). +\] + +Using `chi=2 epsilon_*` gives the exact identities + +\[ +\boxed{M'(p_*)=2\chi(p_*)H'(p_*),} +\] + +and therefore + +\[ +\boxed{F'(p_*)=\chi(p_*)H'(p_*).} +\] + +This is a literal factorization of the finite median density into + +1. the probability mass left in the two charged endpoint sectors, and +2. the speed at which their **conditional odds** rotate through equality. + +The root theorem in the main manuscript works by controlling item 2 through a ratio comparison; full-law concentration is governed by whether item 1 stays substantial away from `p_c`. The two questions are algebraically different already at finite size. + +## 4. Bernoulli score form + +For `N` sites let `X` be the occupied-site count and + +\[ +S_p=\frac{X-Np}{p(1-p)} +\] + +be the product-Bernoulli score. For each sector with positive probability, + +\[ +\frac d{dp}\log P_j(p)=E[S_p\mid r=j]. +\] + +Thus + +\[ +\boxed{ +L'(p) +=E[S_p\mid r=2]-E[S_p\mid r=0].} +\] + +At the root, + +\[ +F'(p_*) +=\frac{\chi(p_*)}{4} +\left(E[S_{p_*}\mid r=2]-E[S_{p_*}\mid r=0]\right). +\] + +This representation may be useful for finite exact transfers or rare-event samplers because it says precisely which two conditional ensembles determine the condition number of the matching root. + +## 5. Asymptotic interpretation in elongated geometries + +In a separated two-window geometry, the interior of the interval between the births is overwhelmingly rank one. At the matching root one therefore expects + +\[ +\chi(p_*)=1-P_1(p_*)\to0. +\] + +The root can nevertheless remain sharply selected because the endpoint **odds ratio** `P_2/P_0` changes exponentially with geometry. The factorization above makes this compatible with an increasingly flat mixture CDF: + +- the mixture may converge to a broad two-atom law or an interior plateau rather than concentrate; +- the finite zero-charge point still converges to `p_c` under the much weaker shortest-period condition of the main root theorem. + +This is the algebraic core of the phrase **balance without concentration**. + +In the extreme-elongation limit the mixture approaches `1/2` throughout every fixed interior `p`, so `F'` vanishes there. There is no contradiction with finite uniqueness: the limiting plateau loses the information carried by the exponentially rare charged sectors. + +## 6. Numerical implication + +Estimating the root by treating `F` as an ordinary well-conditioned median can become inefficient when `chi` is tiny. The identity + +\[ +F'=\chi H' +\] + +says that a tiny absolute error in `F` must be compared to a tiny derivative, whereas a targeted endpoint-sector odds calculation works directly with the object that selects the root. + +This does not authorize an unweighted ratio estimator on rare sectors; it only identifies the correct conditioning. Any computational method must retain the covariance/rare-event error model. + +## 7. Source-field generating function + +The exact topological source may be written + +\[ +Z_p(h,z;\chi_{slope}) +=P_2e^h+P_0e^{-h}+P_1H_p(z;\chi_{slope}), +\] + +as in `projective-homology-gas-20260914.md`. Then `D` is the charge coupled to `h` and the paired component count/slope live in the neutral sector. At `h=0,z=1`, + +\[ +\partial_hZ=M, +\qquad +\partial_h^2\log Z=\chi. +\] + +No continuum/CFT interpretation is required for this source decomposition. diff --git a/docs/manuscripts/geometric-balance/rare-topology-fugacity-and-mass-clock-20260914.md b/docs/manuscripts/geometric-balance/rare-topology-fugacity-and-mass-clock-20260914.md new file mode 100644 index 000000000..a3d452a93 --- /dev/null +++ b/docs/manuscripts/geometric-balance/rare-topology-fugacity-and-mass-clock-20260914.md @@ -0,0 +1,358 @@ +# Rare-topology fugacity, witness grading, and the mass clock + +Date: 2026-09-14 + +Status: synthesis / conjectural programme built from already existing #760/#780/#800/#774 inputs. The exact identities and author-level bounds cited below keep their original status. The new content is the proposed common expansion parameter and its cross-regime consequences. + +## 1. A single small parameter is already appearing in several places + +Fix strictly subcritical square-site NN probability `p kappa(p). +``` + +Thus, up to polynomial/subexponential factors, define the **rare-topology fugacity** + +```text +epsilon_w(p) = exp[-w kappa(p)]. +``` + +The proposal of this note is that several apparently separate finite-width effects are naturally graded by the number of *additional disjoint essential witnesses* they require relative to the baseline object. + +This is not a claim that all coefficients are universal, nor that every rare event is literally independent. It is a proposed exponential bookkeeping principle. + +## 2. Witness-number grading + +A complete essential component itself needs one horizontal essential witness, hence + +```text +nu_w = poly/subexp(w) * epsilon_w. +``` + +Now consider an event already conditioned on the existence of one such macroscopic object. + +### One extra witness + +If a correction requires one additional independent/disjoint essential witness, its **relative** scale should be + +```text +poly/subexp(w) * epsilon_w. +``` + +Three current objects fit this pattern. + +1. **Lineage merger.** The new #780 double-witness/BK argument proves + +```text +limsup w^-1 log(B_w/nu_-) <= -kappa(p0) +``` + +for the factorial merger density `B_w`. Thus the already-proved upper exponent is exactly the one-extra-witness exponent. + +2. **Slow visible doublet.** Draft #800 finds, for NN `p=1/4`, + +```text +Delta gamma_w / (w nu_w) ~= 1.7--1.9, w=3,...,7, +``` + +with nearly opposite residues. This strongly suggests a tunnelling matrix element generated by one rare topological event. + +3. **Periodic mass locality.** #760 reduces `gamma_w-kappa` to wrap-sensitive irreducible pieces/decorations. If the minimal defect is one additional transverse wrap, its natural exponential cost is again `epsilon_w`. + +The coefficients and polynomial powers need not agree. The conjecture is only that the *exponential rate* is controlled by the same one-extra-witness cost. + +### Two or more extra witnesses + +The same logic predicts a hierarchy. Absolute two-component interaction intensities are `O(epsilon_w^2)` up to polynomial factors; relative to the leading one-component intensity they are `O(epsilon_w)`. Higher connected/factorial corrections should be organized by further powers when no cheaper shared topology exists. + +A violation is scientifically useful: it would identify a cheaper multi-winding network or a failure of the disjoint-witness reduction. + +## 3. A two-state tunnelling model for the #800 slow doublet + +The opposite residues in #800 are naturally reproduced by a nearly reducible two-dimensional slow block. Schematically, after factoring the common longitudinal mass, + +```text +R_w = rho_bar,w [ I - epsilon_w L_w + o(epsilon_w) ], +``` + +where `L_w` has two nonzero eigenvalues of order one after an allowed polynomial rescaling of `epsilon_w`. + +Then + +```text +Delta gamma_w = poly(w) epsilon_w [c(p)+o(1)]. +``` + +If the scalar source/readout is approximately antisymmetric between the two slow states, the two residues are equal and opposite to leading order, exactly as observed. + +A particularly sharp version is + +```text +lim_w -w^-1 log Delta gamma_w = kappa(p), +``` + +and, under the stronger renewal identification, + +```text +Delta gamma_w / [w nu_w] -> c_split(p) in (0,infinity). +``` + +The second statement is deliberately stronger and may fail through a different polynomial prefactor even if the first survives. + +### Existing numerical consistency + +For NN `p=1/4`, #800 gives + +```text +w=3..7: Delta gamma/(w nu) = 1.83,1.69,1.77,1.94,1.67. +``` + +Using those same numbers, `-log(w nu_w)/w` is already close to a stable value near one over the available widths, consistent with a common exponential mass scale. This is only a diagnostic; it is not a new estimate of `kappa`. + +## 4. Stronger locality conjecture for #760 + +The zero-Fourier argument already shows that pure periodization cannot move the plane mass. Therefore `gamma_w-kappa` comes only from a piece that actually sees a periodic image. + +The witness-grading refinement is + +```text +lim_w -w^-1 log[gamma_w(p)-kappa(p)] = kappa(p) +``` + +whenever the leading wrap-sensitive defect is exactly one additional essential transverse excursion and its coefficient is nonzero. + +This is stronger than the current `O(exp(-c w))` target. It can fail in two informative ways: + +- a cheaper defect gives exponent `kappa`. + +Thus the exponent itself diagnoses the topology of the leading periodic defect. + +## 5. The merger theorem identifies where pure fragmentation must stop + +The #780 no-merger proof is exponentially strong only because `w kappa(p0)->infinity` at fixed strict subcritical `p0`. + +Its leading relative bad-event factor has the same schematic form + +```text +Pr(merger in a 1/nu window) <= poly(w) exp[-w kappa(p)] + superexp tail. +``` + +Therefore the natural criterion for a **pure fragmentation regime** is not simply `p infinity. +``` + +Conversely, if + +```text +w kappa(p_w) = O(1), +``` + +the extra-witness penalty is no longer asymptotically small and there is no reason for the split-only process to survive. Merger/coalescence and topological hard-core effects can remain at order one. + +This gives a concrete boundary between two programmes that had previously been discussed separately: + +```text +w kappa -> infinity : rare barriers, asymptotic Poisson, pure cuts, slow topological tunnelling; +w kappa = O(1) : near-critical split-merge / interacting homology regime. +``` + +In the standard near-critical scaling `pc-p ~ lambda w^(-3/4)` and `kappa(p) ~ C (pc-p)^(4/3)`, + +```text +w kappa(p) ~ C |lambda|^(4/3). +``` + +Hence pure fragmentation is the `lambda -> -infinity` tail of the near-critical theory, not the full finite-`lambda` process. + +This is a useful negative conclusion: one should **not** attempt to continue the fixed-subcritical fragmentation generator all the way through the critical window by merely changing its clock. + +## 6. The remaining #780 clock is a Palm-score / mass-slope problem + +For a complete component with occupied count `K`, distinct external vacant boundary count `B`, and `q=1-p`, the exact activity identity is + +```text +d log nu_w / dp = E_Palm[S_w], +S_w = K/p - B/q. +``` + +Define the normalized score speed + +```text +v_w(p) = (1/w) d log nu_w / dp. +``` + +Since `-w^-1 log nu_w -> kappa`, the natural bridge conjecture is + +```text +boxed: +v_w(p) -> -kappa'(p) +``` + +locally on compact strict-subcritical intervals at differentiability points of `kappa`. + +Equivalently, + +```text +(1/w) E_Palm[K/p-B/q] -> -kappa'(p). +``` + +This identifies the **macroscopic intensity clock speed** with a microscopic component-Palm composition score. + +If the convergence is locally uniform and `v(p0)=-kappa'(p0)>0`, then + +```text +p_w(x) = p0 + x/[v(p0) w] +``` + +gives + +```text +log[nu_w(p_w(x))/nu_w(p0)] -> x, +nu_w(p_w(x))/nu_w(p0) -> exp(x). +``` + +This is exactly the affine intensity clock required by the simplest #780 process statement. Importantly, a full Ornstein--Zernike amplitude theorem is not needed. + +## 7. A weaker proof target than the full prefactor theorem + +Pointwise exponential-rate convergence by itself does not imply derivative convergence. A sufficient route is: + +1. locally uniform convergence + +```text +f_w(p)=-(1/w)log nu_w(p) -> kappa(p); +``` + +2. local equicontinuity of `f_w'`. + +The exact second Ward identity gives + +```text +(log nu_w)'' + = Var_Palm(S_w) - E[K]/p^2 - E[B]/q^2. +``` + +Thus a strict-subcritical additive-functional estimate + +```text +(log nu_w)'' = O(w) +``` + +uniformly on compact `p` intervals would make `f_w'` locally equi-Lipschitz. At differentiability points of `kappa`, standard compactness then upgrades the rate convergence to + +```text +f_w'(p) -> kappa'(p), +``` + +which is the desired score bridge. + +This target is substantially weaker than proving + +```text +nu_w=A(p) w^-1/2 exp[-w kappa(p)](1+o(1)) +``` + +with a differentiable amplitude. It asks only for an `O(w)` susceptibility bound for the physical activity score. + +## 8. The clock scale automatically crosses over to the critical thermal scale + +The local parameter increment that changes the intensity by an order-one factor is + +```text +delta p_clock ~ 1/[w (-kappa'(p))]. +``` + +If + +```text +kappa(p) ~ C (pc-p)^(4/3), +``` + +then + +```text +-kappa'(p) ~ (4C/3)(pc-p)^(1/3). +``` + +At a near-critical distance + +```text +pc-p ~ w^-3/4, +``` + +we get + +```text +delta p_clock ~ w^-3/4. +``` + +So the fixed-subcritical `1/w` intensity window and the critical `w^-3/4` thermal window are not unrelated scales: they are two limits of the same mass-slope clock + +```text +1/[w(-kappa'(p))]. +``` + +The exact Palm score then supplies the microscopic estimator of that clock speed. + +This is the main cross-regime conjecture of the note. + +## 9. Relation to the near-critical Ward scaling in #774 + +The same formula predicts the change in score magnitude. At fixed strict subcritical `p`, + +```text +E_Palm S_w ~ -w kappa'(p) = O(w). +``` + +In the critical window `pc-p~w^-3/4`, + +```text +-w kappa'(p) ~ w^(3/4), +``` + +which is exactly the score scale already expected from the near-critical complete-component Ward identity. + +Thus the fixed-subcritical clock and the near-critical `K/B` Ward programme are consistent pieces of one crossover, not separate coincidences. + +## 10. Minimal next checks + +No exponential-length new simulation is needed for this synthesis. + +1. **Score/mass bridge.** Reuse existing two-fugacity activity outputs to form `v_w(p)=E S/w` at one or two strict-subcritical parameter values. Compare only against independent `kappa` difference-quotient/interval information from #761/#760; do not infer `kappa` from the same `nu_w` table. + +2. **Slow doublet exponent.** If direct tagged-operator eigenvectors become available, test `-log Delta gamma_w/w` against an independent mass interval before fitting a universal ratio. + +3. **Merger crossover.** Do not spend compute proving mergers are absent deeper in the fixed-subcritical regime; the theorem already does that. The informative future experiment is at a controlled finite `w kappa`, where the witness expansion predicts order-one corrections. + +4. **Mass locality.** If #760 identifies the leading wrap defect, check whether it is one-witness or symmetry-suppressed before choosing an exponential rate. + +## 11. Claim boundary + +Exact/current inputs: + +- `d log nu/dp = E_Palm(K/p-B/q)`; +- the second score/Ward identity; +- the fixed-subcritical exponential rate of `nu_w`; +- #780's double-witness upper bound on merger; +- #760's zero-Fourier periodization mechanism; +- #800's stored slow-doublet diagnostics. + +New conjectures: + +- witness-number grading by powers of `epsilon_w=exp[-w kappa]`; +- exact exponential rate `kappa` for the slow splitting and leading mass-locality correction; +- score-to-mass-slope convergence; +- `w kappa` as the fragmentation-to-split-merge crossover variable. + +These conjectures are meant to compress several branches into one falsifiable mechanism. A different exponent in any one channel is evidence for a different minimal topology, not a reason to tune the same formula. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/reproduce-genealogy-noise-20260914.md b/docs/manuscripts/geometric-balance/reproduce-genealogy-noise-20260914.md new file mode 100644 index 000000000..4dc6740d9 --- /dev/null +++ b/docs/manuscripts/geometric-balance/reproduce-genealogy-noise-20260914.md @@ -0,0 +1,80 @@ +# note-reproduce —— 第一层：逐字复现 #771 包的自述 + +机器 `DevEnvC_TVVfoB`（账号2，16 vCPU，Python 3.9.9，`PYTHONPATH=/workspace/mo/compat`）。 +工作目录 `/workspace/genealogy/`。**本机 Mac 未做任何计算。** +被核验件：`genealogy-noise-20260914/matching_one_genealogy_noise_20260914/repo/`（原样上传）。 + +## 1. 18 项测试 + +命令（按包内 README 口径，脚本 `sys.path` 自己插入 `scripts/`）： + +``` +cd /workspace/genealogy/in/repo +PYTHONPATH=/workspace/mo/compat PY39COMPAT_MARKER=1 \ + python3 -m unittest discover -s tests -p test_coalescent_bulk_clock.py -v +``` + +结果：**`Ran 18 tests in 2.845s` / `OK`，18/18 通过。** +测试名与 `TEST_OUTPUT.txt` 逐条一致（18 行，`test_01…test_18`，顺序相同）； +唯一差别是计时（本地 1.080 s vs 本机 2.845 s），非判据。 + +（垫片生效证据：每步都打印 +`py39compat: int.bit_count backport active (route=installed, stdlib-only, selftest=OK)`， +自检 `(255).bit_count()==8`。） + +结论：**通过**。证据等级：有限枚举事实 + 有限测试（不构成渐近证明）。 + +## 2. `--output` 再生 vs 远端结果 JSON + +接口核对（先做）：脚本用 `argparse`，只有 `--output`（`type=Path`），**没有硬编码绝对路径**； +`__main__` → `main()`；写盘用 `json.dumps(report(), ensure_ascii=False, sort_keys=True, indent=2, default=encode)`， +即**排版（indent=2）**；无 `--help` 障碍（`-h` 正常）。 + +比对（脚本 `scripts/compare_regen.py`，独立于包）： + +| 比对口径 | 结果 | +|---|---| +| (A) 字面字节：脚本 stdout/`--output` 产物 vs 提交件 | **不相同**（11747 B vs 7482 B）| +| (B) 规范化字节 `dumps(load(...), sort_keys=True, separators=(',',':'))` | **逐字节相同** ✓ | +| (C) 解析后结构相等 | **相同** ✓ | + +规范化后两边 canonical sha256 都是 +`d93c99e5728636b51b93846e9296b7b06b140835125721dd2472c96765ad49aa`（各 7481 B）。 + +⚠️ 口径澄清：`EXECUTION.json` 的两个字段名是 +`json_structure_regenerated_identically` 与 **`json_compact_canonical_bytes_regenerated_identically`** +——措辞**明确限定在 compact/canonical**，与 (B)(C) 相符，**没有过度宣称**。 +但要知道：脚本本身写的是**排版 JSON**，因此**它无法在字面字节层面再生成提交件**； +"逐字节相同"只在规范化口径下成立。规格 §2.2 担心的"排版 vs 紧凑"在这里是**已声明**的， +不是隐瞒。（对照：#780 包的 `EXECUTION.json` 主动写明 "Exact identity is after JSON compact +normalization, not literal pretty-vs-compact bytes." —— 同一种做法。） + +## 3. `EXECUTION.json` 四个计数能否从产物复算 + +四个数**都能**从再生的结果 JSON 直接复算（`report()` 的输出里就有）： + +| 字段 | 声称值 | 复算来源 | 复算值 | +|---|---|---|---| +| `circle_combinatorial_realizations` | 89438 | `Σ_n genealogy[n].realizations` = Σ_{n=2..6}(n−1)!·n! = 2+12+144+2880+86400 | **89438** ✓ | +| `full_history_transition_checks` | 4047 | `Σ_n genealogy[n].full_history_transition_checks` = 1+4+25+251+3766 | **4047** ✓ | +| `partition_probability_checks` | 277 | `Σ_n genealogy[n].eppf_equalities` = 2+5+15+52+203 | **277** ✓ | +| `four_coordinate_generator_checks` | 768 | `report()['four_coordinate_generator_equalities']` = 4(η)×4(a)×4(b)×3(m)×4(n) | **768** ✓ | + +⇒ **不是"只出现在 EXECUTION.json、无法复算"的孤立数**；四者都是产物字段的直接聚合。 + +⚠️ 一处措辞注意：`four_coordinate_generator_checks=768` 是**网格检查条数**（768 条恒等式）， +不是"四坐标"的维度数；`768` 与 `4×4×4×3×4` 的分解见于包内脚本 `closure_stationarity_checks()`， +`EXECUTION.json` 未给分解（可接受，但读者易误读为"四维生成的 768 个分量"）。 + +## 4. 未发现的问题（第一层） + +- 脚本未硬编码路径；未在 Python 3.9 触发语法错误（`from __future__ import annotations` + 无 3.10 语法）。 +- 18 测试与 `TEST_OUTPUT.txt` 逐条匹配；`Ran 18 tests ... OK` 属实。 +- 四个计数均可复算。 + +## 5. 复现命令与产物 + +云上：`/workspace/genealogy/{scripts,out}/` +- `scripts/compare_regen.py` → `out/layer1-compare.json`，`out/regen.json` +本机：`genealogy-out/{regen.json,layer1-compare.json,scripts/compare_regen.py}` +耗时：测试 2.845 s；再生+比对 < 1 s；pip 装 numpy 2.0.2 / sympy 1.14.0 约 11 min（一次）。 diff --git a/docs/manuscripts/geometric-balance/research-frontier-20260914.md b/docs/manuscripts/geometric-balance/research-frontier-20260914.md new file mode 100644 index 000000000..660d48e12 --- /dev/null +++ b/docs/manuscripts/geometric-balance/research-frontier-20260914.md @@ -0,0 +1,156 @@ +# Research frontier after the structural reductions — 2026-09-14 + +Status: proof programmes and conjectures unless a subsection explicitly says otherwise. This note is intended to consume the structural results in `structural-consequences-20260914.md`, not to create a new queue of parallel production tasks. + +## 1. Homological free energy: energy minus opportunity entropy + +For a period lattice `Lambda` of area/index `N` and a primitive period class `u in Lambda/±`, let `tau_p(u)` be the subcritical directional connection cost. A class-`u` winding component has a natural number of transverse opportunities of order + +\[ +h_u\asymp N/|u|. +\] + +If its once-per-component intensity has a form + +\[ +\nu_u(p)\approx C(p,\hat u)|u|^{-\beta(p,\hat u)}e^{-\tau_p(u)}, +\] + +then the leading logarithm of the expected number of such rare components is + +\[ +\log h_u-\tau_p(u)+O(\log|u|). +\] + +This suggests the variational barrier + +\[ +\boxed{ +\Psi_\Lambda(p)= +\min_{u\in\Lambda_{\rm prim}/\pm} +\left\{\tau_p(u)-\log\frac{N}{|u|}\right\}.} +\] + +The leading first-rank birth should occur near `Psi_Lambda(p)=0`; the component prefactor controls only the finer centre correction. + +For the axial torus with periods `(w,0),(0,m)`, the cheapest class is `(w,0)`, so `tau_p(u)=w kappa(p)` and `N/|u|=m`. The condition `Psi≈0` reduces exactly to + +\[ +w\kappa(p)\approx\log m, +\] + +which is the centre relation already used in the exponential-aspect analysis. + +### Deterministic simplification in exponential elongation + +If `u` is a Euclidean shortest period with `|u|=ell`, `h=N/ell`, then any nonparallel `v in Lambda` obeys `|det(u,v)|>=N` and hence `|v|>=h`. At fixed subcritical `p`, norm equivalence gives `tau_p(v)>=c_p h`, while its opportunity entropy is at most `log(N/|v|)<=log ell`. Under `log h/ell -> d>0`, nonparallel classes therefore have exponentially larger energy cost and cannot compete with `±u` in this variational problem. This reinforces the directional-separation lemma already recorded for #765. + +The genuinely new regime is a lattice deliberately tuned so that two or more primitive classes have `tau_p(u)-log(N/|u|)` within `O(1)`. Nonparallel essential components cannot be independent species because intersection/topological rank constraints couple them. A useful language is a **homological hard-core gas**: parallel species can coexist; nonparallel species force intersections/mergers and tend to accelerate the rank-two birth. + +Falsifiable prediction: for a family with a finite set of candidate primitive classes, the first birth is controlled by `min_u Psi_u=0`; if two nonparallel classes are nearly tied, the rank-one plateau should shorten relative to a single-minimizer geometry. If a class with significantly larger `Psi_u` dominates, the opportunity factor or microscopic normalization in the conjecture is wrong. + +## 2. Small-p actual-SITE prefactor theorem via a column transfer operator + +The current branch already supplies three pieces of a possible proof: + +1. `dilute-winding-crossover.md` proves for the actual NN site model that, when `w p^2 -> 0`, + \[ + \nu_w(p)=p^w I_0(2wp)(1+o(1)), + \] + and therefore exhibits a genuine `w^{-1/2}` closure factor in the subregime `wp -> infinity`, `wp^2 -> 0`; +2. `sewing-with-memory.md` writes the **complete-component** activity using exact three-column local weights, so distinct external boundary-site weights are not fundamentally nonlocal; +3. the tagged all-height resolvent gives an exact finite-`w` representation of the complete component and identifies mixing, one transverse mode, and Palm normalization as the real missing hypotheses. + +This suggests proving the first fixed-`p` complete-component prefactor theorem on a nonempty interval `0 Phi(w kappa(p))` can be retained as a marginal hypothesis, but it is insufficient to determine the joint black/white crossover. The missing object is the topologically constrained family `H_x` together with the rank weights. + +## 6. Execution order + +To keep the programme from proliferating again: + +1. integrate the persistent birth reflection, exact same-parameter PGF reduction, and dual-even/odd birth coordinates into the structural layer; +2. make the alternating-barrier span comparison publication-grade and close the reciprocal mean white-span theorem; then upgrade to the exponential/Gamma laws through the existing point-process proof; +3. record marked-Poisson as the common interface for morphology work; +4. pursue the small-`p` transfer-operator theorem as the next genuinely new proof technology; +5. keep the homological free-energy and near-critical `(rank,K)` pictures as unifying maps, not immediate production queues. + +No extra width-16/20/24 prefactor campaign is justified by this note alone, and no continuum-field interpretation is implied. diff --git a/docs/manuscripts/geometric-balance/reverse-audit-irrep-source-action-spectrum-20260914.md b/docs/manuscripts/geometric-balance/reverse-audit-irrep-source-action-spectrum-20260914.md new file mode 100644 index 000000000..d2a30e424 --- /dev/null +++ b/docs/manuscripts/geometric-balance/reverse-audit-irrep-source-action-spectrum-20260914.md @@ -0,0 +1,428 @@ +# 反向审计：从单一机制叙事转向 angular irrep、source quotient 与 topological action spectrum + +Date: 2026-09-14 + +Status: correction / synthesis note. 本文专门审计近期 #771 分支（包括本代理自己写入的若干统一化猜想）在新出现的 #47/#61/#802/#807/#808 与独立 oblique 复核之后哪些仍成立、哪些必须降级、哪些应撤回。除明确标为 exact 的代数外，不把新的组织方式升级成 continuum field identification。 + +## 1. 先给裁决：四个需要纠正的地方 + +### 1.1 `spin4 × x≈4 dressing` 不能解释 H4 projector 后的 residual + +此前提出过：square root 的 `ell^-4, ell^-6, ...` 可以由一个 H4/spin-four sector 被共同的 `x≈4` scalar correction 以 `ell^-2` dressing 产生。这个说法只可以保留在 **H4 coefficient 自己的 radial expansion** 中： + +```text +P4(ell)=a4 ell^-4 [1+c2 ell^-2+...]. +``` + +如果一个 angular projector 精确满足 `L[H4]=0`，那么任何仍然正比于 H4 的 scalar dressing 也被同一个 projector 精确杀掉。故 projector 后仍存活的量不能再由“同一 H4 被 scalar dressing”单独解释。 + +因此旧的强解释： + +```text +post-H4 residual ~= dressed H4 +``` + +撤回。保留的弱版本只是： + +```text +H4 coefficient 本身可以有 ell^-2, ell^-4,... 的 radial dressing。 +``` + +#808 正在测的正是 **angularly orthogonal** residual；若它是真信号，应归入 H0/H8/H12/... 中至少一个，而不是重新塞回 H4。 + +### 1.2 “matching-even/odd continuum field” 语言必须降级 + +有限 matching/complement identity 精确给出的是原图与 matching 图之间的 observable/source pairing。它并不自动构造一个作用在同一 local CFT/OPE space 上的 involution。 + +因此本文以后区分： + +```text +finite pair-exchange S/D projection : exact / empirical lattice sector; +RG tangent parity : hypothesis until a local map J is constructed; +OPE/interchiral parity : stronger hypothesis. +``` + +`alpha<->gamma, beta fixed` 等 pathwise statement 仍可作为有限 pair-exchange 事实使用；不得仅据此称某 continuum field “matching-even/odd”。尤其 `V_<1,3>` 等 lower competitor 不能只靠未经证明的 OPE parity 排掉。 + +### 1.3 #802 已证明 source normalization / source redundancy 不能省略 + +新勘误给出逐配置恒等式 + +```text +H_W = N - 2K + H_B, +``` + +所以两个原先被当作独立的 black-pair / white-pair source 实际只差 constant + thermal score。它们属于同一个 normalized source class，不是两个独立 nonthermal directions。 + +同时，若 transfer 用未正规化行权 `exp(gH)`, physical free energy 必须包含 row normalizer `f(g)`： + +```text +I(g)=f(g)-log lambda_tilde(g). +``` + +漏掉 `f_g` 会把 pure normalization/thermal motion 错当 shape response。 + +### 1.4 单一 `epsilon=e^{-w kappa}` witness fugacity 过强 + +#780 的 BK double-witness theorem 的确给 merger 一个相对 `<= exp[-w kappa+o(w)]` 的上界，但它没有证明等号；#800 的 slow doublet 是 spectral object，不是一个已经识别的概率事件；#760 的 wrap-sensitive correction 也尚未证明其最小 defect action 就是 `kappa`。 + +因此统一猜想 + +```text +all one-extra-witness effects have exponent kappa +``` + +降级。更稳健的对象是 defect-specific **incremental topological action spectrum**，见 §5。 + +## 2. 正确的分析顺序：angular irrep first, radial field second + +对同一 microscopic square model、固定 physical circumference `ell`，charge root 作为 orientation 的 D4-invariant scalar 可写成离散角向展开 + +```text +p_ch(ell,theta) + = P0(ell) + + P4(ell) cos(4theta) + + P8(ell) cos(8theta) + + P12(ell) cos(12theta) + + ... . +``` + +这里 `P0,P4,...` 只是 lattice angular irreps / Fourier coefficients；没有先赋予 CFT field、matching parity 或 radial exponent。 + +研究应分成两个阶段： + +1. **angular spectroscopy**：尽量在 same-circle / same-modulus / same-Smith geometry 上提取 `P4,P8,...`； +2. **radial spectroscopy**：再研究每一个已经分开的 coefficient 随 `ell` 的幂、log、mixing；最后才匹配 continuum map/field。 + +这避免把 “一个 power” 同时解释成 angular spin、CFT dimension 与 pair-exchange parity。 + +### 2.1 两角 H4-null projector 的精确 leakage 公式 + +令 + +```text +h_i = H4(theta_i)=cos(4theta_i), +L[f] = (h1 f(theta2)-h2 f(theta1))/(h1-h2). +``` + +则 exact： + +```text +L[1] = 1, +L[H4] = 0. +``` + +用 Chebyshev 恒等式 + +```text +H8 = 2 H4^2 - 1, +H12 = 4 H4^3 - 3 H4, +``` + +直接得到 + +```text +C8 := L[H8] = -(1+2 h1 h2), +C12 := L[H12] = -4 h1 h2 (h1+h2). +``` + +因此一个两角 projector 要同时近似 notch H8/H12，理想条件是 + +```text +h1 h2 = -1/2, +h1+h2 = 0, +``` + +即 + +```text +h1=+1/sqrt(2), h2=-1/sqrt(2). +``` + +N377 的 `(4,19)` / `(11,16)` 恰好接近这个双条件：它不是任意挑的两角，而是一个近似的 **H4/H8/H12 triple-notch geometry**。其当前 `C8≈0.0036146`、`C12≈-0.1378` 与上述公式一致；H16 并未被同时 notch，因此仍须保留更高 harmonic adversary。 + +### 2.2 命名纠正：`p_perp4` 不是自动的 `p_H0` + +只要 `C8,C12,...` 非零，两角 H4-null combination 严格只能叫 + +```text +p_perp4 = L[p_ch], +``` + +而不是已经纯化的 `H0/scalar root`。 + +#808 的 double-notch 使 scalar 解释更可信，但若最终把结果命名为 “H0” 或 `V_<1,4>`，仍需要： + +- 数值上控制 H8/H12/higher 的允许贡献；或 +- 理论上给这些 harmonic 的 radial 下界/selection rule。 + +否则正确结论只能是 `post-H4 angular-orthogonal residual`。 + +## 3. 新的 working spectrum：不要再把 square 的 `4,6,...` 当成一个序列 + +当前最合理的分层是： + +```text +P4(ell) : leading root harmonic ~ ell^-4, 已有强 deterministic evidence; +P0(ell) : post-H4 scalar candidate，#808 检验 ~ ell^-7; +P8(ell) : 独立 harmonic，radial exponent 尚未由数据确定; +P12... : 更高 adversaries，优先作为 leakage bounds 而非默认新任务。 +``` + +`P4` 内部仍可能有 `ell^-6` 等 dressing；但这些项在 exact H4-null projector 中全部消失。 + +因此 #47 的历史 `L^-7` 与 #808 的 scalar `ell^-7` 候选，不应再用“next spin4 descendant”解释。普通 thermal spin4 quasiprimary tower的下一项也并不自然给 relative `q=3`。若 scalar `ell^-7` 被确认，`x=33/4` (`x-x_t=7`) 是一个结构上匹配的 candidate，但 field identity、log collision 与 pair-exchange tangent action仍未解决。 + +更低的 interchiral/scalar competitor（例如 #61 保留的 `V_<1,3>` adversary）必须继续保留，直到真正构造 RG tangent map 或 radial data 排除。 + +## 4. Source space 的 exact 修正：Bernoulli chaos grading + +这里有一个可以逐配置定义、但不冒充 RG parity 的微观 source grading。 + +在 primal model 参数 `p` 上定义标准化 Bernoulli variable + +```text +xi_v = (n_v-p)/sqrt[p(1-p)]. +``` + +matching/complement model 使用 `q=1-p` 与 `n_hat=1-n`，则 + +```text +xi_hat_v + = (n_hat_v-q)/sqrt[q(1-q)] + = -xi_v. +``` + +因此对任何有限 site set `A`，Hoeffding/Walsh chaos basis + +```text +Psi_A = product_(v in A) xi_v +``` + +在 exact complement pairing 下满足 + +```text +Psi_A -> (-1)^|A| Psi_A. +``` + +这是 microscopic source-function space 的精确 grading，不是 continuum OPE parity。 + +### 4.1 #802 source redundancy 被这个 basis 一眼解释 + +对一条相邻 pair `(i,j)`： + +```text +n_i n_j + = p^2 + p sqrt(pq)(xi_i+xi_j) + pq xi_i xi_j, + +(1-n_i)(1-n_j) + = q^2 - q sqrt(pq)(xi_i+xi_j) + pq xi_i xi_j. +``` + +两者的 degree-2 chaos **完全相同**；差异只有 degree 0 normalization 与 degree 1 thermal score。故 modulo `{1, K}` 后 black-pair 与 white-pair source 必然相同。这正是独立核验发现的 `H_W=N-2K+H_B`。 + +### 4.2 新的 source design rule + +在花算力比较两个“物理源”之前，先做 exact decomposition： + +```text +source = constant + thermal/one-site part + genuine higher chaos. +``` + +- constant：只改 normalizer； +- uniform degree-1：thermal reparameterization； +- higher chaos：才可能提供新的 normalized source direction。 + +因此 #802 下一份真正独立 source，至少应在 quotient `source / span{1,K}` 中线性独立。 + +最简单的 exact pair-exchange-odd nonthermal source来自 **odd chaos degree >=3**，而不是另一个 pair-count source。它可以作为 #61 RG-tangent programme 的 microscopic basis element；但它是否流向某个 continuum “odd field”仍须另外证明。 + +## 5. 从 witness number 改成 incremental topological action spectrum + +固定严格亚临界 `p`。对一个 baseline topological event `B_w` 与额外 defect `E_w`，定义增量 action（若极限存在） + +```text +sigma(E|B) + = lim[-1/w log P(E_w and B_w)/P(B_w)]. +``` + +baseline one-winding 的 action 是 `kappa(p)`。 + +### 5.1 已有 rigor 只给部分 inequality + +#780 对 merger 使用两个 vertex-disjoint essential witnesses与 BK，得到绝对双见证概率至多 `exp[-2 kappa w+o(w)]`；除以 baseline `nu~exp[-kappa w+o(w)]` 后得到 + +```text +sigma_merger >= kappa +``` + +（作为 liminf / upper-probability statement）。**没有证明 equality。** + +### 5.2 #800/#760 应各自有自己的 action + +定义概念上： + +```text +sigma_split : slow-doublet tunnelling action; +sigma_wrap : cylinder/plane mass-locality wrap-defect action; +sigma_merge : lineage-merger incremental action. +``` + +目前只有经验上 `Delta gamma/(w nu)=O(1)` 暗示 `sigma_split≈kappa`；这不是证明。`gamma_w-kappa` 的 leading wrap defect 也可能被 symmetry cancellation 推到更高 action，或由不同 constrained network 控制。 + +故不再默认 + +```text +sigma_split=sigma_wrap=sigma_merge=kappa. +``` + +### 5.3 更大胆但更稳健的统一猜想：Wulff network action ratios + +对每一种 topological defect type `T`，猜想存在一个 constrained Wulff/network action + +```text +sigma_T(p), +``` + +并且在 near-critical isotropic limit + +```text +sigma_T(p)/kappa(p) -> c_T, +``` + +其中 `c_T` 只依赖 defect topology/network geometry，而不必是整数。 + +于是 massive tail 的自然 expansion 应是 + +```text +sum_T exp[-c_T s] * P_T(s), +s=w kappa(p), +``` + +而不是简单的整数 powers `exp[-j s]`。 + +#758 的 loop/branch variational rate本身已经提示网络代价不必等于整数 witness count。 + +`w kappa -> infinity` 仍然是 #780 pure-fragmentation 的一个 **充分** 远尾条件，因为已有 merger upper bound会消失；但不能据此宣布 `s=w kappa` 单独控制所有 spectral/locality corrections。 + +## 6. Clapeyron/source-response 公式的 normalization-safe 版本 + +若 safe transfer 使用未正规化微观行权 `W_tilde(g)`，行配分 normalizer为 + +```text +f(g)=log Z_row(g), +``` + +则 physical free energy是 + +```text +I(g)=f(g)-log lambda_tilde(g). +``` + +令 raw row score + +```text +H_raw=partial_g log W_tilde, +mu=E_Q H_raw. +``` + +exact： + +```text +I_g=f_g-mu. +``` + +所以两个 topological safe phases 的 coexistence差满足 + +```text +Theta_g + = (f4_g-mu4) - (f8_g-mu8). +``` + +只有当 `W` 从一开始就是 normalized physical row transition weight，才可以把它简写为纯 `-mu4+mu8`。 + +因此 normalization-safe Clapeyron equation是 + +```text +dp_ch/dg = -Theta_g/Theta_p. +``` + +这保留了之前的 finite Perron/Feynman--Hellmann结构，但纠正“raw score difference就是 physical numerator”的过强写法。 + +### 6.1 source gauge quotient + +若两个 source 相差 + +```text +constant + a * thermal score, +``` + +则它们描述的是同一 normalized probability family的 gauge/reparameterization。fixed-b normal response应不变，而 root tangent按 thermal坐标改变。 + +这与 #773 的 `source modulo span{1,K}` exact quotient一致，也是 #802 勘误的正确抽象形式。 + +## 7. 计算路线也需要改：N1105 当前不是默认下一步 + +独立 `dpfloor` 审计说明，统一 `p_c` 与 tight solver 后 numerical floor 足够低；但随后真正独立 oblique implementation 的 N325 rehearsal 表明状态空间成本指数爆炸，N1105 在现有 automaton上远超当前资源。 + +因此： + +```text +precision gate : pass after fixes; +algorithmic cost gate : fail for generic N1105 tomography. +``` + +不要把“精度上 GO”误写成“计算上 GO”。 + +当前更高信息/成本比的是 #808 N377 specialized double-notch Phase 0；若它也撞墙，下一步应改 state representation / contraction algorithm，而不是硬扩 N。 + +## 8. 下一轮最值得做的四件事 + +### A. N377 只叫 `post-H4 residual gate` + +Phase 0 先给 state/resource cost。若可算，输出 `p_perp4` 与精确 H8/H12 leakage budget。只有当 higher harmonics在允许 radial规模下不足以解释 residual，才升级 “angular scalar”。 + +### B. 构造 microscopic exchange tangent basis + +用 Bernoulli chaos degree 1/2/3 的少数 translation/D4-symmetrized local sources： + +- degree1 thermal control； +- degree2 pair source control（应验证 black/white quotient redundancy）； +- degree3 genuine nonthermal exchange-odd source。 + +计算 normalized safe-phase source-response矩阵，而不是先给 continuum field命名。这是 #61 的一个可执行 lattice-side tangent map。 + +### C. 对 topological defects 测 action，不测统一 prefactor + +对 #800 slow splitting，优先拿 direct tagged-operator eigenvectors并估计 + +```text +-w^-1 log Delta gamma_w +``` + +与 independent mass interval比较。不要再先拟合 `Delta gamma/(w nu)` 为常数。 + +#760 类似：先识别 minimal wrap defect，再决定其 `sigma_wrap`，不要预设是 `kappa`。 + +### D. angular coefficient 分开后再做 continuum field map + +若最终确认： + +```text +P4 ~ ell^-4, +P0-pc ~ ell^-7, +``` + +再去比较 `x=21/4` spin4 与 `x=33/4` scalar/log-block候选；在此之前不需要继续扩大 operator label list。 + +## 9. 对本代理此前几份 note 的明确状态 + +- `equal-circumference-spin4-projector...`：**保留设计思想**；full tomography 的计算可行性已被 cost-wall改写。 +- `sector-even-dressing-of-spin4-tower...`：**降级**为 H4 coefficient 内部 radial dressing；不再解释 post-H4 orthogonal residual。 +- `dimension-21over4-resonance...`：总维数简并与 angular decomposition **保留**；“matching-odd field”措辞降级为 pair-exchange response，且 scalar `x=21/4` 是否存在需数据决定。 +- `rare-topology-fugacity-and-mass-clock...`：mass-clock/Palm-score桥 **保留为证明目标**；统一 witness powers降级为 defect-specific action spectrum。 +- `topological-clapeyron-response...`：Perron coexistence结构 **保留**，但 raw-source公式必须使用本节 normalization-safe版本。 +- Krushkal/Euler bond-FK notes：finite embedded-bond algebra **保留**；不迁移为 square-site local identity。 +- `topological-source-master-scaling...`：作为组织图保留；其 rare tail应从 integer witness series改成 `sum_T exp[-c_T s]` action spectrum，operator parity语言按 #61 降级。 + +这次审计的目标不是减少猜想，而是把可失败的位置重新放对：先测 lattice irrep / normalized source / network action，再谈 continuum operator identity。 \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/root-response-as-residualized-score-20260914.md b/docs/manuscripts/geometric-balance/root-response-as-residualized-score-20260914.md new file mode 100644 index 000000000..8c95d368d --- /dev/null +++ b/docs/manuscripts/geometric-balance/root-response-as-residualized-score-20260914.md @@ -0,0 +1,304 @@ +# Root response as an exact residualized-score quotient + +Date: 2026-09-14 + +Status: exact finite probability calculus for a normalized source family. It reorganizes #802 after the source-normalization erratum and gives a gauge-invariant definition of the nonthermal source direction. No continuum field identification is assumed. + +## 1. Setup + +Let `P_{p,g}` be a **normalized** finite-torus probability law with baseline `g=0` equal to the Bernoulli site model. Let + +```text +X(omega)=r(omega)-1 in {-1,0,+1}, +M(p,g)=E_{p,g}[X]. +``` + +The finite matching root `p_*(g)` is defined by + +```text +M(p_*(g),g)=0. +``` + +Write the normalized likelihood scores at `g=0` as + +```text +S_p = partial_p log P_{p,0}(omega), +H = partial_g log P_{p,g}(omega)|_(g=0). +``` + +Normalization implies + +```text +E S_p=0, +E H=0. +``` + +For iid Bernoulli sites on `N` vertices, + +```text +S_p=(K-Np)/[p(1-p)]. +``` + +## 2. Root tangent is an exact covariance ratio + +Differentiate `M`: + +```text +M_p = Cov(X,S_p), +M_g = Cov(X,H). +``` + +At the root `E X=0`, these are simply `E[X S_p]` and `E[XH]`. + +Implicit differentiation gives + +```text +boxed: +T_g := dp_*/dg + = - Cov(X,H) / Cov(X,S_p). (2.1) +``` + +The denominator is positive for the monotone rank observable in the interior regime. + +Equation (2.1) is the normalized finite-torus analogue of the safe-Perron Clapeyron formula. It makes no reference to a continuum field or a chosen metric on source space. + +## 3. Residualize the source against the thermal nuisance + +Define + +```text +beta_H + = Cov(X,H)/Cov(X,S_p) + = -T_g. +``` + +Now set + +```text +boxed: +H_perp = H - beta_H S_p. (3.1) +``` + +Then exactly + +```text +Cov(X,H_perp)=0. (3.2) +``` + +So `H_perp` is the source score after subtracting precisely the amount of thermal motion required to keep the root fixed to first order. + +This is the intrinsic content of the “fixed-b normal direction” in #802. No Euclidean/Fisher orthogonality language is needed: it is a nuisance-score quotient defined by the root estimating equation itself. + +## 4. Shape response of any observable is a partial covariance + +Let `A(omega)` be any finite observable with no explicit `p,g` dependence. Along the moving root, + +```text +d/dg E_{p_*(g),g}[A]|_0 + = Cov(A,H) + T_g Cov(A,S_p) + = Cov(A,H_perp). (4.1) +``` + +Thus the pair + +```text +T_g = -beta_H, +N_A(g)=Cov(A,H_perp) +``` + +is an exact tangent/shape decomposition. + +For a smooth functional of the rank law rather than a single `A`, replace `A` by its finite influence function; the same residual-score formula holds. + +This is the probability-theory form of the source quotient already used in #773, now tied directly to the root. + +## 5. Gauge invariance explains the #802 erratum + +Suppose a second source score differs by + +```text +H' = H + c S_p. +``` + +Then + +```text +beta_H' = beta_H + c, +T_g' = T_g - c, +``` + +but + +```text +H'_perp + = H + c S_p - (beta_H+c) S_p + = H_perp. (5.1) +``` + +Hence every fixed-root/shape response is identical. + +A constant added to an **unnormalized** source weight disappears after centering the likelihood score, so the full gauge class is + +```text +H ~ H + constant + c * thermal_score. (5.2) +``` + +This is exactly why two raw source formulas can give different root tangents but no independent shape information. + +## 6. Black-pair / white-pair alias becomes one line + +For a periodic row, the exact source identity is + +```text +H_W = N - 2K + H_B. +``` + +After normalization/centering, + +```text +H_W^c - H_B^c + = -2(K-Np) + = -2 p(1-p) S_p. (6.1) +``` + +Therefore the two sources satisfy + +```text +H_W,perp = H_B,perp (6.2) +``` + +configurationwise as source classes. + +Their root tangents differ only by the thermal gauge term: + +```text +T_W-T_B=2p(1-p), (6.3) +``` + +which is precisely the independent #802 audit result. + +The earlier apparent “two-source discrimination” was therefore impossible in principle after thermal profiling. + +## 7. Relation to Bernoulli-chaos grading + +The centered score quotient should be performed **before** interpreting the source by chaos degree. + +For example, black and white pair counts have the same degree-2 Hoeffding/Walsh component and differ only by degree 0/1 pieces. Equation (6.2) is the score-space expression of that exact fact. + +A genuinely new local source direction must have + +```text +H_perp != 0 +``` + +and be linearly independent of previously tested residualized scores. + +Odd chaos degree three is a natural first exact complement-odd nonthermal candidate, but its continuum RG image remains an empirical/theoretical question. + +## 8. Rank-law normal coordinate as an influence function + +For the canonical rank probabilities `P0,P1,P2`, define + +```text +b = (1/2) log(P0/P2), +d = log[P1/sqrt(P0 P2)]. +``` + +At any interior point their source derivatives are linear functionals of the score `H`. In particular one can write + +```text +b_g = Cov(A_b,H), +d_g = Cov(A_d,H) +``` + +for rank-measurable influence functions `A_b,A_d`. + +Then the fixed-b shape response + +```text +N_g=d_g-(d_p/b_p)b_g +``` + +is exactly + +```text +boxed: +N_g = Cov(A_d,H_perp_b), (8.1) +``` + +where + +```text +H_perp_b=H-[b_g/b_p] S_p. +``` + +This recovers #802's chain-rule formula but exposes its statistical structure: it is a partial covariance after profiling the thermal nuisance. + +## 9. Safe-transfer version + +For a semi-infinite safe transfer, the normalized physical one-row source score should be used. If an implementation starts from unnormalized row weights `W_tilde`, with + +```text +f(g)=log Z_row(g), +I(g)=f(g)-log lambda_tilde(g), +``` + +the physical derivative is + +```text +I_g=f_g-E_Q[H_raw]. +``` + +One should first convert all source columns to physical normalized free-energy derivatives, then perform the same thermal residualization. + +This prevents the omitted-normalizer failure found in the original #802 delivery. + +## 10. A new source experiment with real information gain + +Instead of comparing two motif sources before quotienting them, pre-register a small normalized score basis: + +```text +S_p : thermal control; +H_2 : one genuine degree-2 residualized source; +H_3 : one local degree-3 residualized source with exact complement sign; +``` + +and compute the matrix + +```text +Cov(X,H_a), +Cov(A_shape,H_a,perp), +``` + +or the safe-phase Perron analogues. + +The first row tells which sources move the root; the second tells which change shape after thermal motion is removed. + +This is a direct lattice realization of the #61 RG-tangent programme without prematurely naming local CFT parity. + +## 11. Relation to original-U / Rao--Blackwell work + +The same algebra is the standard nuisance projection of an estimating equation. For an original-U influence function, replacing raw source content by its residual after projection onto the root/thermal nuisance is exactly the kind of object that conditional integration should target. + +Thus the current topological source programme and #578's Rao--Blackwell programme share a mathematical quotient, even though their physical observables are different. + +This does not change #275's frozen source contract. + +## 12. Claim boundary + +Exact finite probability identities: + +```text +T_g=-Cov(X,H)/Cov(X,S_p), +H_perp=H-beta_H S_p, +Cov(X,H_perp)=0, +d/dg E[A] along root = Cov(A,H_perp), +source gauge invariance under H->H+c S_p. +``` + +Programme: + +- use residualized microscopic source basis to estimate an RG tangent map; +- connect particular score blocks to continuum fields only after angular/radial and representation information is available. + +The central correction is methodological: **normalize first, quotient the thermal nuisance second, and only then interpret a source physically.** \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/root-tangential-normal-response-20260914.md b/docs/manuscripts/geometric-balance/root-tangential-normal-response-20260914.md new file mode 100644 index 000000000..70cf9f626 --- /dev/null +++ b/docs/manuscripts/geometric-balance/root-tangential-normal-response-20260914.md @@ -0,0 +1,375 @@ +> # ⚠️ 勘误与独立核验裁定（2026-09-14） +> **性质**：本勘误由独立核验代理 `verify802src` 追加，插在文件最开头；**正文自本区块之后起一字未动**，以保持原始交付可追溯。 +> +> **须更正四处**：`§3.1`「黑对/白对是两个独立非热源」；`§5.1` 的 `N̂` 两列数值、`φ≈2e-4`、以及「法向读出比切向大约 `1/φ≈5×10³` 倍」；`§5.2` 的 `N=0 ⟺ T=−1` 与「`N≈0` 而 `T≠0` 一阶不可能」；`§3.2` 的奇周长归因。 +> **纠正来源**（均为其他参与者独立提出，本代理独立复算确认，四条**一条也未能推翻**）： +> ① 分析方笔记 `docs/manuscripts/geometric-balance/two-observable-response-and-angular-alias-20260914.md` §7； +> ② 本代理核验报告 `docs/manuscripts/geometric-balance/audit-802-source-normalization-20260914.md`； +> ③ 机器可读证据 `results/research-dispatch/audit-802-source-normalization-20260914.json`（②③随本勘误一并推送）。 +> +> **1. `§3.1` 源冗余（逐配置精确）**：行内恒有 `H_W = N − 2K + H_B`（`K`=黑站点数，`H_B/H_W`=水平黑对/白对数，`N`=总站点数），故 `mu_(p,gH_W) = mu_(p_eff,gH_B)`、`logit(p_eff) = logit(p) − 2g` —— 两个「源」是**同一概率族的参数重写**，**不是两个独立非热源**。由此 `N_W = N_B`、`T_W − T_B = 2p(1−p)`。3×3/4×4/3×4 全枚举 **0 违例**；在本笔记**自己的满精度数据**上 `T_W − T_B = 0.48279634012494130`（w=8）对 `2p_c(1−p_c) = 0.48279634012493841`，**偏差 2.9e−15**（w=4..8 全部 ≤1.6e−15）。⇒ `§5.1`「两个非热源给出明显不同的像 ⇒ 可区分」**撤回**。 +> +> **2. `§5.1` 的 `N̂` 列漏了每行配分函数的 `γ`-导数**：引擎安全转移用的是**未正规化**行权 `p^k(1−p)^{w−k} exp(γh)`（见 `sector802_lib.build_matrices`），故物理衰减率为 `I_j = f − log λ̃_j`（`f = log Z_row`），于是 **`N̂_phys = N̂_raw + f_g`**，`f_g = w p_c²`（黑对）/ `w(1−p_c)²`（白对）。更正后（`w=4..8`，**两源逐位相同**，差 ≤1.5e−15）： +> `N̂ = 0.05534176915 / 0.04284515731 / 0.03483750063 / 0.02943453316 / 0.02550667528`；`w·N̂ ≈ 0.22137 / 0.21423 / 0.20903 / 0.20604 / 0.20405`（即 `O(1/w)`）。**原两列（黑 `−1.350050… −2.785276`、白 `−0.608081… −1.301340`）作废。** +> **不受影响**：`Δ`、`Δ'`、`S`、`S'`、S1 控制表、根位移表、尺度指数诊断（`−17/4`、`−1/4`、`−4`）—— 因为公共正规化在 `Δ` 中相消，且 `f(p,0) = f_p(p,0) = 0`。`T` 列**不变**（w=8：`T_black = −0.30954106`、`T_white = +0.17325528`）。 +> +> **3. 连带作废**：`§5.1`「两源 `N̂` 都显著非零、随 `w` 增长（`O(w)`）」与「`φ≈2e−4`，法向读出比切向大约 `1/φ≈5×10³` 倍」。以 `S_g^phys = S_g^raw + 2f_g` 重算后：黑 `φ = −5.00e−2 … −2.09e−2`、白 `φ = +2.55e−2 … +1.13e−2`，即 **`1/|φ| ≈ 20 … 88`**（且**符号与原值相反**）⇒「大约三个半数量级」不成立。同时 `§5.1`「为分离两个源再各跑一排宽度」**不再必要**（两源本就不可分离）。 +> +> **4. `§5.2` 逻辑错**：正确者为 **`N = 0 ⟺ (b_g,d_g) ∥ (b_p,d_p) ⟺ ∂_γ ∥ ∂_p ⟺ T = −a`**（`a` = 源沿 `p` 的等效位移）；**`T = −1` 只在「单位直接 `p` 平移」这一约定下成立**（本笔记 `thermal` 行即该约定，是**约定而非定理**）。故「`N≈0` 而 `T≠0` 一阶不可能」**及 `§6 V2` 同一句撤回**。（反例：`exp(γK)` 是纯 logit 平移 ⇒ `N=0`、`T=−p(1−p)`；3×3 环面实测 `T=−0.240`、`N=1.6e−10`。） +> +> **5. `§3.2` 归因错**：`w_0 = p^K (1−p)^{N−K}` 对**任意** `m`（含奇数）**逐配置平移不变**（全枚举违例恒为 **0**），故「奇长度破坏平移不变性」**错**。奇周期上非零的一阶信号来自**交替源自身在奇周期上不周期相容**（`(−1)^y` 绕圈有接缝：平移 1 给 `H_s(Tω) = −H_s(ω) + 2K_{m−1}`），因而 `⟨H_s⟩ = w p ≠ 0`（`m` 奇）；`m` 偶时 `H_s` 严格奇 ⇒ `b_g` 机器零。⇒「均值为零的场一阶为零 ⇒ 必须推到二阶」**只在偶周期**成立。 +> +> **未受影响**：`§0`–`§4`（除 `§3.1`/`§3.2` 上述两句）、`§5.3` 误差预算、`§6` 除 V2 同句外、`§7` 不能宣称清单、`§8` 交付物。 + +--- + + + +# #802 匹配根的切向—法向联合响应：交付记录 + +2026-09-14。代理 `sector802`（替代 `sectorA2`）。机器 `DevEnvC_NePnUn`（账号2）。 +工作目录 `/workspace/sectorA/`，引擎 `/workspace/sectorA/in/engine/tagged_winding_span.py`（pinned #739 引擎副本）。 +所有计算均在云机执行；本机只做读写/上传/下载/汇总。 + +**证据分级**（全文遵守）：**【精确】**=有理数/代数可证或有限枚举逐位；**【引用】**=照抄前人结论，未独立重算； +**【有限宽度】**=w=4..8 的 5 点或 4 个局部斜率，**不是**渐近指数测量；**【假设】**；**【猜想】**。 + +--- + +## 0. 接口口径（权威出处，逐条核对过） + +| 记号 | 定义 | 出处 | +|---|---|---| +| `I^0_{G,w}(p)` | 安全（无水平同调）转移阵 Perron 根 `λ^0`，`I^0 = -log λ^0` | `fixed-width-charge-free-energy-20260914.md` (1.5)；`fixed-width-charge-perron-slope-20260914.md` (1.6) | +| `Δ_w(p)` | `I^0_{4,w}(p) − I^0_{8,w}(1−p)` | #768 / #775 / #802 原文 | +| `b` | `b = (1/2) log(P0/P2)` | #773 / #802 原文 | +| `d`, `ell_c` | `d = log[P1/√(P0P2)]`；`ell_c = log[P1/(2√(P0P2))] = d − log2` | #773（`d`）；#802（`ell_c`） | +| 热力学识别 | `θ_{w,m}/m → Θ_w(p) = I^0_{4,w}(p) − I^0_{8,w}(1−p)`，`θ = log(P2/P0) = −2b` | #771「Root mechanism」**【引用，未独立重算】** | + +由 `θ = −2b` 与上式直接得到（纯代数）：**`b/m → −Δ_w/2`**；又因 `P1 → 1` 指数快， +`log P1/m → 0`，故 **`d/m → +S_w/2`**，其中 `S_w := I^0_{4,w}(p) + I^0_{8,w}(1−p)`。于是 + +``` +T_g = −b_g/b_p → −Δ_g / Δ'_w (m 抵消) +N_g = d_g − (d_p/b_p) b_g → m · (1/2)[ S_g − S'_w Δ_g / Δ'_w ] (逐行量 N̂ = N/m) +``` + +**`P_{r}` 是环面环境同调秩 `r ∈ {0,1,2}` 的概率**（#771/#773 三态坐标；`X = r−1`）。 +数字 Alexander 对偶 `r_4(ω) + r_8(ω^c) = 2` 给出**精确反射恒等式** +`I^0_{8,w}(1−p) = I^2_{4,w}(p)`，故 `Δ_w(p) = I^0_{4,w}(p) − I^2_{4,w}(p)`： +**同一张 black NN 图、同一密度下 rank-0 与 rank-2 的自由能差。** + +--- + +## 1. S1 硬门槛：控制表逐位复现 ✅ **通过** + +复现对象：`compass-snapshot-64159d56/charge-sector-scaling-hierarchy-20260914.md` §1 表（`w·I4` / `w·I8`，w=4..8）。 +口径（四种候选读法中唯一能对的）： + +``` +w·I4 = w · I^0_{4,w}(p_c) 安全 NN 转移在 **black 密度 p_c** +w·I8 = w · I^0_{8,w}(1−p_c) 安全 matching 转移在 **white 密度 1−p_c** +``` + +| w | `w·I4`(本次) | 差 | `w·I8`(本次) | 差 | 状态数 / 安全态 | +|---:|---:|---:|---:|---:|---:| +| 4 | 0.6808677452 | +4.06e-11 | 0.6677642268 | +6.39e-12 | 38 / 19 | +| 5 | 0.6702211440 | +4.20e-11 | 0.6643381257 | +3.03e-11 | 102 / 51 | +| 6 | 0.6651785592 | +4.87e-11 | 0.6620431213 | −3.25e-12 | 282 / 141 | +| 7 | 0.6622847725 | +1.16e-11 | 0.6604282152 | +2.86e-11 | 786 / 393 | +| 8 | 0.6604615126 | +3.49e-11 | 0.6592750557 | −4.29e-11 | 2214 / 1107 | + +**10 个控制数最大偏差 4.867e-11 < 1e-9，门槛通过。** +控制表只印到 10 位小数（四舍五入误差 ~5e-11），故这实际是**复现到该表全部打印精度**。 +Perron 残差 ≤1.5e-15；Collatz–Wielandt 括号宽度 ≤3.6e-14。 + +> ⚠️ 停止规则未触发。但记录一条**口径陷阱**（见 `issues-found.md` #1）：`w·I8` 是 matching 转移在 +> **white 密度 `1−p_c`** 处取值，不是 `p_c` 处。前任 `probe_control.py` 已经算出正确的数（其 +> `probe-control.json` 里 `A_I0_nonwind_1mpc` 乘 w 就是控制值），但没有把比较行读出即被停止； +> 本次是独立重算，**不是照抄**。 + +**附加控制（本次顺带复现，非任务要求）**：安全块内第二特征值给出的 +`g_{G,w} = w log(λ1/|λ2|)`：NN `7.0963 / 6.7081 / 6.5374 / 6.4492 / 6.3992`（w=4..8，**下降**趋 2π=6.28319）， +matching `5.5350 / 5.9794 / 6.1468 / 6.2190 / 6.2531`（**上升**趋 2π）。 +与 `charge-sector-scaling-hierarchy` §3 的 `Δx_relax = 1` 描述**相容**【有限宽度】。 + +--- + +## 2. S2 `Δ_w(p)` 与解析 `p`-导数 + +导数用 **Feynman–Hellmann 解析式**（§4），非有限差分。`p` 网格 `{0.50, 0.55, p_c, 0.63, 0.70}`， +`q = 1−p`。完整表见 `root-response.json → delta_table`；`p_c` 行： + +| w | `I^0_{4,w}(p_c)` | `dI^0_4/dp` | `I^0_{8,w}(1−p_c)` | `dI^0_8/dq` | **`Δ_w(p_c)`** | **`Δ'_w(p_c)`** | `S_w` | `S'_w` | +|---:|---:|---:|---:|---:|---:|---:|---:|---:| +| 4 | 0.17021694 | +1.224513 | 0.16694106 | +1.241435 | **+3.275880e-03** | **+2.465948** | +3.371580e-01 | −1.692145e-02 | +| 5 | 0.13404423 | +1.147793 | 0.13286763 | +1.158539 | **+1.176604e-03** | **+2.306332** | +2.669119e-01 | −1.074504e-02 | +| 6 | 0.11086309 | +1.091232 | 0.11034052 | +1.098195 | **+5.225730e-04** | **+2.189428** | +2.212036e-01 | −6.962797e-03 | +| 7 | 0.09461211 | +1.046655 | 0.09434689 | +1.051403 | **+2.652225e-04** | **+2.098058** | +1.889590e-01 | −4.747931e-03 | +| 8 | 0.08255769 | +1.010110 | 0.08240938 | +1.013487 | **+1.483071e-04** | **+2.023597** | +1.649671e-01 | −3.377157e-03 | + +**结构性事实（网格上直接可见，无需线性化）**：在 `[0.50, 0.70]` 上 `Δ_w(p)` 对 `p` **单调递增**， +且对每个 w=4..8 都有 `Δ_w(0.55) < 0 < Δ_w(p_c)`。故 + +* **`Δ_w(p)` 在拓扑消去后仍非零**（`p_c` 处全部 w 非零，见上表）； +* **`Δ_w` 的零点被夹在 `(0.55, p_c)` 内**，即固定宽度平衡根 `p*_w < p_c`； +* 用 `p_c` 处的值/导数作局部线性估计：`p*_w − p_c ≈ −Δ_w(p_c)/Δ'_w(p_c)`： + +| w | 4 | 5 | 6 | 7 | 8 | +|---|---:|---:|---:|---:|---:| +| `p*_w − p_c` | −1.328446e-03 | −5.101623e-04 | −2.386802e-04 | −1.264133e-04 | −7.328885e-05 | + +（区间宽 7.3e-5 而 `Δ''` 量级 O(1)，线性化在此尺度上安全；误差远小于表列位数。） + +### 2.1 尺度诊断【有限宽度，5 点，**不是**指数测量】 + +| 量 | 局部 log-斜率 (w=4→8) | `p + c/w` 外推 | 3 参 `C w^{-p}(1+D/w)` 拟合 | 文档声称 | +|---|---|---|---|---| +| `Δ_w(p_c)` | −4.589, −4.452, −4.400, −4.353 | **−3.99**（c=−2.62, rms 0.0123） | −3.967（rms_log 0.0057） | `−17/4 = −4.25` | +| `Δ'_w(p_c)` | −0.2999, −0.2853, −0.2765, −0.2706 | **−0.226**（c=−0.330, rms 5.5e-4） | −0.2222（rms_log 6e-5） | `−1/4 = −0.25` | +| 根偏移 `p*−p_c` | −4.289, −4.166, −4.123, −4.083 | **−3.77**（c=−2.29, rms 0.0118） | −3.683 | `−4`（#650/#768 的「n^{-4} 根移动」） | +| `S_w(p_c)` | −1.0470, −1.0302, −1.0221, −1.0169 | −0.970 | — | — | + +**怎么读这张表**（严格遵守任务 §4）： + +* `Δ'_w` 的指数在 **−0.30 到 −0.27 之间漂移**，外推 −0.226；文档的 `−1/4` **落在漂移区间内 ⇒ 相容**。 +* `Δ_w` 的指数在 **−4.59 到 −4.35 之间漂移**，外推 −3.99。文档的 `−4.25` **落在局部斜率区间内， + 但不落在外推值上** ⇒ **不能判定**：数据同时相容于 `−4` 与 `−4.25`，两者的差别在 4 个斜率上不可分辨。 +* 根偏移的指数漂移 −4.29 → −4.08，外推 −3.77；`−4` 在局部斜率区间内 ⇒ **相容，未认证**。 +* **`Δ_w` 与 `Δ'_w` 来自同一次转移矩阵生产（同网格、同解析导数），是一项数据生产的两条后处理， + 不是两条独立证据**（#801 原话）。根偏移 `−Δ/Δ'` 更是这两条的商，**第三条同源诊断**。 + +**独立路线的一致性**（可写，但要标强度）：`mlen218` 用**另一条路线**（rank 系数表 `M_L`，L=3..6） +得平衡根偏移指数 **−4.15**；本次用安全转移自由能（w=4..8）得局部斜率 −4.29..−4.08。 +两条路线方法不同、宽度不同 ⇒ 这是一次**弱一致性核对**，不是精度提升。 + +--- + +## 3. S3 真实源与条件评分 + +### 3.1 源的定义（具体、可复现、单一物理源同时作用于两个扇区） + +取同一个 Bernoulli site 测度 `w_0(p;ω) = p^K (1−p)^{N−K}`，用**明确的局部权重扰动**加一个 +**水平黑键化学势** `γ`： + +``` +H(ω) = Σ_y Σ_{j mod w} x_{j,y} · x_{j+1,y} （每行内循环相邻的黑占据对总数） +w(p,γ;ω) ∝ w_0(p;ω) · exp(γ H(ω)) +``` + +这是「明确定义的局部权重扰动」（#802 允许的源类别），且**对黑白两个扇区是同一个源**： +black G4 转移里 mask = 黑占据集，源因子 `e^{γ·h(mask)}`； +white G8 转移里 mask = 白占据集（连通用 matching 规则），源因子 `e^{γ·h(complement(mask))}`。 +另设一个对照源 `H'(ω) = 白相邻对` 作为**第二个**非热源，用于「两个候选像是否相同」的对比。 + +实现方式：把安全转移的每条边按 `(j, k, h)` 三元组聚合计数（`k = |mask|`，`h` = 源标记）， +于是 `R(p,γ)`、`∂R/∂p`（解析）、`∂R/∂γ`（解析）对任意 `(p,γ)` 都是**精确**求值，无需重跑自动机。 + +### 3.2 条件源评分 —— 有限环面上**逐位验证**【精确（有限枚举）】 + +`#802` 点 3 给的式子 + +``` +b_g = (E[H|0] − E[H|2]) / 2 +ell_c,g = E[H|1] − ( E[H|0] + E[H|2] ) / 2 +``` + +在小环面（`w,m ∈ {3,4}`，最多 4×4=65536 配置）上**全枚举**验证： + +* 秩算法：BFS 生成森林带 `Z²` 势，`r = dim_Q span{每个基本圈绕数向量}`（取值 0/1/2）。 +* **Alexander 正则性**：222720 次检查，**0 违例**。 +* **反射恒等式**：`P8_p(k) = P4_p(2−k)`（两者都以 **black 密度 p** 标号），12 个案例 **最大绝对差 0.000e+00**。 +* **转置对称**：`P4(w,m) == P4(m,w)`，最大差 3.3e-16。 +* **条件评分公式**：与 `b(γ), ell_c(γ)` 的 Richardson 中心差分比较 + （黑对源）`max|b_g(FD) − b_g(公式)| = 4.5e-10`、`max|ell_c,g(FD) − ell_c,g(公式)| = 3.6e-10`（FD 截断噪声量级）。 + ⇒ **公式与实现互证。** +* **均值零场的陷阱（#802 警告的那条）**：取交错行场 `H_s = Σ_y (−1)^y K_y`。 + - 在 **m 为偶**（4×4、3×4 不存在，见下）的环面上，一阶响应**恒为 0**： + `b_g = −2.7e-15 … +6.3e-15`，`ell_c,g = −1.4e-14 … −2.8e-15`（机器零）。这是 + 「平移不变 ⇒ 一阶响应为零」的**精确推论**，不是巧合。 + - 在 **m 为奇**（3×3、4×3）的环面上不为零（`b_g ≈ −0.49 … −0.64`）：因为 `Σ_y(−1)^y ≠ 0`， + 周期方向上的平移不变性被奇长度破坏。 + ⇒ **结论**：平均零的列/行场**在平移不变背景下一阶必为零**，想用它区分自旋选择**必须**推到二阶， + 否则会把「对称性强制的零」误读成「该候选被排除」。 + +--- + +## 4. S4 Feynman–Hellmann 与闭合幅度 + +对安全块（nonnegative，Perron 根简单），`l^T r = 1`： + +``` +∂_g log λ_j = l_j^T (∂_g T_j) r_j / λ_j ⇒ dI^0/dγ = −(l^T (∂R/∂γ) r) / (ρ · l^T r) +``` + +**实现正确性的独立证书（精确代数恒等式）**：用「行内黑占据数」标记 `γ_pop`（G4 时 = `k`，G8 时 = `w−k`）， +可证 + +``` +G4: dI0/dx = +( dI0/dγ_pop + w x ) / ( x(1−x) ) +G8: dI0/dx = −( dI0/dγ_pop + w(1−x) ) / ( x(1−x) ) +``` + +这是 `x(1−x) d/dx[x^k(1−x)^{w−k}] = (k − wx) x^k(1−x)^{w−k}` 的直接推论。 +**在 w=4..8 的全部网格点、两张图上最大相对偏差 4.13e-13。** 这检验的是 **FH 实现**（而非物理）。 +另有 Richardson 中心差分对照：`Δ'_w` 的解析值 vs FD 最大相对差 **5.6e-12**（检查用，不作证书）。 + +> **闭合幅度：本次没有计算。** `P_r ≈ A_r e^{−m I^r}` 里的 `A_r`（contact/seam/归一化常数） +> 进入 `b` 和 `d` 的 O(1) 项，因而把根位置与形状响应移动 O(1/m)。本次只做首阶（m→∞）识别。 +> 要闭合必须显式算 `A_0/A_2` 与 `A_1`，**这是一个明确的缺口**。 + +--- + +## 5. S5 `(T, N)` 联合预测与误差预算 + +### 5.1 联合预测（`T_g = −Δ_g/Δ'_w`；`N̂_g = N_g/m`，`N_g` 是 §0 原文接口量，对 m 线性） + +| 源 | w | `Δ_g` | `S_g` | **`T_g`** | **`N̂_g`(逐行)** | `N_g/S_g` | **`φ_g`** | +|---|---:|---:|---:|---:|---:|---:|---:| +| **thermal**（`H=K`） | 4..8 | `=Δ'_w` | `=S'_w` | **−1.00000000** | **+0.00000000** | 1 | **1** | +| **black-pair** `γ` | 4 | +7.685985e-01 | −2.705374 | −0.31168482 | −1.350050 | 0.49919 | +1.950e-03 | +| | 5 | +7.156537e-01 | −3.431123 | −0.31029950 | −1.713894 | 0.49943 | +9.717e-04 | +| | 6 | +6.782987e-01 | −4.148657 | −0.30980639 | −2.073250 | 0.49980 | +5.200e-04 | +| | 7 | +6.496076e-01 | −4.861471 | −0.30962331 | −2.430001 | 0.50005 | +3.024e-04 | +| | **8** | **+6.263864e-01** | **−5.571598** | **−0.30954106** | **−2.785276** | 0.50048 | **+1.876e-04** | +| **white-pair** `γ'` | 4 | −4.219521e-01 | −1.213267 | +0.17111152 | −6.080813e-01 | 0.34778 | −2.386e-03 | +| | 5 | −3.978350e-01 | −1.571014 | +0.17249684 | −7.864337e-01 | 0.35324 | −1.180e-03 | +| | 6 | −3.787490e-01 | −1.919390 | +0.17298995 | −9.602972e-01 | 0.35752 | −6.275e-04 | +| | 7 | −3.633270e-01 | −2.262290 | +0.17317303 | −1.131556 | 0.36118 | −3.634e-04 | +| | **8** | **−3.505989e-01** | **−2.602094** | **+0.17325528** | **−1.301340** | 0.36432 | **−2.249e-04** | + +`φ_g := (S'_w Δ_g)/(Δ'_w S_g)` = 「形状漂移中被热重新调定解释掉的份额」。 + +* `T` 对黑对源收敛到 **−0.30954**，对白对源收敛到 **+0.17326**（逐步稳定到 ~1e-3，**相容于收敛**）。 +* `N̂` 两个源都**显著非零**，且随 w 增长（源本身逐行 O(w) 强，故 `N̂ = O(w)` 是约定效应， + 不是病态）。**所以「`N_g ≈ 0` 且 `T_g ≠ 0`」在这一批源上不出现。** +* `φ` 极小（~2e-4，且随 w 下降）：形状漂移 ~99.98% 是**扇区偶（common）分量**， + 携带根移动信息的是扇区奇分量 `Δ`。这正是「内禀曲线对切向盲」的**定量版本**： + 法向读出比切向（根移动）部分大约 `1/φ ≈ 5×10³` 倍。 + +### 5.2 关于 `N=0` 的结构性结论（一阶，精确） + +`N = 0 ⟺ (b_g, d_g) ∥ (b_p, d_p) ⟺ g` 与 `∂_p` 平行 `⟺ T = −1`。 +即：**在 `(b,d)` 坐标（概率单纯形的一个微分同胚）下，一阶「法向响应为零」与「纯热重新调定」是同一件事**。 +因此题面 V2 想要的「`N≈0` 而 `T≠0`」**在线性阶上不可能**；真正的盲点内容是 §5.1 的**倍数比 `φ`**。 +（这只在 `(b,d)` 的一阶框架内成立；`φ` 的 w-依赖性用了有限宽度数据。【有限宽度】） + +### 5.3 误差预算（三类分开） + +| 类别 | 量 | 值 | +|---|---|---| +| **数值** | Perron 特征对相对残差（w=4..8，两图） | ≤ **1.5e-15** | +| | Collatz–Wielandt 括号宽度 | ≤ 3.6e-14 | +| | FH 解析导数 vs Richardson 中心差分（`Δ'_w`） | rel ≤ **5.6e-12** | +| | FH 实现 vs 精确代数恒等式 | rel ≤ **4.13e-13** | +| | 控制表复现偏差 | ≤ 4.867e-11（受控制表 10 位打印限制） | +| **采样** | — | **恒为 0**：全程精确转移矩阵 + 有限环面全枚举，无 Monte Carlo | +| **有限宽度（系统性）** | `Δ_w` 指数局部斜率漂移 | −4.589 → −4.353（每步 ~0.05） | +| | `Δ'_w` 指数局部斜率漂移 | −0.2999 → −0.2706 | +| | 外推值与漂移区间的差距 | `Δ`: 外推 −3.99 vs 区间 −4.59..−4.35（**差 0.36**） | +| | `T` 的 w 稳定性 | 黑对 −0.31168→−0.30954（变化 6.9e-3） | + +**若目标函数有下界 `d0 > 0`**：本题 `|Δ'_w| ≈ 2.02…2.47`，故 `|root error| ≤ ε_Δ / d0` +以 `ε_Δ ≈ 5e-11`（控制级）算得 **≤ 2.5e-11**，即根位移认证到 ~1e-11 —— 比 `Δ'_w` 自身的数值误差大 10 倍余量， +说明**认证扇区差 `Δ_w` 比认证两个大本征值各自的精度更划算**（#802 点 4 的原话）。 + +--- + +## 6. 裁定 V1–V4 + +### V1 `Δ_w(p)` 是否在拓扑消去后仍非零、且在 `p_c` 邻域有非零 p-导数？—— **是，两者都是。** + +* `Δ_w(p_c)`: **+3.275880e-03 / +1.176604e-03 / +5.225730e-04 / +2.652225e-04 / +1.483071e-04**（w=4..8），全部非零。 + 用有限宽度看其 w-趋势：局部斜率 −4.59…−4.35【有限宽度】。 +* `Δ'_w(p_c)`: **+2.4659 / +2.3063 / +2.1894 / +2.0981 / +2.0236**，全部非零，且**符号恒正**； + 局部斜率 −0.30…−0.27【有限宽度】。 +* **额外（本次直接得到的）**：`Δ_w(p)` 在 `[0.50,0.70]` 上单调递增，零点夹在 `(0.55, p_c)` 内， + 即**固定宽度平衡根 `p*_w` 从下方收敛到 `p_c`**，位移量级 `−7.3e-5 (w=8) … −1.3e-3 (w=4)`， + 指数漂移 −4.29…−4.08、外推 −3.77 ⇒ **与「w^{-4} 根移动」相容，未认证**。 + +### V2 同一实际源 `g` 下 `(T_g, N_g)` 的联合预测？`N_g` 是否显著非零？ + +* **纯热源**（`H = K`）：`(T, N̂) = (−1, 0)` **精确**（数值上 `N̂` 逐位为 0，`φ = 1`）。 + 这是**「纯热漂移」这个允许的成功结果**的一个实例，而且是精确恒等式，不是拟合。 +* **两个非热源**（黑相邻对 / 白相邻对化学势）：`T = −0.30954 / +0.17326`（w=8，w-稳定）， + `N̂ = −2.785 / −1.301`。**`N_g` 显著非零。** +* **但「`N≈0` 且 `T≠0`」在一阶不可能**（§5.2）：法向为零 ⟺ 源恰为热向 ⟺ `T=−1`。 + 题面设想的那个「切向盲点实例」在 `(b,d)` 一阶框架里退化；盲点真正的内容是 §5.1 的 `φ ≈ 2e-4` + —— **内禀（法向）读出比根移动（切向）分量大约三个半数量级**，这才是「`O_L(b)` 小不能排除根机制」的定量含义。 + +### V3 哪些候选的 `(T,N)` 预测**像不相同**（可区分）？哪些只剩自由幅度且像相同？ + +诚实回答，分两层： + +**(a) 我能真正实现的源之间：可区分。** +* 一切**纯热方向**的源（任何在平移不变背景下只改变黑/白密度的均一扰动；以及偶长环面上任何**均值为零**的列/行场，已证一阶为零） + 的 `(T,N)` 像**坍缩为同一个点 `(−1, 0)`** ⇒ **彼此完全不可区分，只剩自由幅度** + （`|T|=1` 固定，`N=0` 固定，没有任何残余形状信息）。 +* 两个非热源给出**明显不同**的像：`(T,N̂) = (−0.3095, −2.785)` vs `(+0.1733, −1.301)`（w=8）， + 且 `T` 的符号都不同 ⇒ **可区分**。 + +**(b) 题面点名的 CFT 标签（scalar 8-arm / thermal spin±4 / 低阶竞争者）：现有读出不能区分。** +* 本次**没有**为任何 CFT 候选算出实际矩阵元，因此**没有它们的 `(T,N)` 预测**； + 能给的只是**约束的形状**：任何「领先修正严格平行于热切向」的候选，其 `(T,N)` 像恒为 `(−1,0)`， + 与其它同类候选**完全重合**；只有法向投影非零的候选才落在别处。 +* 要把 8-arm 与 spin±4 分开，缺的是**同一个物理源下、两个扇区中 level-4 非导数类（或 6-arm / identity 族）的差矩阵元** + —— 那正是 #768 的模块计算，不在本次范围。 +* **警告（已用数值证明）**：不能用均值为零的列/行场去「挑自旋」，因为它在平移不变背景上一阶恒为零（§3.2）。 + 也**不能**用尺寸幂相同的标签表充当区分（#802 原文）。 + +### V4 现有读出能否分离这些候选？ + +**对「源」这一层：能**（两个非热源给出不同的 `(T,N)`，且 `T` 的符号不同）。 +**对「CFT 标签」这一层：不能**，如实写。缺的东西是明确的： + +1. `P_r` 的 **闭合幅度** `A_r`（contact/seam/归一化）—— 本次未算，它把根与形状移动 O(1/m)； +2. 两个扇区中 **level-4 非导数类（以及 6-arm / identity 家族）的实际差矩阵元** —— #768 的任务； +3. 若要判 `−17/4`，需要**更多宽度**（本次 5 个宽度只给 4 个局部斜率，见 §2.1）， + 而 #650/#WORK-NOW 明确取消了 L5–8 自动阶梯 ⇒ **本次按规矩不追加宽度**。 + +--- + +## 7. 不能宣称什么（红线自查） + +1. **不是 critical exponent measurement。** `Δ_w` 的 `−17/4`、`Δ'_w` 的 `−1/4`、根偏移的 `−4` + 都只有 **5 个宽度 / 4 个局部斜率 / 2 种外推**，且外推与局部斜率区间不一致（§2.1）。只能写「相容」。 +2. **`Δ_w`、`Δ'_w`、根偏移是同一项数据生产的三条后处理**，不是三份独立证据（#801）。 +3. **`b/m → −Δ_w/2`、`d/m → +S_w/2` 的热力学识别引自 #771**（【引用，未独立重算】）。 + 本次独立验证的只有它依赖的**反射恒等式**（有限环面，逐位）。 +4. **S1 是复现别人的控制表**，不是新物理；且只认证到该表的 10 位打印精度。 +5. **`(T,N)` 是一阶（线性响应）**；contact 项与高阶导数本次**没有**处理（它们在二阶出现，§4）。 +6. **源 `γ` 是模型内部的权重扰动**（水平黑键化学势），不是实验可调参数，也不等于 #275 的 original-U 源。 + #275 的源、正规化、moving-root 合同**未改动**。 +7. **小环面暴力枚举验证的是「条件评分公式」**（精确代数 + 有限枚举），**不是** m→∞ 识别，也不是物理。 +8. **`P_r` 的具体几何**（长环面）本次只通过 #771 的引文使用；本次未独立构造 rank-跟踪转移阵。 +9. 未做 L5–8 阶梯，未靠尺寸幂或内禀 `C_L(b)` 判机制（遵守 #650/#768/#802）。 +10. 未改任何原件（STATUS / pc claim / #275 U 合同 / 仓库写操作一概未做；GitHub 只读）。 + +--- + +## 8. 交付物与复现 + +云端 `/workspace/sectorA/`： + +| 文件 | 内容 | +|---|---| +| `scripts/sector802_lib.py` | 安全转移自动机、(k,h) 聚合、FH Perron、CW 括号 | +| `scripts/s1_control.py` | S1 控制表复现（硬门槛） | +| `scripts/s2_s3_response.py` | `Δ_w(p)` 表 + 解析导数 + 三个源的 `(T,N)` | +| `scripts/s4_torus_bruteforce.py` | 小环面全枚举：Alexander / 反射 / 条件评分公式 / 交错场 | +| `scripts/s6_finalize.py` | 后处理、尺度拟合、误差预算、裁定汇总 | +| `scripts/s7_telemetry.py` | 状态数 / 峰值 RSS / 墙钟 / 残差 | +| `out/control-reproduce.json` | S1 逐位对照 | +| `out/root-response-raw.json` | 原始 `(w,p,mark)` 全量 | +| `out/root-response.json` | **主交付（机器可读）** | +| `out/torus-bruteforce.json` | 环面验证 | +| `out/telemetry.json` | 成本实测 | +| `out/s1.log s2.log s4.log s7.log` | 运行日志 | + +**telemetry（实测）**：w=8 单图峰值 RSS **312 MB**、建自动机 **18.7 s**、特征分解 **~1 s**； +安全态 1107（总态 2214），安全转移 265565（G4）/ 204063（G8）。 +全流程（S1+S2/S3+S6）在 16 vCPU / 32 GiB 上合计 < 6 分钟墙钟（OPENBLAS_NUM_THREADS=4）。 +环境：`PYTHONPATH=/workspace/mo/compat`（Py3.9 `int.bit_count` 垫片，自检 OK），numpy 2.0.2 / scipy 1.13.1。 + +**未复现已验证的对拍**：扇区引擎 vs 暴力枚举（`out/validate-bruteforce.json`，`all_ok=true`，diff ~1e-16） +由前任 `sectorA` 完成，本次**引用**不重做（任务 §0 明确）。 diff --git a/docs/manuscripts/geometric-balance/rotational-symmetry-ladder-for-charge-shifts-20260914.md b/docs/manuscripts/geometric-balance/rotational-symmetry-ladder-for-charge-shifts-20260914.md new file mode 100644 index 000000000..ec1b82bb2 --- /dev/null +++ b/docs/manuscripts/geometric-balance/rotational-symmetry-ladder-for-charge-shifts-20260914.md @@ -0,0 +1,286 @@ +# A rotational/Kac-module selection rule for semi-infinite charge-root shifts + +2026-09-14. Corrected conjectural synthesis for the sector-odd charge correction. This note replaces the earlier naive rule `s_*=lcm(2,k)`: point-group symmetry alone is not enough. One must also remove descendants that are null or redundant in the percolation thermal conformal family. + +The correction was forced by the level-two null vector of the thermal field `phi_{2,1}` and by the new same-model oblique-cylinder spin-four controls. + +## 1. Charge-root shift as a sector-odd thermal-family anisotropy + +Assume the leading primal/matching sector difference at criticality comes from a nonredundant chiral descendant of the thermal primary. Percolation has + +\[ +h_t=\bar h_t=5/8, +\qquad x_t=5/4. \tag{1.1} +\] + +A chiral descendant at level `s` and its reflected anti-chiral partner have total dimension + +\[ +x_{odd}=x_t+s \tag{1.2} +\] + +and spins `+s,-s`. A real lattice perturbation uses the reflection-even combination and hence produces an angular harmonic such as `cos(s theta)`. + +The critical sector mismatch scales as + +\[ +\Theta_w(p_c)\asymp w^{1-(x_t+s)}, \tag{1.3} +\] + +while the thermal slope scales as + +\[ +\Theta'_w(p_c)\asymp w^{1-x_t}. \tag{1.4} +\] + +Therefore a surviving level-`s` anisotropy gives + +\[ +\boxed{p_w-p_c\asymp w^{-s}.} \tag{1.5} +\] + +The exponent is a descendant **level/spin only after null and redundant states have been removed**. + +## 2. The thermal level-two state is not an independent spin-two field + +The thermal primary is the degenerate Kac field + +\[ +\phi_t=\phi_{2,1}. \tag{2.1} +\] + +Its level-two singular-vector relation is + +\[ +\left( +L_{-2}-\frac{3}{2(2h_t+1)}L_{-1}^2 +\right)|t\rangle=0. \tag{2.2} +\] + +For `h_t=5/8`, + +\[ +\boxed{L_{-2}|t\rangle=\frac23L_{-1}^2|t\rangle.} \tag{2.3} +\] + +Thus after quotienting by the null state there is no independent level-two quasiprimary. The would-be spin-two thermal descendant is only a derivative/coordinate redundancy. + +This removes the previous proposed `C2 -> Delta=2` mechanism. + +It also aligns with the geometric warning already present in the first version of this note: ordinary spin-two anisotropy mostly changes the continuum metric. The Virasoro null relation shows more sharply why there is no independent thermal-family spin-two correction to promote after metric calibration. + +Primary null-vector references are standard Virasoro/Kac theory; the generic `phi_{2,1}` level-two relation is also used in BPZ differential-equation derivations. + +## 3. First nonredundant low levels of the `phi_{2,1}` quotient + +At a purely counting level, the Verma dimensions at levels `n=0,1,2,3,4` are + +\[ +1,1,2,3,5. \tag{3.1} +\] + +Removing the level-two null module subtracts the partition numbers at level `n-2`, leaving + +\[ +1,1,1,2,3. \tag{3.2} +\] + +Modulo total `L_{-1}` derivatives, the first new quasiprimary content is therefore + +- no independent level-two quasiprimary; +- one level-three quasiprimary; +- one new level-four quasiprimary. + +This is the representation-theoretic reason the point-group rule must start from the **actual thermal Kac module**, not from all integer spins. + +A full logarithmic-module treatment may refine higher levels; only the low-level null structure needed for the selection statements below is used here. + +## 4. Corrected point-group ladder + +Let `C_k` be the microscopic rotational subgroup. The candidate level must both + +1. be invariant under the point group, `s=0 mod k`; +2. exist as a nonnull/nonredundant thermal-family quasiprimary. + +Reflection does **not** by itself exclude odd spin: it exchanges the `+s` and `-s` chiral descendants, and their real sum is reflection even. + +For the low symmetries relevant here this gives + +| microscopic rotational symmetry | first thermal-family candidate | predicted charge-root shift | +|---|---:|---:| +| no nontrivial rotation (`C1`, reflection allowed) | spin 3 | `w^-3` | +| `C2` / `D2` | spin 4 | `w^-4` | +| `C3` / `D3` without 60-degree rotation | spin 3 | `w^-3` | +| `C4` / `D4` | spin 4 | `w^-4` | +| `C6` / `D6` | spin 6 | `w^-6` | + +Exact self-matching/self-duality can annihilate the complete sector-odd amplitude and override the table. + +The old claim `C2 -> w^-2` is withdrawn. + +## 5. Square site: exponent, sign and angular amplitude now all resolve spin four + +The square lattice has `C4`. The axial safe transfer gives + +\[ +p_w^{axis}-p_c\sim-\frac{A}{w^4}, +\qquad A\approx0.30. \tag{5.1} +\] + +The new oblique safe transfers keep the microscopic model fixed and rotate only the homology direction. If the correction has spin four, the leading law is + +\[ +\boxed{ +p_{n,u}^{ch}-p_c +\sim +-\frac{A\cos(4\theta_u)}{(n|u|)^4}.} \tag{5.2} +\] + +This prediction is supported quantitatively by several independent directions. + +### Axis `(1,0)` + +`cos(4theta)=+1`. Widths 8--9 give + +\[ +A_{est}=0.30020,\ 0.29804. \tag{5.3} +\] + +### Diagonal `(1,1)` + +`cos(4theta)=-1`; the root moves to the **opposite side** of `p_c`. At `n=5`, after using physical circumference `ell=5sqrt2`, + +\[ +A_{est}=0.29916. \tag{5.4} +\] + +### Direction `(2,1)` + +\[ +\cos4\theta=-7/25=-0.28. \tag{5.5} +\] + +At `n=4`, + +\[ +A_{est}=0.30077. \tag{5.6} +\] + +### Direction `(3,2)` + +\[ +\cos4\theta=-0.7041420118\ldots \tag{5.7} +\] + +and already at `n=2`, + +\[ +A_{est}=0.29956. \tag{5.8} +\] + +These are much more discriminating than an exponent-four fit: a scalar field of dimension `x_t+4` would not produce the observed orientation sign/magnitude law. + +Data and scripts: + +- `diagonal-spin4-charge-root-20260914.md`; +- `scripts/diagonal_charge_transfer.py`; +- `scripts/oblique_charge_transfer.py`; +- `results/geometric-consistency/oblique-spin4-angular-controls-20260914.json`. + +## 6. Kagome/hexagonal symmetry remains a spin-six control + +Jacobsen's kagome bond eigenvalue sequence has the square-like exponent-four amplitude absent and a leading shift compatible with exponent six. The kagome/hexagonal bulk symmetry includes 60-degree rotation, which forbids spin four and allows spin six. + +This remains consistent with the corrected Kac-module rule: + +\[ +C_6:\quad s_*=6. \tag{6.1} +\] + +The triangular-site self-matching model is stronger still: complement symmetry fixes the charge root to `1/2` at every width, so every sector-odd amplitude vanishes. + +## 7. A new distinction: `C3` is not `C6` + +The old table bundled three-fold symmetry plus reflection with six-fold symmetry. That is not generally justified. + +A spin-three pair transforms trivially under a `120 degree` rotation: + +\[ +e^{\pm i3(2\pi/3)}=1, \tag{7.1} +\] + +and the real combination is reflection even. Since a nontrivial level-three thermal quasiprimary survives the level-two null quotient, a genuine `D3` lattice without 60-degree rotation can in principle have + +\[ +\boxed{p_w-p_c\asymp w^{-3}.} \tag{7.2} +\] + +A `D6` lattice forbids it and first admits spin six. + +This gives a sharper future lattice-symmetry test than the earlier `C2` proposal. + +## 8. Pell directions create a same-model spin-four null experiment + +The square harmonic factorizes arithmetically: + +\[ +a^4-6a^2b^2+b^4 +=(a^2-2ab-b^2)(a^2+2ab-b^2). \tag{8.1} +\] + +The exact spin-four zero direction is + +\[ +\frac ab=1+\sqrt2, +\qquad \theta=\pi/8. \tag{8.2} +\] + +Choose primitive Pell approximants satisfying + +\[ +a^2-2ab-b^2=\pm1. \tag{8.3} +\] + +Then + +\[ +\cos4\theta=O(|u|^{-2}). \tag{8.4} +\] + +Consequently the nominal `ell^-4` spin-four root shift is geometrically suppressed to order `ell^-6` along this sequence. The two Pell signs flip the residual spin-four contribution. + +This suggests a powerful same-model experiment: + +- average the `+1` and `-1` Pell subsequences to expose a genuine spin-six term; +- difference them to isolate the residual spin-four anisotropy. + +The existing `(5,2)` control already lies near this null direction (`cos4theta=41/841`) and its charge root is within about `10^-6` of the reference `p_c` at only `n=2`, despite physical circumference about `10.77`. + +A wider Pell computation is a targeted follow-up, not a generic angle scan. + +## 9. Revised general rule + +The correct conjectural selection principle is + +\[ +\boxed{ +\Delta_{charge}=s_*, +\quad +s_*=\min\{s>0:\ s\text{ is point-group allowed and is a nonredundant thermal-family quasiprimary level}\}.} \tag{9.1} +\] + +This is stronger and safer than the old arithmetic rule `lcm(2,k)`. + +It separates three mechanisms that must not be conflated: + +1. continuum metric/stress-tensor anisotropy; +2. Virasoro-null or total-derivative descendants; +3. genuine sector-odd thermal-family anisotropy. + +Only the third sets the charge-root shift after the first two are removed. + +## 10. Claim boundary + +The `phi_{2,1}` level-two null vector is standard CFT structure. The oblique charge roots are deterministic lattice calculations. Point-group invariance of a spin harmonic is exact. + +The identification of the leading charge correction with a thermal-family quasiprimary, the higher-level logarithmic-module content, the `D3 -> 3` and Pell spin-six separation predictions remain conjectural. The previous `C2 -> 2` prediction is explicitly superseded by this corrected note. diff --git a/docs/manuscripts/geometric-balance/round2-claim-ledger-20260914.md b/docs/manuscripts/geometric-balance/round2-claim-ledger-20260914.md new file mode 100644 index 000000000..ddc0bbcda --- /dev/null +++ b/docs/manuscripts/geometric-balance/round2-claim-ledger-20260914.md @@ -0,0 +1,260 @@ +# 2026-09-14 continuation claim ledger + +Status ledger for stacked Draft PR #771. The purpose is to keep exact topology, author-level probability proofs, conditional consequences and research conjectures visibly separate while the branch is still moving. + +This ledger covers the main claims produced in the geometric-balance continuation. Other contemporaneous branch files should be reviewed on their own terms if they are not named here. + +## Status vocabulary + +- **EXACT**: finite deterministic/probability identity, or elementary finite-state/convex consequence with no asymptotic model input beyond stated definitions. +- **AUTHOR-PROOF**: a complete proof is written on the branch, but it uses published inputs and/or author-level results from #739 and still needs independent proof review before publication. +- **CONDITIONAL**: deduction is complete once one explicitly named hypothesis/theorem interface is granted. +- **CONJECTURE / PROGRAMME**: falsifiable research mechanism or proposed proof route, not promoted as a theorem. +- **LITERATURE BOUNDARY**: audited statement about what an external theorem does and does not imply for square-site Matching-One. + +## A. Exact finite topology and persistent birth structure + +| Claim | Status | Main file | Consequence | +|---|---|---|---| +| Digital-Alexander birth reflection `T1_8(1-U)=1-T2_4(U)`, `T2_8(1-U)=1-T1_4(U)` | EXACT | `structural-consequences-20260914.md` | Entire 4/8 birth process is paired samplewise, not only in law. | +| In rank one, complementary NN/matching homology lines coincide | EXACT | `projective-homology-gas-20260914.md` | Same projective slope on the two colours. | +| Rank-one slope is born at `T1` and remains frozen until `T2` | EXACT | `persistent-slope-marked-birth-20260914.md` | Direction is a persistent first-birth mark; it cannot rotate during the plateau. | +| `(G,L,C)_8 =d (G,L,-C)_4` with `G=T2-T1`, `C=(T1+T2-1)/2` | EXACT | `persistent-slope-marked-birth-20260914.md` | NN/matching share the full plateau-width/slope law; only the centre coordinate is complement-odd. | +| Same-parameter count support is only `(0,1)`, `(1,0)`, `(K,K)` | EXACT | `structural-consequences-20260914.md` | Generic bivariate Poisson/copula models are inadmissible. | +| Rank-one state is `(slope,K)` and different slope species have global hard-core exclusion | EXACT | `projective-poisson-hardcore-crossover-20260914.md` | Multi-direction models must enforce torus intersection topology. | +| Neutral count fluctuations are one-dimensional; `W4-W8` is bounded | EXACT | `neutral-gas-limit-collapse-20260914.md` | Every growing-scale joint fluctuation limit lies on the diagonal; extensive LDP rate is `+infinity` off diagonal. | +| Finite reflection dominance `1-T2 <=st T1` | EXACT | `finite-reflection-dominance-20260914.md` | `P2(1-p)<=P0(p)`, `M(p)+M(1-p)<=0`, finite root `>=1/2`. | + +Finite controls: `scripts/persistent_alexander_birth_reflection.py` exhausts all `L=3` configurations and all `9!` strict orders; at `L=4` it exhausts configurations and checks 20,000 fixed-seed orders, with zero violations of the recorded identities. + +## B. Exact common-window charge coordinates + +| Claim | Status | Main file | Consequence | +|---|---|---|---| +| `chi=P0+P2`, `theta=log(P2/P0)`, `M=chi tanh(theta/2)` | EXACT | `charge-neutral-crossover-coordinates-20260914.md` | Root location and charged-sector rarity are separated coordinates. | +| Matching root is exactly `theta=0` | EXACT | same | Common-window modelling should evolve a charge fugacity, not two independent colour means. | +| Joint PGF `G(s,t)=chi[e^{theta/2}s+e^{-theta/2}t]/(2cosh(theta/2))+(1-chi)H(st)` | EXACT | same | Minimal same-parameter state is one scalar susceptibility, one scalar fugacity, and one positive-integer neutral count law. | +| At the root, `M'=chi theta'/2`, `F'=chi theta'/4` | EXACT | same | Algebraic mechanism for balance without concentration. | +| `chi_8(p)=chi_4(1-p)`, `theta_8(p)=-theta_4(1-p)`, neutral law transported unchanged | EXACT | `charge-fugacity-reflection-20260914.md` | Exact 4/8 involution in crossover coordinates. | +| Graph inclusion gives `theta_8(p)>=theta_4(p)`, hence `theta_4(p)+theta_4(1-p)<=0` | EXACT | same | Strong charge-field reflection constraint on any scaling ansatz. | + +## C. First-exit / Wulff geometry + +| Claim | Status | Main file | Review dependency | +|---|---|---|---| +| `B_S(t)<1` certifies finite exponential susceptibility | AUTHOR-PROOF | `vector-first-exit-domain-20260914.md` | Site-BK first-exit skeleton. | +| True exponential-moment domain is the interior of the polar/Wulff body; large finite boxes exhaust compact interiors | AUTHOR-PROOF | same | Standard uniform directional subcritical exponential estimate. | +| Certified bodies may be convex-hulled; support function, not radial intercept, equals `tau` | EXACT / AUTHOR-PROOF | same | Convexity is exact; domain equality uses the preceding input. | +| Torus winding upper bound `P(r>0)<=N C_T sum_{lambda!=0} exp[-h_T(lambda)]` | AUTHOR-PROOF | `first-exit-torus-winding-upper-20260914.md` | Canonical minimum homology witness is a quotient-vertex-simple cycle; local first-exit witnesses are disjoint SITE variables. | +| With `T=(1-eps)K_p`, full-period bound uses complete period cost `(1-eps)tau_p(lambda)` | AUTHOR-PROOF | same | Depends on Wulff-domain exhaustion. | +| Correlation-norm successive minima organize rank/slope suppression | AUTHOR-PROOF | `correlation-norm-successive-minima-20260914.md` | Deterministic norm-ball packing + torus upper bound. | + +Important correction preserved in the branch: a direct global comparison of two quotient cycle arcs with two independent full-plane connection events is **not** used; periodic lifts can reuse the same Bernoulli variable. The local first-exit construction is the repaired route. + +## D. Arbitrary-shape fixed-p homological free energy + +For fixed subcritical `p`, put + +`rho_n = min_{lambda in Lambda_n\{0}} tau_p(lambda)`. + +If `rho_n->infinity` and `log N_n/rho_n->alpha`, then + +`(1/rho_n) log P_p(r>0) -> -max(1-alpha,0)`. + +Equivalently, + +`log P_p(r>0)=-(rho_n-log N_n)_+ + o(rho_n)`. + +Status: **AUTHOR-PROOF** in `general-period-homological-free-energy-20260914.md`. + +Upper side: first-exit theta bound + correlation-norm lattice packing. Lower side: finite angular net of fixed local connection seeds + deterministic correction + Harris association + finite-group translation packing. No fixed direction or OZ prefactor is required. + +This promotes the earlier heuristic `connection energy - log opportunities` to a two-sided fixed-parameter theorem at exponential scale. + +## E. Directional centres and mass monotonicity + +| Claim | Status | Main file | +|---|---|---| +| Fixed primitive direction: `(1/n)log P(r>0)=-max{tau_p(u)-d,0}` for `Lambda=`, `log m/n->d` | AUTHOR-PROOF | `fixed-direction-exponential-centres-20260914.md` | +| Varying shortest directions `u_n/|u_n|->e`, `log(N/|u_n|)/|u_n|->d`: rate is `-max{tau_p(e)-d,0}` | AUTHOR-PROOF | `varying-direction-exponential-centres-20260914.md` | +| Any nonparallel period in the exponential shortest-period regime is at least transverse height `h=N/ell`, hence direction competition is superexponentially suppressed | EXACT geometry + AUTHOR-PROOF probability | `exponential-homology-class-selection-20260914.md` | +| First-birth slope equals the shortest-period projective line with probability `->1` in that regime | AUTHOR-PROOF | same | +| `p -> tau_{G,p}(e)` strictly decreases for every direction | AUTHOR-PROOF | `varying-direction-exponential-centres-20260914.md` | +| Quantitative FK comparison `tau_p(e)-tau_q(e) >= (q-p)tau_q(e)/rho_FK` uniformly in direction | AUTHOR-PROOF | `direction-uniform-mass-slope-20260914.md` | +| Lower/upper centres satisfy `tau_4,a(e)=d`, `tau_8,1-b(e)=d` | AUTHOR-PROOF | `varying-direction-exponential-centres-20260914.md` | + +Issue impact: the fixed/varying direction centre problem in #765 is reduced to proof review rather than a missing direction case. + +## F. Strict NN/matching directional separation + +| Claim | Status | Main file | +|---|---|---| +| Matching facial enhancement beats a positive NN `p` sprinkling for two-terminal events | AUTHOR-PROOF | `matching-enhancement-mass-gap-20260914.md` | +| `tau_8,p(e) 1` needs stationarity/Palm mean spacing, not Poisson | AUTHOR-PROOF | same | +| Poisson anchor process upgrades normalized white span to `Exp(1)` and uniform-location span to `Gamma(2,1)` | AUTHOR-PROOF conditional on parent process theorem | `poisson-tessellation-consequence-20260914.md` | +| Infinite-volume boundary density `beta(q)=(1-q)theta(q)/q` | EXACT | `supercritical-white-slab-bulk-20260914.md` | +| Mean rewards `(nu E L, nu E N/w, nu E B/w)->(1,theta_8(q),(p/q)theta_8(q))` | AUTHOR-PROOF | `white-component-mean-reward-20260914.md` | +| Samplewise `N/(wL)->theta`, `B/(wL)->(p/q)theta` and conditional slab CLT | CONJECTURE | `supercritical-white-slab-bulk-20260914.md` | + +The mean boundary/volume ratio `E B/E N -> p/(1-p)` is now theorem-level on the branch even though the stronger samplewise slab LLN remains open. + +## I. Fixed-width charge free energy: why the root survives a wide plateau + +For fixed circumference `w`, let `Q^G_{w,m}(p)` be the open-strip probability of no horizontal essential component. Concatenation and an empty separator row give a free energy + +`I^0_{G,w}(p)=lim_{m->infinity}-(1/m)log Q^G_{w,m}(p)`. + +The torus rank-zero probability has the same rate. Digital Alexander gives + +`theta_{w,m}(p)/m -> Theta_w(p)=I^0_{4,w}(p)-I^0_{8,w}(1-p)`. + +Status: **EXACT free-energy existence + AUTHOR-PROOF transfer/Perron regularity** in `fixed-width-charge-free-energy-20260914.md`. + +Consequences: + +- unique fixed-width charge-coexistence point `p_w^ch` from `Theta_w=0`; +- finite-torus roots `p^*_{w,m}->p_w^ch` as `m->infinity`; +- a diagonal use of the parent root-consistency theorem gives `p_w^ch->pc` as `w->infinity`. + +Interpretation: the matching root balances two rare **topological void-sector free energies**. It is not balancing the black/white complete-component intensities, which are equal in the infinite-cylinder alternation identity. This is the spectral explanation of balance without birth-law concentration. + +Review hotspot: primitive Perron block / strict monotonicity for the fixed-width safe-homology transfer should be audited if this is promoted as a production estimator. + +## J. Projective slope / modular positive control + +| Claim | Status | Main file | +|---|---|---| +| Embedded even-spin projective slope harmonic is a literal lattice observable | EXACT | `projective-slope-harmonic-control-20260914.md` | +| 4/8 complement transports the full harmonic exactly | EXACT | same | +| Pinson--Arguin primitive-sector baseline yields a parameter-free critical continuum harmonic | EXACT given merged PR #213 formula | same | +| Modular covariance `A_s(gamma tau)=(|c tau+d|/(c tau+d))^s A_s(tau)` | EXACT sector-reindexing consequence | `projective-slope-modular-covariance-20260914.md` | +| Elliptic-point selection rules, e.g. `A_4(rho_hex)=0` | EXACT continuum symmetry consequence | same | + +This is explicitly a positive-control channel for #585/#589; it is **not** original-U and must not be inserted as its surrogate. + +## K. Sewing, prefactor and locality + +| Claim | Status | Main file | +|---|---|---| +| Finite-state cyclic matrix kernel with one simple Perron band gives `e^{-kappa w}/sqrt(2 pi D w)` and unit logdet residue | EXACT matrix theorem | `matrix-sewing-unit-residue-20260914.md` | +| Finite local memory alone cannot create a continuously varying extra residue | EXACT within that matrix class | `sewing-amplitude-diagnostic-20260914.md` | +| Pure periodization preserves zero-Fourier Perron mass exactly; nonzero `gamma_w-kappa` must come from wrap-sensitive decorations/irreducible pieces | EXACT mechanism statement / CONDITIONAL SITE conclusion | `periodic-mass-locality-mechanism-20260914.md` | +| Exponential decoration locality would imply `gamma_w-kappa=O(e^{-cw})` | CONDITIONAL | same | + +This reduces #760 to a concrete decoration-locality lemma rather than another width fit. + +## L. Loop/branch morphology and #758/#762 + +| Claim | Status | Main file | +|---|---|---| +| For candidate `I(A)=min_r[tau(1,2r)-kappa+kappa(A-r)]`, strict convexity gives fixed bulge saturation and an exactly linear tail | CONDITIONAL on the candidate variational formula | structural notes / `loop-branch-linear-tail-tests-20260914.md` | +| Small-`A` overlap forces `D^{-1}=partial_yy tau(1,0)=kappa+kappa_angle''(0)` | CONDITIONAL consistency relation | same | +| Since `r_*<1/2`, #762 tests at `A=1,2` are necessarily in the linear-branch regime if the candidate is correct | CONDITIONAL | `loop-branch-linear-tail-tests-20260914.md` | +| Under near-critical normalized Wulff isotropy, `r_*=1/(2sqrt3)` and the whole normalized rate is explicit | CONDITIONAL / UNIVERSALITY HYPOTHESIS | `near-critical-isotropic-loop-branch-rate-20260914.md` | + +The explicit isotropic target is + +`I(A)/kappa -> sqrt(1+4A^2)-1` below `1/(2sqrt3)`, and `A+sqrt(3)/2-1` above it. In particular `I(1)/kappa->sqrt(3)/2` and `I(2)-I(1)->kappa`. + +Square-site rotational restoration is **not** claimed as a theorem. + +## M. Near-critical literature boundary and next theorem engine (#740/#767) + +`near-critical-oz-literature-boundary-20260914.md` is **LITERATURE BOUNDARY**, not a SITE theorem. + +Audited result: D'Alimonte--Manolescu v3 proves for square-lattice random-cluster (q=1 = Bernoulli **bond**) a uniform near-critical two-point comparability + +`P(0<->r e) asymp pi_1(xi)^2 sqrt(xi/r) exp(-r/xi)`. + +What transfers safely as architecture: + +- renewal blocks at correlation-length scale; +- Gaussian sewing / Brownian transverse scale `D asymp xi`; +- strict Wulff geometry; +- explicit endpoint arm insertions. + +What does **not** transfer as a theorem: + +- square-site uniform near-critical OZ; +- exact relative-error amplitude; +- complete-component insertion residue; +- square-site critical arm exponents. + +`near-critical-loop-insertion-diagnostic-20260914.md` is **CONJECTURE / DIAGNOSTIC**. It separates: + +- pure cyclic closure: `nu_w ~ xi^{-1}s^{-1/2}e^{-s}`; +- two-endpoint-dressed closure: `nu_w ~ pi_1(xi)^2 s^{-1/2}e^{-s}`; + +with `s=w/xi`. The statistic `J_emp=xi sqrt(s)e^s nu_w` is designed to distinguish insertion semantics near criticality; fixed-`p` data cannot. + +`site-near-critical-renewal-programme-20260914.md` is **PROGRAMME**. It isolates five square-SITE gates: sub-characteristic 4/8 RSW, one-arm stability (or an explicit near-critical arm insertion), uniform killed renewal/barriers, nondegenerate transverse variance, then complete-component cyclic insertion. This is intentionally smaller than proving conformal universality. + +## N. Common-window crossover (#767) + +What is now exact: + +- support and projective slope constraints; +- charge coordinates `(chi,theta,H)`; +- 4/8 reflection and graph-inclusion inequalities; +- separated Poisson-window boundary conditions. + +What remains genuinely open: + +- a square-site near-critical scaling law for `(chi,theta,H)` in the merging regime `rho->infinity`, `log rho=o(w)`; +- the complete-component insertion factor at correlation-length scale; +- a proof-level square-site near-critical renewal theorem or an explicit universality input strong enough to replace it. + +`projective-poisson-hardcore-crossover-20260914.md` is a **CONJECTURAL closure** only for geometries in which several nonparallel period classes genuinely remain competitive. It should **not** be used in the exponential shortest-period regime, where `exponential-homology-class-selection-20260914.md` proves deterministic slope selection. + +## O. Corrections / superseded shortcuts + +1. **Quotient two-arc shortcut rejected.** A torus simple cycle cannot be globally treated as two independent planar connections because periodic lifts can reuse Bernoulli variables. Replaced by local first-exit BK witnesses. +2. **Same-p two-subcritical-free-energy phase diagram rejected.** Because `pc(G8)=1-pc(G4)`, black NN at `p` and complementary matching at `1-p` are not both subcritical away from criticality. `two-free-energy-rank-phase-diagram-20260914.md` now contains the corrected one-sided lower/upper formulation. +3. **Fixed-p OZ amplitude is not extrapolated to criticality.** Near-critical bond-FK already shows a moving critical-arm insertion; square-site complete-component insertion remains separate. +4. **Direction centre does not imply direction window.** Varying-direction centre is coordinate-free; varying-direction Gumbel still needs uniform local component controls. + +## P. Issue-level status after this continuation + +- **#765 directional centres:** main mathematical direction problem is author-level closed; review/acceptance remains. +- **#766 vector first-exit/Wulff:** main theoretical certificate and torus application are author-level closed; numerical certification can now be built on them. +- **#763 no-prefactor Gumbel:** axial parent proof retained; fixed primitive directions extended; varying directions remain a local-uniformity problem. +- **#764 black/white gap law:** reciprocal mean, Poisson gap laws and mean `(L,N,B)` rewards substantially closed; samplewise white-slab LLN/CLT remains. +- **#767 common near-critical crossover:** exact finite state space/coordinates are closed; square-site merging scaling law remains open. +- **#760 periodic mass locality:** reduced to decoration locality / Perron perturbation. +- **#758/#762 morphology:** candidate mechanism now has sharp linear-tail and conditional isotropic no-parameter tests, but the LDP mechanism itself remains conjectural. +- **#740 prefactor/sewing:** finite-memory sewing theorem and near-critical insertion alternatives are separated; actual SITE complete-component residue remains open. + +## Q. Recommended review order + +1. Persistent Alexander / projective slope exact identities and finite controls. +2. Vector first-exit theorem and torus cycle application. +3. General-period fixed-p free energy and direction-uniform FK slope inequality. +4. Matching enhancement pivotal conversion and all-direction strict mass gap. +5. Alternating black/white Palm mean results. +6. Fixed-direction Poisson--Gumbel extension. +7. Fixed-width charge free energy / Perron interpretation. +8. Only then review conditional near-critical and morphology mechanisms. + +This ordering maximizes the amount of downstream material validated by each proof audit and keeps conjectural near-critical work from blocking the geometric/probability core. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/safe-frontier-balanced-ternary-encoding-20260914.md b/docs/manuscripts/geometric-balance/safe-frontier-balanced-ternary-encoding-20260914.md new file mode 100644 index 000000000..72027a782 --- /dev/null +++ b/docs/manuscripts/geometric-balance/safe-frontier-balanced-ternary-encoding-20260914.md @@ -0,0 +1,169 @@ +# Balanced ternary / parenthesis encoding for safe site frontiers + +2026-09-14. Constructive combinatorics programme suggested by the exact match with the dilute periodic TL zero-string basis. + +## 1. The target code is extremely simple + +Let + +\[ +\mathcal W_w +=\left\{\sigma\in\{\circ,(,)\}^w: +\#(=\#)\right\}. \tag{1.1} +\] + +Then + +\[ +|\mathcal W_w| +=[z^0](1+z+z^{-1})^w=T_w, \tag{1.2} +\] + +the central trinomial coefficient. + +Jacobsen's dilute periodic TL zero-string basis begins at width two as + +```text +circle circle, (), )( +``` + +and has generating function exactly (1.2). The periodic basis allows `)(` because an arc may cross the declared periodic cut without being a noncontractible closed loop. + +The transparent site safe-frontier state count is the same `T_w`. This suggests a direct all-width code + +\[ +\boxed{\mathcal S^{safe}_w\leftrightarrow\mathcal W_w.} \tag{1.3} +\] + +## 2. Why occupied-run boundaries are the right positions + +For a cyclic binary occupancy mask `b_0,...,b_{w-1}`, let + +\[ +\partial b=\{i:b_i\ne b_{i-1}\} \tag{2.1} +\] + +be its set of occupied/vacant transition edges. If the mask has `r` occupied runs, then + +\[ +|\partial b|=2r. \tag{2.2} +\] + +All safe frontier connectivity beyond the mask itself lives in the explored past and can only distinguish how these `2r` run boundaries are connected through the annulus. + +The machine enumeration says a FIXED mask has + +\[ +\frac12\binom{2r}{r} \tag{2.3} +\] + +safe states. + +Now fix only the transition set `partial b`, not which complementary binary mask is called occupied. Exactly two masks have that transition set: `b` and `1-b`. Their combined number of safe states is therefore + +\[ +2\times\frac12\binom{2r}{r}=\binom{2r}{r}. \tag{2.4} +\] + +But this is exactly the number of ways to place `r` left parentheses and `r` right parentheses on the `2r` transition positions, putting `circle` everywhere else. + +Thus the counts match **locally for every fixed transition-edge set**, not only after summing over masks: + +\[ +\boxed{ +\{\text{safe states for the complementary pair }b,1-b\} +\longleftrightarrow +\{\text{balanced parenthesis signs on }\partial b\}.} \tag{2.5} +\] + +Summing over all even transition sets recovers the central trinomial coefficient. + +## 3. Interface-loop construction of the map + +The natural map should use interfaces rather than occupied-cluster blocks directly. + +Take a finite explored half-cylinder realizing a safe state. Draw the black/white interfaces on the medial lattice. Every transition edge in `partial b` creates one interface endpoint at the top boundary. Since horizontal occupied homology has not yet appeared, the relevant interface connectivity is representable by contractible/through-cut annular arcs without an occupied deck-invariant cycle. + +Choose the periodic cut. Encode a top endpoint by + +- `(` if its interface arc opens in the chosen reduced annular ordering; +- `)` if it closes; +- `circle` if no interface endpoint sits at that frontier position. + +Noncrossing interface arcs force equal numbers of openings and closings. An arc crossing the periodic cut produces a prefix imbalance and hence words such as `)(`; this is allowed and is exactly why ordinary Dyck words/Catalan counting is too small. + +This gives the candidate + +\[ +\Phi_{int}:\mathcal S^{safe}_w\to\mathcal W_w. \tag{3.1} +\] + +Complementing occupied/vacant sites leaves the interface curves fixed but reverses their orientation convention, corresponding naturally to + +\[ +(\leftrightarrow). \tag{3.2} +\] + +Hence a complementary pair of masks accounts for the two global parenthesis orientations in (2.4). + +## 4. Inverse construction + +Given a balanced periodic parenthesis word: + +1. use the standard periodic/dilute-TL rule to draw its unique reduced noncrossing annular arc connectivity; +2. choose which side of the interfaces is occupied from one reference boundary interval; global reversal gives the complementary mask; +3. thicken occupied regions on one side of the arcs; +4. route the arcs sufficiently deep in the square-lattice half-cylinder so that their discrete realizations are mutually separated; +5. read off occupied-component connectivity and deck gains at the top row. + +Because the arcs are noncrossing and no noncontractible closed interface/zero-block is introduced, the resulting occupied state should be safe. + +The main proof obligation is step 4: show every reduced annular arc diagram can be realized by NN SITE geometry for an arbitrary declared top mask without accidental extra contacts. Depth is free, so a channel-routing proof should be possible. + +## 5. Relation to the type-B code + +Balanced periodic parenthesis words are another standard model of the annular/type-B noncrossing connectivity module. The no-zero-block type-B partition description in `safe-frontier-central-trinomial-20260914.md` and the ternary word code here should be related by the usual conversion between signed annular blocks and reduced parenthesis arcs. + +Thus the current picture is + +\[ +\boxed{ +\mathcal S^{safe}_w +\leftrightarrow +\mathcal W_w +\leftrightarrow +NC_B^{nozero} +\leftrightarrow +\mathcal B^{dTL}_{s=0}.} \tag{5.1} +\] + +The ternary word is probably the most practical computational representation; the type-B partition is the cleanest block-connectivity representation. + +## 6. Computational payoff if the bijection is proved + +The current reference implementation discovers states by BFS and hashes connectivity/gain tuples. A direct ternary code would remove state discovery entirely: + +1. enumerate all balanced ternary words of length `w`; +2. index them by combinatorial rank; +3. implement a row update directly on the interface word; +4. reject creation of a forbidden noncontractible loop. + +This should give a smaller-memory transparent transfer and may make widths beyond the current `w=8--9` controls much easier. + +It would also put the Bernoulli SITE kernel directly into the conventional dilute-TL connectivity language, simplifying comparison of subleading spectra and symmetry sectors. + +## 7. Exact tests already available + +Any proposed map must preserve all of: + +- total count `T_w`; +- fixed-mask multiplicity `binom(2r-1,r-1)`; +- width-two states `circle circle,(),)(` after the declared interface convention; +- G4/G8 equality of the safe state vocabulary through the checked widths; +- exact charge-root polynomials at `w=2,3,4` after local transition weights are assigned. + +These make the bijection unusually falsifiable. + +## 8. Claim boundary + +The balanced-word counting and dilute-TL basis dimensions are exact. The local count (2.4) follows from the machine-checked fixed-mask multiplicity through current widths. The explicit interface map, all-width fixed-mask multiplicity, and discrete inverse realization remain to be proved. diff --git a/docs/manuscripts/geometric-balance/safe-frontier-central-trinomial-20260914.md b/docs/manuscripts/geometric-balance/safe-frontier-central-trinomial-20260914.md new file mode 100644 index 000000000..cda2c5e73 --- /dev/null +++ b/docs/manuscripts/geometric-balance/safe-frontier-central-trinomial-20260914.md @@ -0,0 +1,159 @@ +# Safe homology frontiers and no-zero-block type-B noncrossing partitions + +2026-09-14. Structural observation from the fixed-width charge-transfer oracle, sharpened by a precise combinatorial match. + +This note separates machine-checked enumeration from a still-missing constructive topology/combinatorics bijection. It is not needed for the Perron free-energy theorem; it explains why the safe state space is much smaller and more rigid than a generic connectivity-partition transfer space. + +## 1. Safe frontier state + +Process the cylinder one horizontal row at a time. A frontier state records occupied sites in the current row, their explored connectivity, and integer horizontal deck gains. A transition is rejected immediately when two routes inside one component imply inconsistent deck gain; that inconsistency is exactly nonzero horizontal winding. + +`fixed_width_charge_transfer.py` constructs this state space by exhaustive BFS from the empty frontier, with no state cap or pruning heuristic. + +Contract each occupied run in the current cyclic row to one frontier object. If there are `r` occupied runs, the nontrivial state information is the way explored components connect these `r` boundary objects through the past half-cylinder, together with their relative deck lifts. + +## 2. Exact enumeration through width eight + +For BOTH NN (`G4`) and matching (`G8`) adjacency, the reachable safe-state counts are + +| `w` | safe states | +|---:|---:| +| 2 | 3 | +| 3 | 7 | +| 4 | 19 | +| 5 | 51 | +| 6 | 141 | +| 7 | 393 | +| 8 | 1107 | + +Moreover the two graphs have the same set of encoded safe states at every one of these widths. Their transition maps/weights differ; their reachable state vocabulary does not. + +The count is exactly the central trinomial coefficient + +\[ +\boxed{T_w=[z^0](1+z+z^{-1})^w +=\sum_{j=0}^{\lfloor w/2\rfloor}\binom{w}{2j}\binom{2j}{j}.} +\tag{2.1} +\] + +The implementation asserts (2.1). Its asymptotic is + +\[ +T_w\sim\frac{\sqrt3}{2\sqrt{\pi w}}3^w. \tag{2.2} +\] + +## 3. Refinement by occupied runs + +Fix a nonempty binary occupancy mask on the cyclic frontier. Let `r` be the number of occupied runs. A fully occupied row is not safe because its horizontal edges already create winding. + +Exhaustive enumeration gives, for every fixed mask tested through `w=8`, + +\[ +\boxed{N_{\rm safe}(\text{fixed mask with }r\text{ runs}) +=\binom{2r-1}{r-1}=\frac12\binom{2r}{r}.} \tag{3.1} +\] + +Thus `r=1,2,3,4` give `1,3,10,35`. The number of cyclic binary masks with exactly `r` occupied runs is + +\[ +2\binom{w}{2r}. \tag{3.2} +\] + +Multiplying (3.1) and (3.2), then adding the empty state, gives exactly (2.1). So the central-trinomial agreement is refined by the number of cyclic occupied runs; it is not only a total-count coincidence. + +## 4. The combinatorial class is now much more specific + +A type-B noncrossing partition on signed objects `+/-[r]` is invariant under sign reversal and has at most one sign-invariant **zero block**. The total type-B noncrossing count is + +\[ +\binom{2r}{r}. \tag{4.1} +\] + +More specifically, the number with **no zero block** and exactly `k` pairs of nonzero blocks is known to be + +\[ +\binom{r}{k}\binom{r-1}{k-1}. \tag{4.2} +\] + +Summing over `k`, Vandermonde gives + +\[ +\boxed{ +|NC_B^{\rm nozero}(r)| +=\sum_{k=1}^r\binom{r}{k}\binom{r-1}{k-1} +=\binom{2r-1}{r-1}.} \tag{4.3} +\] + +Equation (4.3) is **exactly** the safe-state multiplicity (3.1). Thus the previous vague phrase “one half of type-B Catalan objects” can be replaced by a precise target: + +\[ +\boxed{\text{safe frontier with }r\text{ runs} +\quad\leftrightarrow\quad +NC_B^{\rm nozero}(r).} \tag{4.4} +\] + +The no-zero-block condition also has the correct topology: a sign-invariant/zero block is the natural finite signed-partition image of an explored component whose lift meets one of its deck translates, i.e. a component that already carries horizontal homology and should have been rejected from the safe state space. + +## 5. Proposed explicit universal-cover map + +Label the `r` occupied runs in cyclic order. Lift the processed cylinder to an infinite strip and choose one fundamental copy of every frontier run. For each explored component touching the frontier: + +1. collect the chosen frontier runs whose selected lifts belong to one lifted component; +2. also record which frontier-run lifts in the adjacent deck copy belong to that same lifted component; +3. encode the latter as the signed partners of the former. + +Deck translation sends every lifted component to another component and induces block negation. If the explored state is safe, no lifted component is fixed by deck translation, so no block can be a zero block. Planarity/noncrossing of disjoint explored components in the cylinder should give the type-B noncrossing condition. + +This gives a natural injective candidate map + +\[ +\Phi:\{\text{safe frontier states on a fixed }r\text{-run mask}\} +\longrightarrow NC_B^{\rm nozero}(r). \tag{5.1} +\] + +The integer gain stored by the transfer state is exactly the data telling whether another boundary run is met in the same fundamental lift or in the neighbouring deck lift. + +## 6. What is still missing for a theorem + +Two points require a constructive proof rather than count matching. + +### 6.1 No hidden higher-deck ambiguity + +One must prove that a safe planar frontier component cannot require additional independent data beyond the signed two-copy type-B picture. Equivalently, after choosing an appropriate lift of each connected component, all frontier contacts relevant to the canonical state must be represented by the signed annular diagram without losing possible gain information. + +The finite BFS strongly suggests this is automatic from noncrossing plus absence of a deck-invariant component, but it should be proved geometrically. + +### 6.2 Surjectivity / lattice realizability + +Given an arbitrary no-zero-block type-B noncrossing partition, construct a sufficiently deep explored square-lattice half-cylinder realizing it without creating horizontal homology. The noncrossing diagram supplies disjoint topological arcs; the remaining work is to route those arcs on the discrete lattice while respecting the declared occupied frontier mask. + +If both steps are proved, (4.3) immediately yields the all-width central-trinomial formula rather than merely explaining it numerically. + +## 7. NN versus matching vocabulary + +Matching diagonals alter which safe state can be reached in one row, but before winding is created they should not change the annular connectivity vocabulary. Every safe matching diagonal lies in a contractible local face configuration and should admit a finite NN detour in the explored past without changing the signed annular partition/deck gains. + +This leads to the all-width conjecture + +\[ +\boxed{\mathcal S^{safe}_{4,w}=\mathcal S^{safe}_{8,w}.} \tag{7.1} +\] + +The machine check through `w=8` supports (7.1). Once the type-B bijection is proved separately for both adjacencies, (7.1) follows immediately because both are then canonically identified with the same `NC_B^{nozero}` object for each occupancy mask. + +## 8. Interface to the charge-transfer calculation + +The combinatorial identification supplies several cheap controls. + +- A future implementation returning a different state count has a frontier/gain convention mismatch before any eigenvalue is trusted. +- The common G4/G8 vocabulary means the charge criterion compares two positive kernels on one canonical signed-partition state space. +- Central-trinomial growth gives an honest complexity estimate for the transparent reference implementation. +- A successful type-B recoding may permit a much more compact/direct transition generator than the current BFS state discovery. + +## 9. Literature boundary + +The total type-B noncrossing count `binom(2r,r)` is classical. The no-zero-block refinement (4.2) appears explicitly in the type-B noncrossing literature and sums to `binom(2r-1,r-1)`. What is not taken from that literature is the proposed identification with safe percolation frontier connectivities; that remains the repository's topology/combinatorics bridge to prove. + +## 10. Claim boundary + +The finite counts, run-number refinement, and G4/G8 state-set equality through the checked widths are machine-checked facts. The published combinatorial count of no-zero-block type-B noncrossing partitions is exact. The actual bijection (5.1), its surjectivity, and all-width adjacency-independent vocabulary remain author-level conjectures pending constructive proof. diff --git a/docs/manuscripts/geometric-balance/safe-frontier-dilute-tl-bridge-20260914.md b/docs/manuscripts/geometric-balance/safe-frontier-dilute-tl-bridge-20260914.md new file mode 100644 index 000000000..faaec9a41 --- /dev/null +++ b/docs/manuscripts/geometric-balance/safe-frontier-dilute-tl-bridge-20260914.md @@ -0,0 +1,197 @@ +# Safe site frontiers, no-zero type-B noncrossing partitions, and the dilute periodic TL zero-string module + +2026-09-14. Literature/combinatorics bridge. This note does not claim that the Bernoulli safe transfer has already been algebraically conjugated to Jacobsen's dilute O(N) transfer matrix. It identifies a strikingly exact common connectivity module and states the remaining intertwining problem. + +## 1. Three descriptions with the same state count + +The transparent site safe transfer has, for widths + +\[ +w=1,2,3,4,5,6,7,8,\ldots, \tag{1.1} +\] + +state counts + +\[ +1,3,7,19,51,141,393,1107,\ldots \tag{1.2} +\] + +(after the trivial width-one convention is aligned). + +These are central trinomial coefficients + +\[ +T_w=[z^0](1+z+z^{-1})^w, \tag{1.3} +\] + +with generating function + +\[ +\boxed{ +\sum_{w\ge0}T_w x^w +=\frac1{\sqrt{(1+x)(1-3x)}}.} \tag{1.4} +\] + +Independently, Jacobsen's dilute periodic Temperley--Lieb treatment defines the zero-string transfer module `T^(0)` and gives exactly + +\[ +\boxed{ +f_0(x)=\frac1{\sqrt{(1+x)(1-3x)}} +=\sum_{n\ge0}a_nx^n, +\qquad a_n=\dim T^{(0)}.} \tag{1.5} +\] + +He explicitly records + +\[ +\dim T^{(0)}=3\text{ at }n=2, +\qquad +\dim T^{(0)}=7\text{ at }n=3, \tag{1.6} +\] + +and the asymptotic + +\[ +a_n\sim\frac12\sqrt{\frac3{\pi n}}3^n. \tag{1.7} +\] + +This is identical to the safe-frontier count and asymptotic. + +A third description is supplied by type-B noncrossing partitions with no zero block. For a frontier occupancy mask with `r` occupied runs, their count is + +\[ +|NC_B^{nozero}(r)| +=\binom{2r-1}{r-1}, \tag{1.8} +\] + +which is exactly the safe multiplicity for that mask. Summing over cyclic masks gives (1.3). + +Thus the evidence points to one combinatorial object seen in three languages: + +\[ +\boxed{ +\text{safe site frontier} +\ \leftrightarrow\ +NC_B^{nozero} +\ \leftrightarrow\ +\text{dilute periodic TL zero-string connectivity}.} \tag{1.9} +\] + +## 2. Why “dilute” is the right TL adjective + +The ordinary periodic Potts/TL `s=0` reduced sectors have + +\[ +\frac12\binom{2n}{n}\sim4^n \tag{2.1} +\] + +open states and the same number of closed states. Every loop strand position is present. + +The safe SITE frontier, however, has explicit vacant positions in the current row. Its state growth is `~3^w/sqrt(w)`, not `~4^w/sqrt(w)`. This is exactly the combinatorics of a **dilute** connectivity basis in which a strand position may be empty. + +Jacobsen's dilute O(N) reduced states use the symbol `circle` for an empty site; at `n=2`, the zero-string basis is + +```text +circle circle, (), )( +``` + +of dimension three. This is the same structural trichotomy as the width-two safe SITE frontier: empty, contractible pairing, and across-cut pairing with no winding cycle. + +The equality of generating functions therefore has a direct state-semantic explanation, not merely a matching integer sequence. + +## 3. Topological meaning of the zero-string condition + +In the dilute periodic loop module, `s=0` means there is no string propagating between the two time slices. Noncontractible loops are controlled separately by the winding-loop weight. + +In the safe SITE transfer, a state is retained precisely while no occupied component has acquired nonzero horizontal homology. In the universal-cover/type-B language this is the no-zero-block condition. + +The detailed dictionary should therefore identify + +- occupied frontier runs with dilute boundary objects; +- site-component connectivity/deck gains with dilute annular pairings; +- a deck-invariant/zero block with a noncontractible loop/string configuration excluded from the safe module. + +The exact local conventions differ from Jacobsen's O(N) model; the claim here is about the connectivity module, not equality of Boltzmann weights. + +## 4. Why this may explain the remarkable small state space for square SITE + +Jacobsen notes that for some problems, specifically including square-site percolation, the actual number of states needed can be even smaller than the generic dense Potts count `1/2 binom(2n,n)`. + +The Bernoulli SITE formulation exposes a natural reason: the appropriate topological frontier includes genuine vacancies and therefore lives on a dilute annular connectivity module whose dimension is central-trinomial rather than central-binomial. + +This provides a conceptual explanation for why a direct site/homology transfer can reproduce the same semi-infinite root sequence with only + +\[ +1,3,7,19,51,\ldots \tag{4.1} +\] + +safe states rather than a generic `~4^w` dense TL basis. + +## 5. Remaining proof: basis bijection + +A full theorem should construct an explicit bijection + +\[ +\Phi_w:\mathcal S^{safe}_w\to\mathcal B^{dTL}_{w,s=0} \tag{5.1} +\] + +which preserves annular connectivity. + +The most promising route factors it through the type-B description: + +\[ +\mathcal S^{safe}_w +\to NC_B^{nozero} +\to\mathcal B^{dTL}_{w,s=0}. \tag{5.2} +\] + +For a fixed occupied mask, contract its runs and use the signed universal-cover partition described in `safe-frontier-central-trinomial-20260914.md`. Standard dilute-TL diagrams already supply an annular noncrossing representation with vacancies. The missing step is to align their conventions for the cut/deck gain and prove surjectivity at the discrete square-site level. + +## 6. Stronger remaining proof: generator intertwining + +Basis equivalence alone does not imply the two transfer matrices are the same. The deeper target is an intertwiner for local updates. + +Let + +\[ +K^{site}_{4,w}(p),\qquad K^{site}_{8,w}(p) \tag{6.1} +\] + +be the two substochastic safe Bernoulli kernels. A dilute-TL representation has local vacancy/connectivity operators. Seek maps of the form + +\[ +\Phi K^{site}_{G,w}(p)\Phi^{-1} +=\mathcal T^{dTL}_G(\rho_1(p),\ldots,\rho_9(p)) \tag{6.2} +\] + +for some nonintegrable local weights `rho_i(p)`, possibly after a simple diagonal gauge transformation. + +If (6.2) holds, several observations become structural rather than empirical: + +- central-trinomial state count; +- annular/topological sector semantics; +- Perron magnetic scaling; +- descendant relaxation spectrum; +- the fact that the same connectivity vocabulary supports both NN and matching kernels. + +There is no reason to assume the resulting `rho_i(p)` lie on Jacobsen's integrable dilute O(N) manifold; the point is algebraic representation, not exact solvability. + +## 7. A possible representation-theoretic payoff + +Jacobsen observes in the integrable dilute O(N) model that, for a special noncontractible-loop weight, the spectrum of the one-string sector can embed massively into the zero-string sector. He explicitly suggests a representation-theoretic explanation. + +Our SITE problem exhibits a different but related phenomenon: two microscopic graphs (NN and matching) act on what appears to be the same dilute annular connectivity module, while digital Alexander duality exchanges their topological endpoint sectors. + +This suggests that the rapid charge-root convergence may be profitably studied at the module/intertwiner level rather than only through CFT asymptotics. In particular, the sector-even cancellations responsible for the eigenvalue method could have an algebraic shadow in the common annular module. + +This is a research direction, not a conclusion. + +## 8. Literature boundary + +Jacobsen, arXiv:1507.03027, equations (72)--(73), gives the dilute periodic TL dimensions and generating functions. His ordinary Potts open/closed reduced sectors instead have `1/2 binom(2n,n)` states each. The type-B no-zero-block enumeration is classical combinatorics. + +The identification of the square-site safe frontier with the dilute zero-string connectivity module, and especially the proposed local-generator intertwining, are new bridge conjectures here. + +## 9. Claim boundary + +The equality of dimension sequences/generating functions is exact. The state semantics line up strongly and the width-two/three bases match the expected dilute pattern. A formal all-width bijection and a transfer-generator conjugacy have not yet been proved. diff --git a/docs/manuscripts/geometric-balance/safe-transfer-pdtl-magnetic-sector-20260914.md b/docs/manuscripts/geometric-balance/safe-transfer-pdtl-magnetic-sector-20260914.md new file mode 100644 index 000000000..47a3ed3f2 --- /dev/null +++ b/docs/manuscripts/geometric-balance/safe-transfer-pdtl-magnetic-sector-20260914.md @@ -0,0 +1,229 @@ +# The safe/void transfer as the periodic dilute-TL magnetic `alpha=0` sector + +2026-09-14. Literature-grounded sector dictionary plus independent square-site transfer checks. This note sharpens `safe-frontier-dilute-tl-bridge-20260914.md`: the common connectivity module is not only a central-trinomial zero-string space. The **particular twist selected by the no-horizontal-homology condition** is naturally + +\[ +\boxed{d=0,\qquad \omega=e^{i\gamma}=i,\qquad \alpha=\omega+\omega^{-1}=0.} \tag{1} +\] + +At criticality this standard module has exactly the percolation magnetic conformal weight. For triangular-site percolation the periodic dilute Temperley--Lieb representation is an integrable lattice theorem. For the square-site Bernoulli safe transfer the claim is a topological/module dictionary, not local Yang--Baxter integrability. + +## 1. The published periodic dilute-TL module + +Morin-Duchesne, Kluemper and Pearce study critical triangular-site percolation through the Yang--Baxter-solvable dilute `A_2^(2)` loop model. Its periodic standard modules are labelled by + +\[ +W_{N,d,\omega},\qquad \omega=e^{i\gamma}, \tag{1.1} +\] + +where `d` is the number of defects/through-lines and `omega` is the periodic twist. + +For the periodic ground state they obtain + +\[ +(h,\bar h) += +\left( +\Delta_{\gamma/\pi,d/2}, +\Delta_{\gamma/\pi,-d/2} +\right), \tag{1.2} +\] + +with + +\[ +\Delta_{r,s} +=\frac{(3r-2s)^2-1}{24}. \tag{1.3} +\] + +In the zero-defect module the noncontractible loop fugacity is + +\[ +\boxed{\alpha=\omega+\omega^{-1}=2\cos\gamma.} \tag{1.4} +\] + +The same paper's zero-defect dimensions are the central trinomial numbers. + +Primary source: A. Morin-Duchesne, A. Kluemper, P. A. Pearce, *Critical site percolation on the triangular lattice: From integrability to conformal partition functions*, arXiv:2211.12379v2. + +## 2. Why the safe transfer chooses `alpha=0` + +The square-site safe transfer retains a frontier state until an occupied component closes a horizontally noncontractible cycle. The moment such a cycle appears the transition is rejected. + +In a periodic loop representation this is exactly the specialization in which a noncontractible closed loop has zero statistical weight: + +\[ +\boxed{\alpha=0.} \tag{2.1} +\] + +Equation (1.4) then gives + +\[ +\omega=\pm i, +\qquad +\gamma=\pm\frac\pi2\pmod{2\pi}. \tag{2.2} +\] + +The two signs are conjugate descriptions of the same real zero-loop-weight condition. Choose `omega=i`, `gamma=pi/2`. + +The frontier itself carries no propagating noncontractible defect/string, so + +\[ +\boxed{d=0.} \tag{2.3} +\] + +Thus the topological safe condition picks the published module + +\[ +\boxed{W_{N,0,i}.} \tag{2.4} +\] + +This is much sharper than saying only that the state count resembles a dilute zero-string module. + +## 3. The magnetic exponent follows without fitting + +Set + +\[ +r=\gamma/\pi=1/2, +\qquad s=d/2=0. \tag{3.1} +\] + +Then + +\[ +\Delta_{1/2,0} +=\frac{(3/2)^2-1}{24} +=\frac5{96}. \tag{3.2} +\] + +Therefore + +\[ +\boxed{h=\bar h=5/96, +\qquad x=h+\bar h=5/48.} \tag{3.3} +\] + +This is precisely the percolation magnetic scaling dimension observed independently in the safe Perron excitation: + +\[ +w I^0_w(p_c)\longrightarrow2\pi\frac5{48}. \tag{3.4} +\] + +So the `x_m=5/48` limit is not an arbitrary successful CFT assignment after the fact. It is the conformal ground state of the exact periodic dilute-TL module selected by the no-noncontractible-loop condition. + +## 4. The central-trinomial state count is the same module dimension + +The transparent safe automaton has dimensions + +\[ +1,3,7,19,51,141,393,1107,3139,\ldots \tag{4.1} +\] + +for successive widths after the width-one convention is aligned. These are + +\[ +T_N=[z^0](1+z+z^{-1})^N. \tag{4.2} +\] + +Morin-Duchesne--Kluemper--Pearce's periodic zero-defect standard module has the same central-trinomial dimension sequence. + +The earlier note `safe-frontier-dilute-tl-bridge-20260914.md` explains the combinatorial route through vacancies and type-B/no-zero-block annular connectivity. The present note adds the twist/fugacity information: it is specifically the `alpha=0` member of that zero-defect module which realizes the safe topological semantics. + +## 5. Independent descendant check from square-site momentum + +The module dictionary makes a sharp prediction for the first descendants of the magnetic primary. The level-one states + +\[ +L_{-1}|m\rangle, +\qquad +\bar L_{-1}|m\rangle \tag{5.1} +\] + +have + +\[ +\Delta x=1, +\qquad +s=+1,-1. \tag{5.2} +\] + +The transparent square-site safe transfer was diagonalized independently of the dilute-TL formula. Its first excited Perron eigenvalue is a twofold degenerate pair. Restricting the exact one-column translation operator to this eigenspace gives eigenvalues + +\[ +\boxed{e^{+2\pi i/N},\qquad e^{-2\pi i/N}} \tag{5.3} +\] + +to numerical precision for `N=5,...,9`. + +The scaled energy gap simultaneously obeys + +\[ +N\log(\lambda_0/|\lambda_1|)\longrightarrow2\pi. \tag{5.4} +\] + +Thus the first relaxation mode is not merely numerically close to `2pi/N`: its lattice momentum is exactly `+/-1`, as predicted by the level-one magnetic descendants of `W_{N,0,i}`. + +Machine-readable controls and the reproduction script are in + +- `scripts/safe_transfer_momentum_spectrum.py`; +- `results/geometric-consistency/safe-transfer-momentum-spectrum-w5-w9-20260914.json`. + +## 6. What this does and does not identify on the square lattice + +The square NN/matching safe kernels are not claimed to be Yang--Baxter-integrable dilute `A_2^(2)` transfer matrices. Their local Bernoulli row weights differ from the integrable triangular-site weights. + +The robust identification is instead + +```text +annular connectivity module = dilute periodic zero-defect vocabulary, +noncontractible-loop rule = alpha=0 / omega=i, +UV continuum ground sector = magnetic x=5/48, +first descendants = Delta x=1, momentum +/-1. +``` + +In other words the square transfer is a nonintegrable lattice regularization acting on the same topological module and flowing to the same `c=0` magnetic sector. + +This distinction matters: it licenses the sector dictionary and conformal quantum numbers without pretending that square-site finite-width eigenvalues satisfy the triangular model's Bethe equations. + +## 7. Upgrade of the integrable-field-theory programme + +`integrable-potts-magnetic-charge-scaling-20260914.md` previously left the critical magnetic twist/source unidentified. Equation (2.4) removes that ambiguity at the UV lattice/CFT endpoint: + +\[ +\boxed{\text{target UV sector}=W_{N,0,i},\quad \alpha=0,\quad c_{eff}=-5/4.} \tag{7.1} +\] + +The genuinely missing bridge is now narrower: + +> Find the **off-critical thermal/massive finite-volume continuation of the `alpha=0`, zero-defect magnetic sector**, or an equivalent excited/twisted Potts NLIE whose UV limit is `W_{0,i}` and whose two thermal signs are related by Potts duality. + +This is no longer a search over arbitrary twists. + +The 2002 Dorey--Pocklington--Tateo Potts TBA/NLIE gives finite-size thermal flows over continuous `q<=4`, but the specific `q=1` `alpha=0` magnetic-sector finite-volume continuation is not supplied in the material inspected here. The critical periodic dilute-TL paper fixes exactly which endpoint that continuation must reach. + +## 8. Consequence for the dual-odd charge function + +The charge scaling function compares two microscopic realizations of the same continuum magnetic module at opposite thermal signs: + +\[ +\mathcal F(X)=\mathcal E_{W_{0,i}}(X)-\mathcal E_{W_{0,i}}(-X).\tag{8.1} +\] + +This makes its leading oddness structurally natural. It also clarifies the role of the square-lattice sector-odd spin-four correction: the `w^-17/4` mismatch is not a difference of two unrelated primary sectors; it is a lattice-regularization difference **inside the same magnetic module**. + +That is precisely why sector-even primary and ordinary irrelevant contributions can cancel so efficiently from the charge root. + +## 9. Why higher raw safe eigenvalues should not all be assigned to one Verma tower + +The first descendant check is exceptionally clean. Higher raw eigenvalues of the finite safe Perron block, however, need not form a single irreducible Virasoro Verma module. Periodic dilute-TL standard modules are reducible/indecomposable at the percolation root of unity, and the finite connectivity space may contain several conformal families and logarithmic extensions. + +Therefore the correct next step is **momentum/module resolved spectroscopy**, not fitting every raw energy gap to an integer. + +A higher-level identification should be accepted only when its energy, translation momentum and standard-module/character multiplicity all agree. + +## 10. Claim boundary + +The published periodic standard-module weight formula and the `d=0`, twist-labelled framework are literature facts. `alpha=0 <-> omega=+/-i` is exact algebra. The magnetic weight `5/48` follows exactly from the published formula. The square-site first-descendant momentum/gap check is a deterministic transfer calculation. + +For triangular critical site percolation the periodic dilute-TL representation is integrable and explicit. For square-site NN/matching percolation, the statement is a topological/UV module identification and universality bridge; a full all-width basis/generator intertwiner remains to be proved. diff --git a/docs/manuscripts/geometric-balance/sector-even-dressing-of-spin4-tower-20260914.md b/docs/manuscripts/geometric-balance/sector-even-dressing-of-spin4-tower-20260914.md new file mode 100644 index 000000000..75264750a --- /dev/null +++ b/docs/manuscripts/geometric-balance/sector-even-dressing-of-spin4-tower-20260914.md @@ -0,0 +1,188 @@ +# Sector-even `x=4` dressing as a mechanism for the square `4,6,...` charge-root ladder + +Date: 2026-09-14 + +Status: mechanism conjecture obtained by combining two already-visible layers of #771. No new transfer output is introduced here. It is intended to distinguish mechanisms that happen to produce the same pseudo-critical exponent. + +## 1. Two existing observations should be combined, not treated as unrelated exponents + +The fixed-width charge hierarchy already separates: + +1. a **large common sector-even correction** in the average magnetic gap, naturally diagnosed as a per-row `ell^-3` term and hence an irrelevant field of dimension approximately `x_e=4`; +2. a much smaller **sector-odd spin-four correction** in the primal/matching difference, with critical per-row scale `ell^-17/4` and root shift `ell^-4`. + +The first has RG correction exponent + +```text +omega_e = x_e - 2 = 2. +``` + +The second has the square angular harmonic `cos(4 theta)` and is currently interpreted as a thermal-family spin-four sector with total dimension `x_t+4=21/4`. + +The new empirical compression in `equal-circumference-spin4-projector-20260914.md` shows that axis, diagonal and `(2,1)` root amplitudes are all well described, at their two largest available circumferences, by + +```text +A_est(ell) = A_inf + A_2 / ell^2, +``` + +with nearly the same `A_inf` and `A_2` across orientations. The direct critical free-energy amplitudes show the same `ell^-2` dressing. + +The exponent of that dressing is exactly the exponent already carried by the common `x≈4` sector-even correction. + +## 2. Minimal scaling-field mechanism + +Let `g4` denote the leading sector-odd spin-four scaling field and let `ge` denote the leading sector-even scalar/lattice irrelevant field with `omega_e=2`. + +The charge-sector difference is odd under the primal/matching exchange, so a contribution linear in `ge` alone cancels from the difference. But the mixed term `g4 ge` is still sector-odd and retains the spin-four angular character. + +At the level of the physical critical charge gap, the minimal expansion is therefore + +```text +E_charge^phys(pc;ell,theta) + = cos(4theta) ell^(-17/4) + [ B0 + + B2 ell^-2 + + B4 ell^-4 + + ... ] + + angular-orthogonal terms, +``` + +where the `B2` term may be generated by one insertion of the common `x=4` field, the `B4` term by repeated dressing / other scalar corrections, and so on. + +The thermal derivative has its own scalar dressing, + +```text +partial_p E_charge^phys + = ell^(-1/4) + [ C0 + C2 ell^-2 + C4 ell^-4 + ... ] + + angular-orthogonal terms. +``` + +Taking the zero then gives + +```text +p_root-pc + = -cos(4theta) ell^-4 + [ A0 + A2 ell^-2 + A4 ell^-4 + ... ] + + angular-orthogonal terms. +``` + +Thus the square pseudo-critical sequence can naturally contain powers + +```text +ell^-4, ell^-6, ell^-8, ... +``` + +without requiring a new sector-odd primary or a new spatial spin for every exponent. + +This is exactly the kind of compression requested by the research compass: several finite-size powers may be consequences of one microscopic distinction plus one already-visible common correction. + +## 3. Numerical coherence already present + +Using only committed deterministic transfer outputs, the two-point root-amplitude fits give + +```text +axis: A_inf=0.289943, A_2=0.656237 +diagonal: A_inf=0.286325, A_2=0.641543 +(2,1): A_inf=0.292143, A_2=0.690466 +``` + +while the critical free-energy amplitudes give approximately + +```text +axis: B_inf=0.9774, B_2=2.830 +diagonal: B_inf=0.9653, B_2=2.769 +(2,1): B_inf=0.9847, B_2=2.965. +``` + +An independent axis fit of the physical thermal-slope coefficient over widths `4..8` gives roughly + +```text +C(ell) = C_inf + C_2/ell^2, +C_inf ~= 3.377, +C_2 ~= 1.74. +``` + +Then + +```text +B_inf/C_inf ~= 0.289, +``` + +which reproduces the root leading amplitude. Expanding the ratio to the next order also gives an `A_2` of the same order as the directly fitted values. + +None of these two-width fits is an asymptotic proof. Their value is that **three unrelated orientations share not only the leading `cos4theta` amplitude but approximately the same first radial correction**. + +## 4. Why this matters for the Jacobsen square/kagome comparison + +A pseudo-critical exponent `6` can now have two different meanings. + +### Square lattice + +The square lattice already permits the spin-four sector. A subleading `ell^-6` root correction can arise from + +```text +spin-four odd field × sector-even omega=2 dressing, +``` + +and should therefore retain the same `cos(4theta)` angular character. + +### Kagome / genuine C6 symmetry + +The leading spin-four coupling is forbidden by six-fold rotation. A leading exponent `6` there can instead come from a genuinely allowed spin-six thermal-family anisotropy (subject to the same module/matrix-element caveats already recorded elsewhere). + +Therefore + +> **equal pseudo-critical exponents do not imply equal correction operators.** + +In particular, the square subleading `6` and kagome leading `6` should not be identified merely because the powers coincide. + +This resolves an ambiguity in the informal phrase “the correction ladder is `4,6,...`”: on the square lattice the ladder may be a dressed spin-four family, while on C6 lattices the first surviving angular sector can genuinely be spin six. + +## 5. Falsification by angular projection + +The conjecture has a simple signature: + +- the square `ell^-6` correction generated by scalar dressing must remain in the **same `cos4theta` angular sector**; +- an angular-orthogonal exponent-six mechanism need not. + +The exact-equal-circumference `(4,3),n=2` versus axis `w=10` experiment proposed in `equal-circumference-spin4-projector-20260914.md` is therefore the first gate. If the critical mismatch, thermal slope and root all factorize with the same `cos4theta` ratio at `ell=10`, the data support dressing rather than a large angular-orthogonal competitor. + +If the equal-length residual is non-negligible, its angular character should be identified before any width ladder is commissioned. + +## 6. Pell-node consequence becomes sharper + +For Pell directions approaching the spin-four node `theta=pi/8`, + +```text +cos(4theta)=O(|u|^-2). +``` + +If the square exponent-six term is merely the sector-even dressing of spin four, it carries the same `cos4theta` factor. Along a Pell sequence: + +```text +leading spin4 root term: cos4theta * ell^-4 -> O(ell^-6), +dressed spin4 ell^-6 term: cos4theta * ell^-6 -> O(ell^-8). +``` + +By contrast, a **genuine angular-orthogonal exponent-six** contribution can remain `O(ell^-6)` at the node and compete directly with the geometrically suppressed leading spin-four term. + +This changes the interpretation of the Pell experiment. It is not merely a way to “see the next power”; it can separate + +```text +same-spin dressing +vs +a genuinely different angular sector. +``` + +The existing `(5,2), n=2` point already lies close to the node. A second width for `(5,2)` is therefore justified only after the equal-circumference gate shows a residual worth resolving, or if a theoretical candidate predicts a specific node-surviving term. + +## 7. Operator boundary + +The mechanism above does **not** identify the common `x≈4` correction with a unique normalized CFT operator, nor does it prove that the mixed coefficient `g4 ge` is nonzero. It only notes that: + +1. an `omega=2` sector-even correction is already visible independently; +2. multiplying/dressing a sector-odd spin-four response by such an even field preserves sector parity and angular spin; +3. the currently observed cross-orientation `ell^-2` amplitude drift has exactly that structure. + +A module/OPE calculation could later determine whether the mixed term is allowed/nonzero. Until then this is a tighter mechanism conjecture than assigning each even finite-size power to a separate field. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/sector-odd-kac42-conjecture-20260914.md b/docs/manuscripts/geometric-balance/sector-odd-kac42-conjecture-20260914.md new file mode 100644 index 000000000..ee9f860ef --- /dev/null +++ b/docs/manuscripts/geometric-balance/sector-odd-kac42-conjecture-20260914.md @@ -0,0 +1,83 @@ +# Scalar Kac-(4,2) interpretation: demoted after rotational-symmetry check + +2026-09-14. CORRECTION / RETRACTION OF THE PRIMARY IDENTIFICATION IN THE EARLIER VERSION OF THIS NOTE. + +The numerical fact that the square-site charge-sector mismatch is consistent with a correction dimension + +\[ +x_{odd}=21/4 \tag{1} +\] + +remains useful. What is withdrawn is the earlier claim that the **most natural operator identification** is the scalar diagonal Kac field `(4,2)`, for which + +\[ +h_{4,2}=21/8,\qquad h+\bar h=21/4. \tag{2} +\] + +That dimension match is real but insufficient. + +## Why the scalar identification is now disfavored + +Jacobsen's semi-infinite-cylinder data show two different lattice-symmetry patterns in the SAME percolation continuum theory: + +- square-site percolation: the leading pseudo-critical correction has exponent `Delta_1=4`; +- kagome bond percolation: the `Delta=4` amplitude is absent and the leading correction is `Delta_2=6`. + +Jacobsen explicitly suggests that the three-fold rotational symmetry of the kagome lattice, replacing the square lattice's four-fold symmetry, may force the first amplitude to vanish. + +A scalar correction of dimension `21/4` is invariant under both `C4` and `C3/C6`. Rotational symmetry therefore gives no natural reason for its amplitude to exist on the square lattice but vanish on the kagome lattice. This is a substantive negative clue, not a minor aesthetic objection. + +Accordingly the scalar `(4,2)` field is no longer the leading hypothesis. + +## What survives from the old note + +The finite-size scaling arithmetic remains: + +\[ +x_t=5/4,\qquad +\Theta_w'(p_c)\asymp w^{-1/4}, \tag{3} +\] + +and the observed square-site root shift + +\[ +p_w^{ch}-p_c\asymp w^{-4} \tag{4} +\] + +implies that the first correction which is ODD under exchange of the two topological magnetic sectors has total scaling dimension + +\[ +\boxed{x_{odd}=x_t+4=21/4.} \tag{5} +\] + +The direct transfer diagnostic + +\[ +\Theta_w(p_c)\asymp w^{-17/4} \tag{6} +\] + +is consistent with the same value. + +The open question is therefore the **spin / lattice-symmetry representation** of that `x=21/4` correction, not its dimension alone. + +## Replacement conjecture + +The current leading hypothesis is recorded in + +`sector-odd-spin4-anisotropy-20260914.md`: + +> the first square-lattice sector-odd correction has spin four and total dimension `x_t+4=21/4`; `C4` symmetry permits it, while `C3/C6` symmetry forbids it. The next allowed rotational correction has spin six and produces a pseudo-critical exponent six, matching the kagome pattern. + +This is structurally more compatible with Jacobsen's square/kagome comparison. + +## Status of the Kac coincidence + +Equation (2) should now be treated only as a **dimension coincidence / secondary alternative**. It could still become relevant if a logarithmic module mixes a scalar `(4,2)` state with the actual spinful lattice correction, but no such mechanism has been shown here. + +Any future attempt to restore the scalar interpretation must answer the rotational-selection objection explicitly. + +## Literature boundary + +Primary comparison: J. L. Jacobsen, arXiv:1507.03027. He finds the square-site exponent `4`, the kagome leading exponent `6`, and explicitly proposes rotational symmetry as a possible reason the kagome `Delta=4` amplitude vanishes. He does not identify either correction with a specific CFT field. + +This note therefore records a self-correction: numerical exponent matching alone was not enough to justify the scalar Kac-(4,2) assignment. diff --git a/docs/manuscripts/geometric-balance/sector-odd-spin4-anisotropy-20260914.md b/docs/manuscripts/geometric-balance/sector-odd-spin4-anisotropy-20260914.md new file mode 100644 index 000000000..83a82e425 --- /dev/null +++ b/docs/manuscripts/geometric-balance/sector-odd-spin4-anisotropy-20260914.md @@ -0,0 +1,200 @@ +# Sector-odd rotational anisotropy as the source of the 4,6,... pseudo-critical exponents + +2026-09-14. CONJECTURAL CFT/lattice-symmetry interpretation, replacing the earlier scalar Kac-(4,2) guess. + +The starting facts are unusually restrictive: + +1. square-site and kagome-bond percolation are expected to have the same continuum percolation CFT; +2. Jacobsen's semi-infinite-cylinder eigenvalue criterion has leading pseudo-critical exponent `4` on the square lattice; +3. on the kagome lattice the amplitude of that `4` correction vanishes and the leading exponent is `6`; +4. Jacobsen explicitly suggests the change from four-fold to three-/six-fold lattice rotation as the reason for the missing kagome amplitude; +5. on the square-site safe transfer, the critical open/closed sector mismatch scales consistently with `w^-17/4`, while the thermal derivative scales as `w^-1/4`. + +A scalar correction does not naturally explain item 3. A spinful lattice-anisotropy correction does. + +## 1. Thermal family and sector parity + +The percolation thermal field has + +\[ +x_t=5/4, +\qquad h_t=\bar h_t=5/8. \tag{1.1} +\] + +Under primal/dual exchange the thermal perturbation changes sign: moving one side of criticality for the primal model corresponds to the opposite side for the matching/complementary description. It is therefore natural for the **sector-odd** correction spectrum to live in the thermal conformal family rather than the identity family. + +The relevant lattice observable is + +\[ +\Theta_w(p_c) +=I^0_{4,w}(p_c)-I^0_{8,w}(1-p_c), \tag{1.2} +\] + +the difference of the two magnetic/topological excitation energies. Sector-even irrelevant fields can be large in each term and cancel from (1.2). + +## 2. Square lattice: first allowed anisotropic thermal descendant + +A level-`s` chiral descendant of the thermal primary has weights + +\[ +(h_t+s,\bar h_t) +\quad\hbox{or}\quad +(h_t,\bar h_t+s), \tag{2.1} +\] + +and therefore + +\[ +x=x_t+s, +\qquad +\text{spin}=\pm s. \tag{2.2} +\] + +A reflection-invariant lattice perturbation uses the real combination of the two spins. + +On a square lattice, `C4` invariance permits spin `s` only when + +\[ +s\equiv0\pmod4 \tag{2.3} +\] + +among nontrivial rotational anisotropies. The first possibility is therefore `s=4`: + +\[ +\boxed{ +x_{odd,\square}=x_t+4=21/4, +\qquad \text{spin}=4.} \tag{2.4} +\] + +Its contribution to a **per-row** cylinder excitation energy scales as + +\[ +\Theta_w(p_c)\sim A_4 w^{1-x_{odd}} +=A_4 w^{-17/4}. \tag{2.5} +\] + +The thermal response scales as + +\[ +\Theta'_w(p_c)\sim B w^{-1/4}. \tag{2.6} +\] + +Hence the zero of the sector difference moves by + +\[ +\boxed{ +p_w^{ch}-p_c +\sim-\frac{A_4}{B}w^{-4}.} \tag{2.7} +\] + +This reproduces the square-site pseudo-critical exponent four without requiring a scalar field of dimension `21/4`. + +## 3. Kagome / triangular rotational selection + +The kagome and triangular geometries have six-fold spatial symmetry in the bulk; Jacobsen phrases the relevant change relative to the square basis as three-fold rotational symmetry. In either description, a spin-four perturbation is not invariant: + +\[ +e^{i4(2\pi/3)}\ne1, +\qquad +e^{i4(\pi/3)}\ne1. \tag{3.1} +\] + +So the square spin-four amplitude must vanish on a symmetry-preserving kagome/triangular realization: + +\[ +\boxed{A_4^{\rm kagome}=0.} \tag{3.2} +\] + +With inversion/reflection excluding odd-spin anisotropies, the first allowed thermal-family anisotropy is spin six: + +\[ +\boxed{ +x_{odd,\hex}=x_t+6=29/4, +\qquad \text{spin}=6.} \tag{3.3} +\] + +It gives + +\[ +\Theta_w(p_c)\sim A_6 w^{1-29/4}=A_6 w^{-25/4}, \tag{3.4} +\] + +and after division by the same thermal response, + +\[ +\boxed{p_w-p_c\sim w^{-6}.} \tag{3.5} +\] + +This is exactly the leading exponent Jacobsen finds for kagome bond percolation after setting the square-like `A_1` term to zero. + +## 4. Why the ordinary square spin-four identity-family field does not already give the root shift + +Square critical lattice models generically possess lower-dimensional spin-four / identity-family anisotropy corrections. A familiar candidate has total dimension around four and produces ordinary corrections in each finite-size energy. + +The safe-transfer data indeed show a much larger common correction in the average magnetic excitation energy, consistent with a per-row `w^-3` term (`x=4`). + +But the matching root is controlled by the **difference** + +\[ +I^0_4-I^0_8. \tag{4.1} +\] + +The two sectors represent the same continuum magnetic primary. A sector-even identity-family anisotropy can therefore have equal amplitudes and disappear from (4.1). What survives must be both + +1. allowed by lattice rotation; +2. odd under primal/dual topological-sector exchange. + +A spin-four descendant of the thermal family satisfies exactly that structural request. + +## 5. A more precise operator statement to test + +The conjecture should not be phrased as the literal operator `partial^4 epsilon`. Total derivatives can integrate to zero, and Virasoro descendants mix. The correct target is: + +> **Sector-odd spin-four conjecture.** The lowest-dimension `C4`-invariant, reflection-even, primal/dual-odd irrelevant scaling field with a nonzero matrix-element difference between the two magnetic cylinder sectors is a spin `±4` quasi-primary/descendant in the thermal Virasoro module, of total dimension `x_t+4=21/4`. + +For six-fold lattices the corresponding first allowed field is the spin `±6` thermal-module descendant of dimension `x_t+6=29/4`. + +This formulation leaves room for logarithmic mixing inside the `c=0` theory while making the symmetry and dimension claims falsifiable. + +## 6. Immediate falsification tests + +### 6.1 Square orientation test + +A genuine spin-four amplitude changes phase/sign under rotation of a weak anisotropic perturbation according to `e^{i4 theta}`. Introduce a controlled rectangular/diagonal anisotropy while keeping the continuum thermal coordinate fixed and test the angular harmonic of the sector mismatch. + +### 6.2 Six-fold lattice test + +For a C6-symmetric percolation realization, the direct sector mismatch at criticality should have no `w^-17/4` term. Its first rotational thermal-family contribution should instead scale as `w^-25/4`, yielding a pseudo-critical `w^-6` shift. + +Jacobsen's kagome data already supply strong qualitative support for this selection rule; a direct fit of the **sector free-energy difference**, rather than only the root, would be the sharper test. + +### 6.3 Symmetry breaking on kagome/triangular + +Add a small perturbation that reduces C6/C3 to C2 or C1 while preserving critical tuning. If the missing exponent four is a spin-four anisotropy channel, an amplitude linear in the symmetry-breaking coupling should reappear. + +### 6.4 Self-matching triangular site control + +Critical triangular-site percolation is exactly self-matching at `p=1/2`. In a period convention that preserves the matching identification, the primal/dual charge-sector difference should vanish much more strongly than on the non-self-matching square lattice. This provides a clean control separating duality-odd amplitudes from rotationally allowed but sector-even corrections. + +## 7. Relation to Jacobsen's correction ladder + +On the square lattice, further spin-four thermal-family descendants and/or higher derivative levels can generate exponents `6,8,...` after the leading spin-four contribution. On a six-fold lattice the spin-four amplitude is removed, making exponent six the first visible term. + +The precise origin of every later even power is not fixed by this note. The robust claim is narrower: + +\[ +\boxed{\text{rotation selection naturally distinguishes square }4 +\text{ from kagome }6.} \tag{7.1} +\] + +That distinction is not explained by a scalar `(4,2)` assignment. + +## 8. Literature boundary + +Jacobsen, arXiv:1507.03027, reports `Delta_1=4` for square-site percolation, `Delta_2=6` as the leading kagome-bond correction, and explicitly proposes rotational symmetry as a possible reason the kagome `A_1` amplitude vanishes. He does not identify a specific irrelevant CFT field. + +The general symmetry principle that square-lattice anisotropy permits spin multiples of four whereas triangular/hexagonal symmetry permits multiples of six is standard lattice-CFT finite-size-scaling logic. The specific identification with a **primal/dual-odd thermal-family spin-four descendant of dimension `21/4`** is a new conjecture here and requires operator/sector verification. + +## 9. Claim boundary + +The symmetry selection `C4` allows spin four while `C3/C6` forbids it is exact group theory. The observed square/kagome correction exponents are literature facts. The assignment of the sector-odd mismatch to thermal-family spin `4` / `6` descendants is conjectural, albeit more structurally compatible with the lattice comparison than the withdrawn scalar Kac-(4,2) guess. diff --git a/docs/manuscripts/geometric-balance/self-matching-triangular-charge-control-20260914.md b/docs/manuscripts/geometric-balance/self-matching-triangular-charge-control-20260914.md new file mode 100644 index 000000000..eac54b8ab --- /dev/null +++ b/docs/manuscripts/geometric-balance/self-matching-triangular-charge-control-20260914.md @@ -0,0 +1,146 @@ +# Triangular-site self-matching as an exact zero/oddness control for charge scaling + +2026-09-14. Exact topological control; no scaling theory is needed for the principal statements. + +## 1. Self-matching triangulations + +For independent site percolation on a planar triangulation, every face already has all pairs of its vertices adjacent. The site matching graph therefore adds no new face diagonal. In particular the standard triangular lattice is self-matching: + +\[ +G^\star_{match}=G_{tri}. \tag{1.1} +\] + +On an honest periodic triangular triangulation, digital Alexander duality gives configurationwise + +\[ +\boxed{ +r_G(\omega)+r_G(\omega^c)=2.} \tag{1.2} +\] + +At `p=1/2`, `omega` and `omega^c` have the same law. Hence + +\[ +\boxed{P_0(1/2)=P_2(1/2)} \tag{1.3} +\] + +for every finite honest torus. + +Thus the matching/root observable has the exact finite value + +\[ +\boxed{p^*_{\Lambda}=1/2} \tag{1.4} +\] + +with no finite-size drift. + +## 2. Fixed-width charge free energy + +Let `I^0_{tri,w}(p)` be the no-horizontal-homology strip free energy. Because the primal and matching graphs are literally the same graph, + +\[ +\Theta_w^{tri}(p) +=I^0_{tri,w}(p)-I^0_{tri,w}(1-p). \tag{2.1} +\] + +Therefore + +\[ +\boxed{\Theta_w^{tri}(1/2)=0} \tag{2.2} +\] + +for every width `w`, and strict monotonicity gives the unique semi-infinite charge root + +\[ +\boxed{p_w^{ch}=1/2\quad\text{for all }w.} \tag{2.3} +\] + +This is the probability/digital-Alexander version of the size-independent factor that appears in exactly solvable/self-dual cases of the graph-polynomial/eigenvalue method. + +## 3. Exact finite oddness in logit coordinate + +Put + +\[ +h=\log\frac p{1-p}. \tag{3.1} +\] + +Self-matching gives + +\[ +\Theta_w^{tri}(h) +=I^0_{tri,w}(h)-I^0_{tri,w}(-h). \tag{3.2} +\] + +Hence, at every finite width, + +\[ +\boxed{ +\Theta_w^{tri}(-h)=-\Theta_w^{tri}(h).} \tag{3.3} +\] + +In particular ALL even logit derivatives vanish exactly at the root: + +\[ +\boxed{ +\partial_h^{2k}\Theta_w^{tri}(0)=0, +\qquad k=0,1,2,\ldots.} \tag{3.4} +\] + +This is a much stronger positive control than root `1/2` alone. Any transfer implementation of a self-matching site lattice that produces a nonzero even charge derivative at `h=0` has a complement, sector, or normalization bug before it has a new finite-size effect. + +## 4. Consequence for the square/matching dual-odd scaling function + +The square NN graph is not literally self-matching, so (3.3) does not hold at finite width. Nevertheless both square NN and its matching graph flow to the same percolation fixed point, with the complement map reversing the thermal scaling field. + +The new square safe-transfer data show precisely the pattern expected if the **leading universal continuum charge function inherits (3.3)**: + +- `Theta_h ~ w^-1/4` after differentiation; +- `Theta_hhh ~ w^(5/4)`; +- the individual sector second derivatives are `O(sqrt(w))`, but their charge difference `Theta_hh` is only `O(w^-1/2)` in the current widths. + +Thus triangular self-matching supplies the exact finite model for the parity that square/matching should recover asymptotically after sector-even lattice corrections are removed. + +This strengthens the interpretation in `dual-odd-charge-scaling-function-20260914.md`: the conjectured continuum oddness is not an arbitrary fit symmetry but the universal remnant of exact matching duality. + +## 5. Consequence for sector-odd irrelevant channels + +At the exact self-matching point, the full charge free-energy difference vanishes: + +\[ +I^0_{primal,w}(1/2)-I^0_{matching,w}(1/2)=0 \tag{5.1} +\] + +for every finite width. Therefore every contribution that is ODD under primal/matching sector exchange must cancel in the finite-size expansion. + +In particular, if the square-lattice `w^-17/4` mismatch is produced by a sector-odd spin-four thermal-family anisotropy, its analogue has exactly zero amplitude in the self-matching triangular-site problem. But the statement is stronger: **all** sector-odd amplitudes vanish, regardless of spin. + +This makes triangular site a control for sector parity, not by itself a way to distinguish spin four from a scalar odd operator. + +## 6. Three-lattice discrimination strategy + +The clean comparison is therefore: + +| lattice/model | rotation | self-matching? | expected charge-shift pattern | +|---|---|---|---| +| square site | `C4` | no | spin-4 odd correction allowed; observed `Delta=4` | +| kagome bond / hexagonal Bravais symmetry | `C6` (Jacobsen discusses three-fold basis symmetry) | no | spin-4 forbidden; observed leading `Delta=6` | +| triangular site | `C6` | yes | every sector-odd amplitude zero; root exactly `1/2`, charge function exactly odd | + +This separates two mechanisms that would otherwise be conflated: + +1. **rotational selection**, which removes particular spins; +2. **exact primal/matching self-duality**, which removes the entire odd sector at the self-dual thermal point and enforces the finite logit parity (3.3). + +A scalar `x=21/4` explanation of the square `Delta=4` has no natural account of the square/kagome contrast, whereas the spin-four hypothesis does. + +## 7. Symmetry-breaking experiment + +The sharpest future test would start from a six-fold non-self-matching model (kagome bond is the existing literature example) and add a controlled anisotropy that lowers rotational symmetry while retuning to criticality. + +The spin-four hypothesis predicts that once `C6/C3` no longer forbids spin four, a `Delta=4` charge-root correction should reappear with amplitude proportional to the appropriate spin-four component of the anisotropy for weak perturbations. + +By contrast, changing a self-matching triangular-site model while preserving exact self-matching would still keep the entire sector-odd charge difference zero. To activate the test there one must break self-matching as well as rotational symmetry. + +## 8. Claim boundary + +The finite root `1/2`, fixed-width charge-free-energy identity, and exact logit oddness are consequences of self-matching plus complement symmetry. The square/kagome/triangular comparison as evidence for a spin-four thermal-family correction is a research interpretation. No new triangular transfer computation is required to establish the exact zero/oddness controls. diff --git a/docs/manuscripts/geometric-balance/sewing-amplitude-diagnostic-20260914.md b/docs/manuscripts/geometric-balance/sewing-amplitude-diagnostic-20260914.md new file mode 100644 index 000000000..6acaf7c43 --- /dev/null +++ b/docs/manuscripts/geometric-balance/sewing-amplitude-diagnostic-20260914.md @@ -0,0 +1,118 @@ +# Sewing amplitude diagnostic: band multiplicity versus genuine insertion + +2026-09-14. Consequence of `matrix-sewing-unit-residue-20260914.md`. The goal is to make the remaining SITE prefactor ambiguity falsifiable without confusing three distinct mechanisms: diffusion scale, multiplicity of soft bands, and a genuine component/mark insertion. + +## 1. One simple band + +For the pure cyclic object + +\[ +L_w=w[z^wy^0]\{-\log\det(I-A(z,y))\}, \tag{1.1} +\] + +one isolated simple Perron band with root `R=e^kappa` and transverse diffusion `D` gives + +\[ +L_w\sim\frac{e^{-\kappa w}}{\sqrt{2\pi D w}}. \tag{1.2} +\] + +There is no extra endpoint/Perron-overlap residue. Such factors enter only the analytic part of the determinant factorization and disappear from the coefficient of the logarithmic singularity. + +Thus, after `D` is defined by the same twisted band, the dimensionless residue is exactly one. + +## 2. Several dominant simple bands + +Suppose instead that the determinant has finitely many dominant simple bands `lambda_j` with the same real exponential rate `R`, no Jordan singularity, and aperiodic real saddles at `theta=0`, with + +\[ +\log R_j(\theta)=\log R+\frac12D_j\theta^2+O(\theta^4). \tag{2.1} +\] + +The logarithm splits additively: + +\[ +-\log\det(I-A) +=\sum_j-\log(1-\lambda_j)+\text{analytic}. \tag{2.2} +\] + +Each simple band contributes unit logarithmic residue, so + +\[ +\boxed{ +L_w\sim +\frac{e^{-\kappa w}}{\sqrt{2\pi w}} +\sum_j D_j^{-1/2}.} \tag{2.3} +\] + +If all dominant bands have the same `D`, the apparent residue relative to one-band normalization is the **integer band multiplicity**. + +For periodic/complex saddles, phases and residue-class oscillations must be kept explicitly; averaging them into an arbitrary positive amplitude would lose information. This is another reason to diagnose lattice periodicity before fitting one scalar `zeta`. + +## 3. A genuine insertion changes the amplitude continuously + +Now consider a closed coefficient with an insertion, + +\[ +J_w=[z^wy^0]\operatorname{tr} +\left[B(z,y)(I-A(z,y))^{-1}\right], \tag{3.1} +\] + +or the derivative of the log determinant with respect to a physical/mark source. Near a simple Perron pole, + +\[ +(I-A)^{-1} +\sim\frac{r\,l^T}{1-\lambda}, \tag{3.2} +\] + +so the singular amplitude contains + +\[ +l^T B r \tag{3.3} +\] + +and the derivative of the pole location. These quantities can vary smoothly and nontrivially with `p`. Unlike the pure logarithmic coefficient, they are not forced to one. + +Therefore a nontrivial prefactor function does **not** refute a diffusive transverse mode. It diagnoses that the actual SITE observable corresponds to an inserted/marked cyclic object rather than the pure unrooted determinant. + +## 4. Application to complete-component sewing + +The current SITE problem has three logically distinct possibilities. + +### A. Pure cyclic determinant + +A canonical regeneration state makes each complete component exactly one unmarked closed cycle of a Markov-additive kernel. Then + +\[ +\nu_w\sim\frac{e^{-\kappa w}}{\sqrt{2\pi D w}} +\] + +(up to an integer/periodic band multiplicity). `zeta=1` for a single aperiodic band. + +### B. Marked/insertion object + +A convenient cut, seam, regeneration mark, or external-boundary bookkeeping leaves a source `B` on the cyclic kernel. Then the `w^{-1/2}` power and the same `D` can survive while the amplitude is a genuine smooth function. The exact Palm mark-unbiasing formulas in `sewing-with-memory.md` belong here. + +### C. No isolated finite/quasi-compact band + +The effective state remains infinite with another soft mode, no spectral gap, or a continuum of near-leading states. Then even the `w^{-1/2}` power or Brownian range law may fail. + +This classification is sharper than asking only whether a fitted `beta_eff` is near `1/2`. + +## 5. Falsification table + +If future rigorous/numerical certificates establish: + +- `beta != 1/2`: rule out A and the simple form of B; look for C or a different transverse scaling; +- `beta=1/2`, Brownian/curvature `D` consistent, but a smooth non-unit `zeta(p)`: rule out pure A, favour B; +- `beta=1/2`, `zeta` equal to an integer after common `D` normalization: inspect dominant-band multiplicity/periodicity before introducing a new physical amplitude; +- `beta=1/2`, `zeta=1`, and mark corrections vanish under exact unrooting: A becomes a plausible exact sewing target. + +Three-width interpolation cannot distinguish these alternatives. The right evidence is a certified mass, a separately defined diffusion/curvature scale, and a controlled component/mark normalization. + +## 6. Relation to the unit-residue conjecture + +The previous branch conjecture `U` asked whether the actual SITE sewing factor might equal one. The matrix lemma and this diagnostic refine it: + +> unit residue is automatic **after** the component has been identified with a pure cyclic log determinant having one simple soft band. + +The difficult theorem is therefore the component-to-cyclic-object identification, not a second amplitude calculation once that identification is complete. diff --git a/docs/manuscripts/geometric-balance/shared-label-gaussian-sheet-and-lineage-memory-20260914.md b/docs/manuscripts/geometric-balance/shared-label-gaussian-sheet-and-lineage-memory-20260914.md new file mode 100644 index 000000000..319938b46 --- /dev/null +++ b/docs/manuscripts/geometric-balance/shared-label-gaussian-sheet-and-lineage-memory-20260914.md @@ -0,0 +1,225 @@ +# 同一标签下的噪声谱系：Brownian sheet、四坐标闭合与两种记忆速度 + +2026-09-14。继续 #764/#772/#780。目标不是在每个参数重新取得同一种分布，而是保留一份标签从早到晚产生的所有相关。第1–3节为两个固定宽度稀疏 SITE 模型补出多参数评分极限；第4–8节是由此产生的显式随机过程的精确分析；第9节是固定 p、大宽度的条件推广，不混用极限。 + +## 1. 标签时钟与屏障时钟不是同一个时钟 + +用同一批 U_v~Uniform(0,1)，在黑参数 p(t)=εt 时令 v 黑，当 U_v≤εt。本文先取均匀列概率。两种模型是前轮已建立最小模板过程的： + + b=w=8：黑 NN / 白 matching，A_B=1，r=8； + b=w=4：黑 matching / 白 NN，A_B=19，r=4/19。 + +纵坐标 z=ε^b y，屏障时钟 τ=A_B t^b。参数 t 的有限正区间保持不变，然后 ε↓0。 + +固定空间原点，跟踪包含它的白色 essential 簇；若微观原点变黑或进入局部孔洞，单列为异常状态，其概率在这个有限参数区间趋零。不在每个参数另选一个白簇。 + +令 I_t 为极限 cuts 中原点所在的区间，L_ε(t) 为实际完整簇跨度，H_ε(t) 为去除有界端部后的查询孔洞数。记 q(t)=1-εt，K_ε、B_ε 为完整白簇占据数、不同外边界黑站点数，评分为 + + S_ε(t)=K_ε(t)/q(t)-B_ε(t)/(εt)。 (1.1) + +有限 ε 下它不必均值为零。本轮使用的归一化是 + + Y_ε(t)=A_B t^b ε^b L_ε(t)， + Q_ε(t)=sqrt(A_B/w) ε^((b+1)/2) t^((b+1)/2) S_ε(t)。 (1.2) + +再乘一个趋一的 sqrt(q(t)) 不改变结论。 + +## 2. 一个 Gaussian sheet 可以与全部稀有图案一起取极限 + +对有界纵向区间 A 定义 + + W_ε(A,t)= -ε^((b-1)/2)/sqrt(w) + Σ_{ε^b y∈A}Σ_x [1{U_(x,y)≤εt}-εt]。 (2.1) + +负号只为与白色评分一致。单个标签的精确共同参数协方差是 + + Cov(1{U≤εt},1{U≤εu}) + = ε min(t,u)[1-ε max(t,u)]。 (2.2) + +故 W_ε 的极限 W 满足 + + Cov(W(A,t),W(B,u))=|A∩B| min(t,u)。 (2.3) + +W 是空间白噪声、参数方向 Brownian 的 Gaussian 随机测度，即这里所需的 Brownian sheet。对任何有限组区间与参数，独立标签的三角阵 Lindeberg 条件直接成立：单个归一化项趋零，协方差由 (2.2) 给出。 + +还必须证明 W 与稀有 cuts、holes 独立，而不是把它们当不同随机源。将 O(ε^-b) 行分成长度 ell_ε=floor(ε^-1) 的主体块，留至少 2b+3 行缓冲。一个归一化标签块的绝对值≤C ε^((b-3)/2)，趋零；一块含稀有模板的概率≤C ell_ε ε^b。块内的混合特征函数项由“最大归一化块量×稀有概率”控制。对所有块求和后是 O(ε^((b-3)/2))，仍趋零。相交的不同最小模板及大小≥b+1黑簇由前轮 O(ε) 界处理。缓冲区比例趋零，其标签方差和稀有事件期望贡献也趋零。 + +因此有限维联合极限为相互独立的： + + W(A,t)，cuts(dz dτ)，holes(r dz dτ)。 (2.4) + +这是从同一批标签推导的渐近独立。本文不凭有限维证明附加宣称无限参数轴上的过程拓扑收敛。 + +## 3. 从全区间标签评分，转回真正完整簇 + +在前轮“最大参数无复杂黑簇”的好事件上，真实白簇与 cuts 区间只有有界端点偏移。区间内部未查询点是紧的有限个最小孔洞；其余白点及黑边界点都属于完整探索结果。因此在每个有界宏观窗口内，完整评分与全区间逐标签评分之差≤C/(εt)+C H_ε/q(t)。 + +乘以 (1.2) 的因子后，这个差趋零。端点所需宏观窗口先限制在 [-M,M]，再用最小屏障的空区间尾令 M→∞；在有限组正参数上可以同时完成这个步骤。 + +所以本轮得到实际 SITE 的有限维联合弱极限： + + (Y_ε(t),H_ε(t),Q_ε(t)) + -> (Y_t,H_t,Q_t)， + Y_t=τ |I_t|， + Q_t=sqrt(τ/t) W(I_t,t)。 (3.1) + +W 独立于整个 cut/mark 过程，但积分区间由后者决定。给定一个时刻的区间，Q_t~N(0,Y_t)。tagged 协议中 Y_t~Gamma(2,1)，H_t|Y_t~Poi(rY_t)，且 Q_t 与 H_t 条件独立。 + +因此单时刻评分特征函数为 (1+u²/2)^-2，方差2、峰度9/2。这里不是 component-Palm 的 Exp 时钟/Laplace 评分，不能混淆两种取样。关于实际有限 ε 评分高阶矩，本文不从弱极限自动升级；后面所列协方差与四阶矩首先是这个明确极限过程的精确结果。 + +## 4. 完整六变量两时刻变换 + +取 t10，k>1。 (7.3) + +这直接排除了联合 Gaussian 解释。每个时刻单独通过正态性检查，也不能证明整个谱系噪声是 Gaussian OU。随机几何仍藏在多时刻依赖中。 + +## 8. 不保存整片噪声，也能精确闭合为四个坐标 + +设 η=1/b，更一般允许0≤η≤1。取 + + x-=τD-，x+=τD+，y=x-+x+， + h=孔洞数，g=归一化评分。 + +以下在显式极限模型中是精确 Markov 闭合，不是实际有限格子的“四态”表示。四个坐标中两个是正实数、一个非负整数、一个实数。 + +两次离散事件之间： + + dx-=x- ds，dx+=x+ ds， + dg=((1-η)/2)g ds + sqrt(η y) dB_s。 (8.1) + +孔洞以速率 r y 加一。左、右切分分别以速率 x-、x+ 发生。以左切分为例，取 U~Uniform(0,1)，将 x-改成Ux-，并设 v=(Ux-+x+)/y。随后 + + h' |h,v ~ Binomial(h,v)， + g'=v g + sqrt[y v(1-v)] Z，Z~N(0,1)。 (8.2) + +给定当前状态和v，二项抽取与Z独立。右侧同理。 + +为什么过去不再需要保存？给定当前区间与当前Gaussian总和，区间内部的白噪声是均匀均值加Gaussian bridge。所有较早区间都包含当前区间，较早参数的噪声与当前子区间对比量协方差为零；因此它们不再提供额外的空间分配信息。孔洞在给定当前计数后仍为独立均匀位置；历次保留操作不改变这个性质。式 (8.2) 正是对这些内部位置进行积分，而不是生成一份错误的新物理噪声。 + +平稳分布对所有η都相同： + + x-,x+ 独立 Exp(1)， + g|x-,x+ ~ N(0,y)，h|x-,x+~Poi(ry)，条件独立。 (8.3) + +可以用原始Poisson/Gaussian场直接构造，也可以在生成元上验证。对测试函数 x-^a x+^b g^(2d)(h falling n)，条件均值正比 x-^a x+^b y^(d+n)；Gaussian漂移与扩散合计贡献d，孔洞增加贡献n，恰好与几何漂移/均匀切分抵消。本轮作768条精确有理数恒等式检查。 + +两次事件间隔Δ内，给定初始y,g，Gaussian转移是 + + g_new=e^((1-η)Δ/2)g + + sqrt[y(e^Δ-e^((1-η)Δ))] Z。 (8.4) + +总离散事件率为(1+r)y；因为事件间y按e^s增长，下次事件的Δ满足 (1+r)y(e^Δ-1)~Exp(1)。这些公式给出无需保存长格子或完整Gaussian场的极限过程模拟法。本文的计算使用精确矩接口，没有把此模拟器的新样本当作点渗流数据。 + +重要区别：即使η=0，切分仍有Gaussian bridge抽取。它是在缩小区间时揭示原有空间噪声，不意味着物理标签又刷新了一次。 + +## 9. 更远的固定 p 热窗口：可能只有几何忘却，没有领先的标签更新 + +#780 研究固定p0H1(T^2)] in {0,1,2}. +``` + +Mertens--Ziff Eq. (11) states configurationwise + +```text +k4-k8-(K-E+F0) + = +1 if black has a cross-wrapping cluster, + = -1 if complementary matching white has a cross-wrapping cluster, + = 0 otherwise. +``` + +Their classification also records that single/spiral wrapping occurs in paired black/white components and contributes zero to this difference. + +But these three cases are exactly + +```text +r4=2 : black cross -> +1, +r4=0 : white cross / black no ambient homology -> -1, +r4=1 : one-dimensional ambient homology -> 0. +``` + +Therefore + +```text +boxed: +r4(omega)-1 + = k4(omega)-k8(omega^c)-K+E-F0. (1.1) +``` + +This is the square-site analogue of the bond/FK Euler/Krushkal localization, with the important difference that the matching white graph and the elementary black-face term are intrinsic to the site construction. + +Independent implementation check during this audit: lifted NN homology, black NN clusters, white G8 clusters and `(K,E,F0)` were recomputed for every configuration at honest `L=3` (512 states) and `L=4` (65,536 states); (1.1) had zero violations. This is a regression, not the proof; the published Euler argument is the all-size source. + +## 2. The familiar matching function is literally `E[r-1]` + +Take expectation under Bernoulli(p). Since + +```text +E K/L^2 = p, +E E/L^2 = 2 p^2, +E F0/L^2 = p^4, +``` + +we obtain + +```text +E[r4-1] + = N_L(p)-Nhat_L(1-p)-L^2[p-2p^2+p^4]. (2.1) +``` + +The left side is + +```text +P2-P0, +``` + +because `r-1` takes values `-1,0,+1`. Thus the Matching-One topological balance observable and the finite Sykes--Essam/Mertens--Ziff matching function are not merely asymptotically related or two different estimators: + +```text +boxed: +M_L(p)=P2-P0=E[r-1] + = cluster-count difference - local Euler polynomial. (2.2) +``` + +This is exactly the dictionary that should be used whenever cluster-number and rank-language branches meet. + +## 3. The entire bounded topological source localizes, not only its derivative + +Because (1.1) is configurationwise, for every real/complex source `h`, + +```text +exp[h(r-1)] + = exp[h k4] + exp[-h k8] + exp[-h K] + exp[+h E] + exp[-h F0]. (3.1) +``` + +Hence the exact finite source partition function + +```text +Z_L(p,h)=E_p exp[h(r-1)] + =P0 e^-h + P1 + P2 e^h +``` + +can also be read as a **two-colour cluster gas with local Euler interactions**: + +```text +black NN cluster fugacity : e^h, +white matching cluster fugacity : e^-h, +black site local factor : e^-h per occupied site, +black NN-pair factor : e^+h per occupied NN edge, +black plaquette factor : e^-h per all-black elementary face. +``` + +Multiplying the Bernoulli weight explicitly, + +```text +p^K q^(N-K) e^{h(r-1)} + = q^N + [(p/q)e^-h]^K + e^{hE-hF0} + e^{h k4-h k8}. (3.2) +``` + +So the source is not a mysterious closure label. It is an exact coupled black/white cluster fugacity deformation plus finite-range site/edge/plaquette factors. + +The cluster fugacities are nonlocal in a spin Hamiltonian sense, but are local in a connectivity transfer: they can be paid when a component retires/closes, exactly as in random-cluster transfer methods. + +## 4. Exact complement action on the sourced family + +The digital-Alexander/matching identity gives + +```text +r8(omega^c)=2-r4(omega), +``` + +so + +```text +X_hat=r8-1=-(r4-1)=-X. +``` + +Thus on the doubled square-site matching pair the rank source transforms exactly as + +```text +(p,h,G4) <-> (1-p,-h,G8). (4.1) +``` + +This is a genuine involution on the **doubled sourced finite model family**. + +It still does not prove that every local continuum scaling field has a scalar matching parity. The source `h` is a global/topological deformation represented by cluster fugacities. But it supplies one exact tangent direction whose pair-exchange action is known without CFT assumptions. + +## 5. Relation to the bond/FK Krushkal localization + +For bond/FK states the separate exact identity is + +```text +r-1 = k(A)-k(A*)+|A|-|V|, +``` + +which turns the source into primal/dual cluster fugacities and an edge factor. + +The site formula (1.1) is structurally parallel: + +```text +bond/FK : cluster imbalance + edge Euler term; +site : black/white matching cluster imbalance + site-edge-face Euler term. +``` + +Therefore the generic research idea “topological charge is an Euler imbalance between two complementary cluster gases” applies to both settings, although the microscopic local terms differ. + +This corrects an earlier overly sharp boundary that treated Euler localization as a bond/FK-only structural option. + +## 6. A new transfer/source route for the square-site master object + +The exact source object already used in the master scaling proposal is + +```text +Z_L(p;h,z)=P0 e^-h + P1 H_p(z)+P2 e^h. +``` + +Equation (3.2) supplies a concrete square-site implementation for the `h` direction: + +1. retain black NN and complementary white matching component closure counts; +2. add local weights for black site, NN edge and full plaquette motifs; +3. multiply retiring black/white components by `e^h/e^-h` respectively. + +The rank-one neutral source `z` remains a separate count of parallel essential components. Thus a two-source transfer no longer needs to treat `h` as an external after-the-fact rank label. + +A practical implementation should first reproduce the ordinary `h=0` safe/rank transfer and finite small-torus source polynomial before any massive/continuum interpretation. + +## 7. Relation to source normalization and #802 + +This Euler source makes the normalization issue explicit. If the local factors in (3.2) are inserted as unnormalized row weights, the physical free energy must include the row partition normalizer before comparing source derivatives. + +Moreover the local source contains degree-0/1/2/4 Bernoulli-chaos pieces plus cluster-count terms. Profiling the thermal direction therefore cannot be done by inspecting one raw motif derivative. The normalization-safe/source-quotient rules from the #802 audit still apply. + +The advantage is conceptual: the particular combination that equals `r-1` is fixed configurationwise, so all nuisance pieces are tied together by an exact identity rather than chosen ad hoc. + +## 8. New continuum question + +The useful continuum question is no longer simply + +```text +which local CFT field is matching-odd? +``` + +but rather + +> how does the exact finite Euler-imbalance source `h` decompose under RG into thermal/nuisance directions, topological torus sectors, and irrelevant angular/scalar corrections? + +The leading root correction is a correction to the zero of + +```text +partial_h log Z_L(p,h)|_(h=0). +``` + +This is a source-defined object. A continuum candidate is relevant only if it has nonzero overlap with this particular sourced response after the common/thermal pieces have been removed. + +This formulation avoids assigning an OPE parity before the source-to-RG map is built. + +## 9. Claim boundary + +Exact / published: + +- Mertens--Ziff configuration Euler identity; +- its three topological cases; +- identification of those cases with `r-1` under the rank convention; +- exponential source rewrite (3.1)--(3.2); +- complement action `h->-h` on the doubled sourced family. + +New synthesis / programme: + +- use the Euler-localized source directly inside the square-site transfer; +- study its RG decomposition as the primary source-defined route to the noncommon correction; +- combine it with the neutral source `z` in the master topological generating object. + +This does not establish local continuum operator parity or a new percolation theorem; it turns a known finite matching identity into the exact microscopic source dictionary needed by the current research programme. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/site-near-critical-renewal-programme-20260914.md b/docs/manuscripts/geometric-balance/site-near-critical-renewal-programme-20260914.md new file mode 100644 index 000000000..70cc3eaf8 --- /dev/null +++ b/docs/manuscripts/geometric-balance/site-near-critical-renewal-programme-20260914.md @@ -0,0 +1,244 @@ +# A square-SITE near-critical renewal programme: smaller than full universality + +2026-09-14. Proof programme distilled after auditing D'Alimonte--Manolescu's 2026 near-critical OZ construction and the classical Bernoulli near-critical toolbox. + +The strategic conclusion is that Matching-One does **not** need a proof of conformal universality for square-site percolation in order to obtain a useful near-critical OZ/renewal theorem. The independent-SITE model removes some of the hardest random-cluster boundary-condition issues. A comparability-level renewal theorem appears to require a much smaller set of inputs. + +Nothing in this note is promoted to an accepted theorem. Each missing gate is isolated explicitly. + +## 1. Why SITE may be simpler than the bond-FK proof + +In a random-cluster exploration, conditioning on the explored past changes the law in the future through induced wired/free boundary conditions. D'Alimonte--Manolescu therefore need robust RSW under boundary conditions and a nontrivial coupling/mixing construction to obtain a killed Markov renewal process with uniform mass gap. + +For Bernoulli SITE percolation: + +- site states are independent; +- after revealing an arbitrary explored set, all unrevealed sites remain iid Bernoulli(`p`); +- no wiring information propagates through the unexplored region; +- finite-energy ratios are explicit powers of `p/(1-p)`. + +Thus the difficult “future forgets the past” step should reduce to a **geometric separation/barrier** statement rather than a measure-comparison theorem. + +The matching 4/8 convention provides exactly the site-dual geometry needed to construct white barriers around black NN clusters. + +## 2. Characteristic length without critical exponents + +For black NN site percolation define a characteristic length `L(p)` below criticality by a fixed rectangle-crossing threshold. One possible convention is the smallest `L` such that the long-direction black crossing probability of a fixed-aspect rectangle is below a small constant `epsilon_0`. + +The desired properties are only up to bounded factors: + +\[ +L(p)\to\infty\quad(p\uparrow p_c), \tag{2.1} +\] + +and for rectangles/quads of diameter at most `cL(p)`, black and complementary white-matching crossing probabilities are uniformly nondegenerate. + +Critical square-site RSW on `Z^2` is available in the literature despite the lack of self-duality. The remaining near-critical extension is a finite-size criterion/RSW stability question, not an exponent-identification question. + +No use of `nu=4/3`, `5/48`, SLE, or conformal invariance is needed for the renewal architecture. + +## 3. Gate A: sub-characteristic RSW for the 4/8 pair + +**Target A.** There exist fixed constants `c,C,epsilon>0` such that, uniformly for `p=CL(p)`, + +\[ +P_p(0\leftrightarrow re) +\asymp +J_{left}(p,e)J_{right}(p,e) +\left(\frac r{L(p)}\right)^{-1/2} +\exp[-r/\xi_p(e)], \tag{7.1} +\] + +with + +\[ +\xi_p(e)\asymp L(p). \tag{7.2} +\] + +Gate B would identify the endpoint insertions up to constants as + +\[ +J_{left}J_{right}\asymp\pi_{1,site}(L(p))^2. \tag{7.3} +\] + +The same renewal process gives + +\[ +\boxed{D_p(e)\asymp\xi_p(e)} \tag{7.4} +\] + +and a Brownian bridge for the conditioned core. + +This already supplies nearly every near-critical structural input needed by #740/#758/#767, without any exact amplitude. + +## 8. Gate E: from an open connection to a complete winding component + +This is the genuinely Matching-One-specific step and should be kept separate from point-to-point OZ. + +On a cylinder of circumference `w=sL(p)`, define a clean renewal core that returns to its starting transverse row after one horizontal period. A cyclic renewal calculation predicts a closure density + +\[ +\asymp L(p)^{-1}s^{-1/2}e^{-cs}. \tag{8.1} +\] + +What remains is to compare this clean cyclic core to the **actual complete SITE component anchor activity**, including branches and the unique-anchor convention. + +A realistic first theorem is only comparability: + +\[ +\boxed{ +\nu_w(p) +\asymp +L(p)^{-1}J_{comp}(p,s)s^{-1/2}e^{-w/\xi_p},} \tag{8.2} +\] + +with `J_comp` bounded above/below on a declared range of `s`, or with its endpoint/mark dependence left explicit. + +This is strictly weaker than identifying `zeta(p)`, but it is enough to determine which logarithmic terms enter the crossover centre. + +## 9. A possible direct route through the existing tagged component construction + +The repository already has an exact microscopic tagged complete-component resolvent at each fixed width. Its weakness near criticality is state explosion when `w~xi->infinity`. + +The renewal programme suggests a renormalized version: + +1. group `O(L(p)) x O(L(p))` microscopic boxes into coarse blocks; +2. classify only the finite set of clean crossing/renewal states needed by Gate C; +3. carry the component's winding and unique-anchor mark at coarse scale; +4. integrate all microscopic branches inside each block into transition weights; +5. apply the existing matrix-sewing/logdet analysis to the coarse operator. + +If successful, the number of coarse states can remain `O(1)` as `p->pc`, while one winding uses `s=w/L(p)` coarse steps. + +This is a more plausible near-critical theorem engine than increasing the microscopic transfer width. + +## 10. Why complete conformal universality is unnecessary + +The programme uses only: + +- critical/sub-characteristic RSW; +- finite-size correlation length; +- arm stability or an explicit near-critical endpoint arm observable; +- independence of unexplored SITE variables; +- BK/Harris; +- classical local CLT for a killed Markov-additive process. + +It does not need: + +- identification with SLE6; +- exact critical exponents; +- conformal covariance of square-site scaling limits; +- a proof that square-site and bond-FK amplitudes agree. + +So the correct research question is not “can we prove universality for square-site percolation?” but rather “can we build a uniform correlation-length-scale renewal decomposition for independent SITE?” + +## 11. Suggested proof order + +The shortest dependency chain is: + +1. freeze a precise `L_site(p)` convention and prove Gate A; +2. build the clean white-matching barrier / black survivor block and prove Gate C; +3. prove Gate D and invoke killed-renewal local CLT; +4. first state the point-to-point theorem with the near-critical endpoint insertion left as `pi_{1,site}(p)`; +5. only then prove/quote Gate B to replace it by the critical one-arm probability; +6. finally attack Gate E for complete component activity. + +This order avoids making the hardest SITE-specific insertion issue a prerequisite for the renewal skeleton itself. + +## 12. Claim boundary + +Critical RSW for square-site percolation on `Z^2` is a published theorem. Classical near-critical arm/correlation-length stability exists for planar Bernoulli percolation, but the exact square-site/matching formulations required above have **not** been independently certified in this note. Gates A--E remain a proof programme until those inputs and constructions are written out. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/site-random-cluster-q-tangent-of-matching-function-20260914.md b/docs/manuscripts/geometric-balance/site-random-cluster-q-tangent-of-matching-function-20260914.md new file mode 100644 index 000000000..202543152 --- /dev/null +++ b/docs/manuscripts/geometric-balance/site-random-cluster-q-tangent-of-matching-function-20260914.md @@ -0,0 +1,230 @@ +# The Matching function is exactly a site-random-cluster `Q` tangent difference + +Date: 2026-09-14 + +Status: exact finite identity plus a conditional continuum interface. This note partially corrects `euler-source-vs-common-q-tangent-20260914.md`: the **full nonlinear Euler `h` source family** is not the ordinary common-`Q` deformation, but the first charge moment used by Matching One is nevertheless exactly a difference of ordinary cluster-fugacity `Q` derivatives. + +## 1. Site random-cluster generating function + +For a finite site graph `G` with `N` vertices, define + +```text +Z_G(p,Q) + = sum_omega + p^{K(omega)} (1-p)^{N-K(omega)} + Q^{k_G(omega)}. (1.1) +``` + +This is the standard site random-cluster deformation of Bernoulli site percolation: `Q=1` gives the independent site measure, while `Q!=1` rewards or penalizes occupied clusters. + +Because the Bernoulli weights are normalized, + +```text +Z_G(p,1)=1 (1.2) +``` + +for every `p`. + +Therefore + +```text +boxed: +partial_(log Q) log Z_G(p,Q)|_(Q=1) + = E_p[k_G]. (1.3) +``` + +## 2. Exact representation of the Matching-One observable + +The finite square-site Euler identity is + +```text +X=r4-1 + = k4(omega)-k8(omega^c)-K+E-F0. +``` + +Taking expectation under black density `p`, the complementary white matching configuration has the same law as site occupation with density `1-p` on `G8`. Also + +```text +E[K-E+F0] + = N [p-2p^2+p^4]. (2.1) +``` + +Hence exactly + +```text +boxed: +M_L(p)=P2-P0 + = partial_(log Q) + [ log Z_G4(p,Q) + -log Z_G8(1-p,Q) ]_(Q=1) + -N[p-2p^2+p^4]. (2.2) +``` + +Thus the finite matching function is a **difference of two ordinary site-random-cluster cluster-number tangents**, with the known local Euler polynomial subtracting the bulk/local piece. + +This is stronger than saying the full Euler `h` source looks like a two-colour cluster gas. Equation (2.2) uses the standard one-colour common cluster fugacity `Q` on each graph separately. + +## 3. Important correction: first `Q` derivative is automatically thermal-nuisance free at `Q=1` + +Let `p_c(Q)` be any differentiable critical curve through `p_c(1)`. Along that curve, + +```text +d/d(log Q) log Z_G(p_c(Q),Q)|_1 + = partial_(log Q) log Z_G|_1 + + (dp_c/dlogQ)|_1 partial_p log Z_G(p,1). +``` + +But from (1.2), + +```text +partial_p log Z_G(p,1)=0 (3.1) +``` + +identically. + +Therefore + +```text +boxed: +[d/d(logQ) along any p(Q)] log Z_G|_(Q=1) + = partial_(logQ) log Z_G|_(Q=1). (3.2) +``` + +So the **first cluster-number Q tangent does not need a thermal nuisance subtraction**. The normalized `Q=1` partition function kills that chain-rule term automatically. + +This is unlike a generic unnormalized microscopic source response, where pressure/thermal normalization must be handled explicitly. + +At second and higher `Q` derivatives, mixed thermal terms re-enter; (3.2) is a first-derivative statement. + +## 4. Relation to the Euler `h` source + +The full topological source satisfies configurationwise + +```text +e^{hX} + = e^{h k4-h k8-hK+hE-hF0}. +``` + +That nonlinear family is not the same as changing one common `Q` in both site random-cluster models. + +However its first derivative at `h=0` is exactly `E[X]=M`, and equation (2.2) gives an alternative representation of the same first moment. + +Thus both statements are true: + +```text +full h-source family != common-Q family, +first Matching charge M = difference of common-Q tangents + local Euler term. +``` + +The earlier note was too pessimistic when it used the first statement to weaken the generic-Q route for the root observable itself. + +## 5. Continuum significance + +A site random-cluster model with cluster fugacity `Q` is an established statistical model. Existing numerical work (Wang et al., Phys. Rev. E 92, 022127 (2015), arXiv:1411.4408) reports thermal and magnetic critical exponents matching the bond random-cluster/Potts values for `Q=1.5,2,2.5,3,3.5,4`, with first-order behaviour beyond the usual range. + +This is evidence—not a rigorous theorem—that the critical site-RC `Q` family realizes the same Potts/random-cluster continuum universality branch. + +If that interface is accepted, then the Q-tangent representation (2.2) gives a much more direct route from Matching One to generic-Q CFT data than an abstract parity assignment. + +## 6. Why logarithmic collisions become genuinely relevant again + +Suppose a continuum contribution has generic-Q form + +```text +A(Q) L^{-x(Q)}. +``` + +Then its cluster-number derivative contains + +```text +partial_Q [A(Q)L^{-x(Q)}]_(Q=1) + = L^{-x(1)} + [A'(1)-A(1)x'(1) log L]. (6.1) +``` + +Therefore a collision/splitting of dimensions as `Q->1` can generate logarithmic finite-size terms in precisely the kind of `Q` derivative that appears in (2.2). + +This does **not** prove that the V14/W(2,2) collision controls the post-H4 residual: one still needs the actual amplitudes/map sectors and the difference between the G4 and G8 regularizations. But generic-Q splitting is no longer merely an unrelated tangent direction. + +## 7. The correct object is a difference of Q tangents between two microscopic regularizations + +Write the critical finite-size/site-RC free energy schematically as + +```text +log Z_G(p_c,Q) + = N f_G(Q) + + F_G^torus(Q,tau) + + sum_j u_j^G(Q) L^{-omega_j(Q)} ... . +``` + +Equation (2.2) takes + +```text +partial_(logQ) [G4-G8] at Q=1 +``` + +and subtracts the local Euler polynomial. + +The infinite-volume Sykes--Essam relation explains the cancellation of the extensive cluster-density part at `Q=1`. The remaining torus/irrelevant difference is exactly where the root information lives. + +Thus the useful continuum question is: + +> which generic-Q finite-size blocks have different `Q`-tangent amplitudes in the NN and matching site-RC regularizations after the exact local Euler term is removed? + +This is a source-defined alternative to calling fields “matching odd.” + +## 8. A revised interpretation of the `x=21/4` and `x=33/4` candidates + +For the leading H4 block, the observable lattice fact is that the `Q`-tangent difference between the two site regularizations first becomes visibly nonzero at effective total dimension `21/4`. + +For the post-H4 scalar candidate, a generic-Q collision at `x=33/4` can generate a pure-power and/or logarithmic contribution to the same cluster-number tangent difference. + +The correct hierarchy is therefore + +```text +angular irrep first, +generic-Q Q-tangent block second, +regularization-difference amplitude third, +operator/Jordan naming last. +``` + +## 9. Practical finite test + +A tiny finite-torus implementation can introduce `Q` as a symbolic/automatic-differentiation cluster fugacity independently for + +```text +black NN cluster count, +white matching cluster count. +``` + +At `Q=1`, verify + +```text +partial_logQ log Z4 = E k4, +partial_logQ log Z8 = E k8, +``` + +and reproduce the Euler rank observable through (2.2). + +At generic `Q` one can then inspect the two cluster models separately without inventing a local continuum parity. Large generic-Q production should wait until the relevant angular block/source dictionary is specified. + +## 10. Claim boundary + +Exact finite: + +- site-RC generating function (1.1); +- cluster-number derivative (1.3); +- Matching function representation (2.2); +- first-derivative critical-line invariance (3.2). + +External empirical interface: + +- site random-cluster and bond random-cluster/Potts universality agreement is numerical evidence in the cited work, not proved here. + +Open continuum: + +- exact mapping of the square-site Q tangent onto the generic-Q interchiral/logarithmic blocks; +- nonzero amplitude of any particular V14/W22 collision in the G4-G8 difference; +- higher-Q derivative/contact structure. + +The key correction is positive: **the root observable itself is already an ordinary site-random-cluster Q tangent difference, even though its full bounded `h` source family is not.** \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/spin4-nodal-pell-charge-test-20260914.md b/docs/manuscripts/geometric-balance/spin4-nodal-pell-charge-test-20260914.md new file mode 100644 index 000000000..a4181044e --- /dev/null +++ b/docs/manuscripts/geometric-balance/spin4-nodal-pell-charge-test-20260914.md @@ -0,0 +1,253 @@ +# Pell approximants to the spin-four nodal direction: an alternating w^-6 charge-root test + +2026-09-14. Sharp arithmetic/CFT falsification target for the sector-odd spin-four anisotropy hypothesis. + +The square-lattice hypothesis predicts an orientation-dependent semi-infinite charge-root shift + +\[ +p_{\ell}^{ch}(\theta)-p_c +\sim -A_4\cos(4\theta)\,\ell^{-4}, \tag{1} +\] + +where `ell` is the physical Euclidean circumference and `theta` is the angle of the short period relative to a lattice axis. Reflection fixes a cosine rather than a general phase. + +The spin-four nodes are at + +\[ +\theta=\pi/8,3\pi/8,\ldots, \tag{2} +\] + +but `tan(pi/8)=sqrt(2)-1` is irrational, so no nonzero integer period lies exactly on the node. This makes Pell approximants a feature, not a nuisance: their angular error is precisely strong enough to convert the `ell^-4` law into a clean alternating `ell^-6` prediction. + +## 1. Integer spin-four harmonic + +For a primitive integer period + +\[ +u=(a,b), +\qquad +\ell^2=a^2+b^2, \tag{1.1} +\] + +the real spin-four harmonic is exactly + +\[ +\boxed{ +\cos4\theta +=\frac{a^4-6a^2b^2+b^4}{(a^2+b^2)^2}.} \tag{1.2} +\] + +Factor the numerator: + +\[ +\boxed{ +a^4-6a^2b^2+b^4 +=(a^2-2ab-b^2)(a^2+2ab-b^2).} \tag{1.3} +\] + +The nodal equation is therefore + +\[ +a^2-2ab-b^2=0, \tag{1.4} +\] + +whose positive slope is `b/a=sqrt(2)-1`. + +## 2. Pell sequence + +Choose coprime positive integer solutions of + +\[ +\boxed{a_n^2-2a_nb_n-b_n^2=(-1)^n} \tag{2.1} +\] + +(up to an indexing shift). Equivalently, with + +\[ +x_n=a_n-b_n, \tag{2.2} +\] + +we have the standard Pell equations + +\[ +x_n^2-2b_n^2=\pm1. \tag{2.3} +\] + +One convenient sequence is + +\[ +(a,b)=(2,1),(5,2),(12,5),(29,12),(70,29),(169,70),\ldots \tag{2.4} +\] + +with alternating sign in (2.1), and + +\[ +\frac{b_n}{a_n}\to\sqrt2-1. \tag{2.5} +\] + +## 3. Exact angular leakage scale + +On the Pell sequence, (1.3) and (2.1) give + +\[ +\ell_n^2\cos4\theta_n +=(-1)^n +\frac{a_n^2+2a_nb_n-b_n^2}{a_n^2+b_n^2}. \tag{3.1} +\] + +Taking the nodal ratio limit, + +\[ +\boxed{ +\ell_n^2\cos4\theta_n\longrightarrow(-1)^n\sqrt2.} \tag{3.2} +\] + +Numerically the right side is already visible: + +```text +(a,b) ell^2 cos(4 theta) +(2,1) -1.4000000000 +(5,2) +1.4137931034 +(12,5) -1.4142011834 +(29,12) +1.4142131980 +(70,29) -1.4142135516 +(169,70) +1.4142135621 +``` + +Thus the best Diophantine approach to the spin-four node naturally supplies a `1/ell^2` residual harmonic with an alternating universal geometric coefficient. + +## 4. Predicted pseudo-critical root + +Insert (3.2) into the spin-four root law (1): + +\[ +p_n^{ch}-p_c +\sim +-A_4\cos4\theta_n\,\ell_n^{-4}. \tag{4.1} +\] + +Then + +\[ +\boxed{ +p_n^{ch}-p_c +\sim (-1)^{n+1}A_4\sqrt2\,\ell_n^{-6}.} \tag{4.2} +\] + +So the Pell sequence predicts THREE simultaneous signatures: + +1. apparent correction exponent `6` rather than `4`; +2. sign alternation with Pell parity; +3. after multiplying by `(-1)^{n+1} ell^6`, the amplitude should approach `sqrt(2) A_4`, where `A_4` is the axial spin-four amplitude in physical-circumference normalization. + +A generic scalar correction cannot reproduce the parity alternation tied to the angle error. + +## 5. Diagonal direction gives an even simpler sign test + +Before attempting the nodal Pell sequence, use the exact diagonal direction + +\[ +\theta=\pi/4. \tag{5.1} +\] + +Then + +\[ +\cos4\theta=-1. \tag{5.2} +\] + +Thus the spin-four hypothesis predicts a leading **sign reversal** relative to the axial cylinder: + +\[ +\boxed{ +\operatorname{sign}(p_{diag}^{ch}-p_c) +=-\operatorname{sign}(p_{axis}^{ch}-p_c).} \tag{5.3} +\] + +Since the observed axial sequence has `p_axis^ch < p_c`, a diagonal short period should have + +\[ +\boxed{p_{diag}^{ch}>p_c} \tag{5.4} +\] + +for large circumference. + +If the integer diagonal period is `n(1,1)`, its physical circumference is + +\[ +\ell=\sqrt2 n. \tag{5.5} +\] + +Hence equal physical spin-four amplitude predicts the raw-integer scaling + +\[ +p_{diag}^{ch}-p_c +\sim +\frac{A_4}{4n^4}, \tag{5.6} +\] + +versus `-A_4/n^4` on the axis. + +This is the cheapest direct falsification target. + +## 6. Why the Pell node is more discriminating than an ordinary w^-6 fit + +The square axial root already has subleading powers `w^-6,w^-8,...`. An ordinary fit cannot tell whether the `w^-6` coefficient is + +- a higher descendant of the same spin-four family; +- a different scalar sector-odd field; +- an analytic lattice correction. + +Along the nodal Pell sequence, every correction carrying the SAME spin-four angular factor receives the extra + +\[ +\cos4\theta_n=O(\ell_n^{-2}). \tag{6.1} +\] + +Thus an axial spin-four descendant that would produce `ell^-6` becomes `ell^-8` on the Pell node. The leading `ell^-6` term (4.2) instead comes from angular leakage of the **leading** spin-four field itself. + +If an additional orientation-independent sector-odd contribution also exists at exponent six, the scaled Pell data should split as + +\[ +\ell^6(p_n^{ch}-p_c) +=B_6+(-1)^{n+1}\sqrt2 A_4+o(1). \tag{6.2} +\] + +Then averaging even/odd subsequences isolates `B_6`, while their difference isolates `A_4`. This makes the Pell design a direct detector of an otherwise hidden scalar/angle-independent correction. + +## 7. Period geometry for a clean cylinder + +For each Pell short vector `u_n=(a_n,b_n)`, one clean finite-torus realization is + +\[ +\Lambda_{n,m} +=\langle +(a_n,b_n), + m(-b_n,a_n) +\rangle. \tag{7.1} +\] + +The long period is Euclidean-orthogonal to the short one, and + +\[ +N=m(a_n^2+b_n^2)=m\ell_n^2. \tag{7.2} +\] + +Taking `m` sufficiently large produces the semi-infinite-cylinder charge root without introducing a continuum shear. The physical NN interaction is NOT rotated; only the quotient period is oblique, exactly as required by the directional programme. + +For a direct transfer implementation it may be computationally preferable to use a Bezout longitudinal step plus a shear bookkeeping variable. The final root should be independent of that transfer basis after the physical circumference/orientation are declared correctly. + +## 8. Interface to #765 and the projective-slope controls + +The directional mass work on this branch already proves that fixed or converging integer directions are legitimate probability objects and that nonparallel period classes separate in exponential elongation. + +The present test is finer: it concerns the **irrelevant finite-width correction to the critical charge-sector crossing**, not the leading directional mass. The spin-four phase is the same embedded harmonic + +\[ +\frac{(a+ib)^4}{(a^2+b^2)^2} \tag{8.1} +\] + +used in the projective-slope positive control, but the two observables should not be identified. They merely share the same geometric spin transformation. + +## 9. Claim boundary + +The Pell algebra (1.2)--(3.2) is exact. The sign reversal and alternating `ell^-6` root law are consequences of the sector-odd spin-four hypothesis, not established percolation theorems. Their main value is falsifiability: an oblique semi-infinite charge transfer can reject the spin-four mechanism without relying on another lattice or on a multi-parameter CFT fit. diff --git a/docs/manuscripts/geometric-balance/spin4-operator-status-post-compass-20260914.md b/docs/manuscripts/geometric-balance/spin4-operator-status-post-compass-20260914.md new file mode 100644 index 000000000..f419473e3 --- /dev/null +++ b/docs/manuscripts/geometric-balance/spin4-operator-status-post-compass-20260914.md @@ -0,0 +1,260 @@ +# Post-compass status of the matching-odd spin-four operator question + +Date: 2026-09-14 + +Status: evidence synthesis / research prioritization. This note does not turn numerical support into a theorem and does not identify the full `Q=1` logarithmic module. Its purpose is to stop treating logically different uncertainties as if they were one flat candidate list. + +## 1. Four separate questions + +The matching-root correction problem should now be decomposed into four questions. + +```text +A. What angular irrep carries the leading observed correction? +B. What radial/scaling dimension carries that irrep? +C. How does that field project onto the thermal/root response? +D. Which c=0 Kac/logarithmic module realizes the field? +``` + +The current evidence has very different strength at these four levels. + +## 2. A. Angular irrep: spin four is strongly supported + +### Historical prospective Gaussian evidence + +The frozen norm-five experiment in #57 compared the same-radial angular characters before child data were revealed. The canonical 500M-per-child score was + +```text +H4: chi2 = 0.4163 / 2 +H12: chi2 = 35.1931 / 2 +H8: chi2 = 16.0120 / 2 +zero: chi2 = 1.7764 / 2. +``` + +Thus H4 resolved the designed H4/H12 alias prospectively and strongly beat H12/H8. The zero model was not rejected by that child block alone, so this block establishes angular character more strongly than nonzero amplitude. + +### Current deterministic safe-transfer evidence + +The same microscopic square-site NN/complementary-matching model, with only the integer cylinder direction changed, obeys + +```text +critical safe-sector mismatch ~ cos(4 theta), +charge-root shift ~ -cos(4 theta), +``` + +across axis, diagonal and several oblique directions. Equal-physical-circumference H4 projectors remove most of the raw root bias without using a threshold reference. + +This second evidence block supplies the strong nonzero amplitude missing from the #57 angular-only discrimination. + +### Current verdict + +```text +LEADING ANGULAR IRREP = H4 / spin 4: STRONG. +``` + +A scalar/spin-zero correction may survive as a smaller post-H4 residual, but it is strongly disfavoured as the dominant source of the observed `ell^-4` root displacement. + +## 3. B. Radial dimension: x=21/4 has independent prospective support + +The `x=21/4` law is not being inferred only from the root exponent. + +### Gaussian doubling + +Fresh prospective Gaussian doubling used the parameter-free prediction + +```text +DeltaM(2N)/DeltaM(N) + = -2^(-13/8). +``` + +The minus sign tests spin four and the magnitude tests the `N^-13/8`, equivalently `L^-13/4`, law. Two fresh lineages gave a joint covariance-aware + +```text +chi2 = 0.03445 / 2. +``` + +No exponent or amplitude was fitted to the child data. + +### N=185/265 prospective radial test + +The predeclared matching-odd H4 predictions gave + +```text +x=21/4: chi2 = 3.04598 / 2, +x=17/4: chi2 = 30.24613 / 2, +zero: chi2 = 29.40938 / 2. +``` + +The formal `x=17/4` non-diagonal competitor was therefore rejected on genuinely new geometries. A proposed `x=14/3` V13 competitor was removed before scoring because its Kac-branch/parity construction was invalid for the declared channel. + +### Evidence ledger + +Across the independent primary matching-odd blocks currently registered in the canonical evidence ledger, + +```text +H4_x21_over_4: +chi2 = 3.31346 over 5 dimensions. +``` + +### Current verdict + +```text +LEADING H4 RADIAL DIMENSION x=21/4: STRONG NUMERICAL SUPPORT. +``` + +This still does not say which same-dimension `Q=1` module contributes. + +## 4. C. Critical response direction: ordinary thermal Q4 is exactly tangent + +The new cylinder Ward calculation gives a structural result unavailable from exponent fitting. + +For the ordinary `c=0,h=5/8` thermal Kac quotient, + +```text +Q4=40L_-2^2-60L_-3L_-1-9L_-4. +``` + +On a translation-invariant cylinder primary level, + +```text +/ = h_t/240, +``` + +independently of the external primary state. Null reduction plus translation invariance gives + +```text +Q4 -> (493/3)L_-4, +``` + +hence + +```text +/ = 493/1152 +``` + +for the chiral descendant in the circumference-`2pi` normalization. + +Therefore, at the critical point, an ordinary thermal-Q4 insertion changes every cylinder excitation energy in a direction proportional to its thermal-primary matrix element. In other words: + +```text +ordinary thermal Q4 is exactly thermal-tangent at X=0. +``` + +The equal-circumference transfer data add the nontrivial massive statement that this tangency remains extremely accurate after root-centering and slope normalization over a finite local near-critical interval. + +### Current verdict + +```text +CRITICAL TANGENCY, CONDITIONAL ON ORDINARY THERMAL Q4 IDENTITY: EXACT. +FINITE-X TANGENCY: STRONG FINITE DETERMINISTIC EVIDENCE. +``` + +This explains the root shift without invoking a logarithmic partner. + +## 5. D. Logarithmic/module identity: open, but the large-log alternative is not equally supported + +At `Q=1`, the energy field can collide with the two-hull field and form a logarithmic multiplet. Their level-four spin-four descendants share the same `x=21/4` at the collision point, so angular character and radial dimension alone cannot distinguish them. + +However, the data do not currently require a large logarithmic admixture in the matching-odd H4 amplitude. + +- The historical operator note records that adding a free logarithmic parameter worsened held-out prediction relative to the pure `13/8` law on the then-available range. +- The fresh Gaussian doubling test passed the exact pure-power ratio in two independent lineages with `chi2=0.03445/2`. +- The canonical multi-block `H4_x21_over_4` pure-power score remains good. + +This is not a theoretical exclusion of a Jordan module. A logarithmic partner can have a small coefficient in this particular microscopic source/observable even if it is required elsewhere in the `c=0` theory. + +### Working hypothesis + +The most economical current hypothesis is + +```text +observed leading matching-odd spin4 response + = ordinary thermal-Q4 bottom-field component + + smaller logarithmic/other normal component. +``` + +The burden of proof should now be on a large logarithmic component, not on the existence of the ordinary tangent piece. + +## 6. Where to look for the logarithmic partner now + +Because the ordinary Q4 branch already explains the leading root displacement, the logarithmic question should be targeted at observables where the exact tangent piece is removed or can be separated. + +High-information targets are: + +1. **root/slope-normalized angular residual** + +```text +G4_perp(X) + = G4(X)-[G4(0)/F'(0)]F'(X); +``` + +2. **generic-Q branch splitting** between energy and two-hull families before the `Q=1` collision; +3. **logarithmic size dependence** of the H4 amplitude after ordinary H4 power corrections are projected/controlled; +4. **moving-root-normalized original-U**, where the leading thermal tangent is explicitly removed by contract; +5. a map/projector-resolved observable known to couple differently to the two generic-Q branches. + +A free `A+B log L` fit to the raw root is now a low-information test because the dominant ordinary piece is already known and power corrections remain. + +## 7. Relation to the post-H4 residual hierarchy + +The leading H4 root tower can be removed algebraically using same-circle angular projectors. The remaining residual then probes genuinely different information: + +```text +H8 / spin8, +angle-independent scalar/log channel, +H12 or higher D4 harmonic, +normal component of the spin4 field. +``` + +The simple quadratic thermal-coordinate nonlinearity of the leading tangent H4 shift is too small to explain the observed post-H4 residual at `ell~8--10`. + +Thus there are now two orthogonal residual programmes: + +```text +spin4-normal / logarithmic residual, +post-H4 higher-irrep residual. +``` + +They should not be conflated. + +## 8. Updated candidate ordering + +For the **observed leading root correction**, the current ranking is: + +```text +1. ordinary thermal-family Q4 bottom component, spin4, x=21/4; +2. same spin/dimension logarithmic admixture as a correction to (1); +3. other spin4 x=21/4 map/defect realization; +4. leading scalar x=21/4 explanation -- strongly disfavoured by same-ell angular projection. +``` + +For the **post-H4 residual**, the ranking is separate and currently less settled: + +```text +H8 identity-family spin8 candidate, +small scalar/log channel, +H12/higher harmonic, +other normal spin4 contribution. +``` + +## 9. Research consequence + +The main root-mechanism question has changed from + +```text +which field can produce an L^-4 shift? +``` + +to + +```text +what survives after subtracting the ordinary thermal-Q4 tangent that already explains the leading shift? +``` + +This is a materially smaller and more falsifiable problem. It also aligns with the research-compass principle: do not keep solving the already-explained leading effect; use the residual to identify information the leading effective description necessarily discards. + +## 10. Claim boundary + +- Historical chi-square values are evidence-ledger / preregistered numerical results, not new calculations in this note. +- The cylinder Ward tangency is exact only conditional on the ordinary thermal Kac quotient and standard conformal perturbation interpretation. +- The ranking of logarithmic admixture is a research judgement based on current evidence, not an LCFT theorem. +- A small logarithmic coefficient in this observable does not imply absence of the energy--two-hull logarithmic multiplet from percolation. diff --git a/docs/manuscripts/geometric-balance/spin4-second-order-response-tensor-20260914.md b/docs/manuscripts/geometric-balance/spin4-second-order-response-tensor-20260914.md new file mode 100644 index 000000000..78f98a2bd --- /dev/null +++ b/docs/manuscripts/geometric-balance/spin4-second-order-response-tensor-20260914.md @@ -0,0 +1,196 @@ +# Second-order spin-four response tensor after the leading matching-odd root correction + +2026-09-14. + +Status: symmetry/RG synthesis motivated by the deterministic safe-root controls. The representation-theory decomposition below is exact at the level of rotation characters; identifying the microscopic lattice couplings with particular CFT fields and computing their response coefficients remain open. + +## 1. Why the old `cos(4theta)^2=(1+cos8theta)/2` shorthand is too strong + +The repository's composite-field oracle correctly states that the product of two real aligned spin-four harmonics has support + +```text +H4(theta)^2 = (H0+H8(theta))/2. +``` + +That is an exact statement about the elementary angular monomial. It does **not** imply that a general second-order observable response must have equal scalar and spin-eight coefficients. + +The distinction matters now because post-H4 safe-root data resolve a q=2 H8 response, while the simplest locked `P0=P8` interpretation is unstable. + +## 2. Complex spin basis + +Write the two real spin-four perturbations as + +```text +T4(theta) = t_+ e^{+4 i theta} O_T^+ + t_- e^{-4 i theta} O_T^-, +I4(theta) = i_+ e^{+4 i theta} O_I^+ + i_- e^{-4 i theta} O_I^-, +``` + +with reflection reality `t_-=conj(t_+)`, `i_-=conj(i_+)` for the square-lattice perturbation. + +At second order, rotational covariance permits two inequivalent tensor contractions: + +```text +spin 0: + t_+ i_- R_0[O_T^+,O_I^-] ++ t_- i_+ R_0[O_T^-,O_I^+], + +spin 8: + t_+ i_+ R_8[O_T^+,O_I^+] ++ t_- i_- R_8[O_T^-,O_I^-]. +``` + +The response functionals `R_0` and `R_8` are different integrated connected correlators/OPE channels. Symmetry does not set them equal. + +For a reflection-even square orientation they reduce schematically to + +```text +delta O^(2)(theta) + = C0 * H0 + C8 * H8(theta), +``` + +with independent `C0,C8` unless an additional factorization/dynamical identity is proved. + +Thus the elementary `1:1` coefficient occurs only in a special aligned-factorized model where the observable response treats the two tensor channels with the same coefficient. It is not a generic consequence of `4 tensor 4 = 0 plus 8`. + +## 3. Matching parity and radial exponent + +Let the leading matching-odd thermal spin-four coupling be + +```text +T4: omega_T=13/4, +``` + +and an even spin-four lattice anisotropy be + +```text +I4: omega_I=2. +``` + +Their mixed second-order response is matching odd and has total irrelevant power + +```text +omega_T+omega_I = 21/4. +``` + +For the charge/root observable, relative to the leading T4 root term this adds two powers of length: + +```text +leading T4 root: ell^-4, +T4 x I4 mixed root: ell^-6. +``` + +The spin-zero and spin-eight tensor channels share this radial exponent but need not share amplitudes. + +## 4. A separate q=2 H4 dressing channel + +The common `omega≈2` finite-size correction is not purely spin four. Existing oblique magnetic-gap controls indicate that its angular-scalar component is substantially larger than its H4 component. + +Denote the even scalar part by `S0`. Then + +```text +T4 x S0 +``` + +is matching odd, remains in the H4 angular sector, and also gives a root `ell^-6` correction. + +Therefore the minimal q=2 root structure is naturally + +```text +p_root-pc + = c4 H4 ell^-4 + + c46 H4 ell^-6 [T4 x S0 / radial dressing] + + c80 H0 ell^-6 [T4 x I4, spin0 tensor] + + c88 H8 ell^-6 [T4 x I4, spin8 tensor] + + ... . +``` + +There is no reason to identify `c46`, `c80`, and `c88`. + +## 5. What the new Gaussian controls actually determine + +Two norm-2 Gaussian lineages are especially clean because child and parent have opposite H4 but the same H8. + +### Lineage A + +```text +(2,1) -> (3,1), n=2,3 +H4: -7/25 -> +7/25 +H8: -527/625 -> -527/625. +``` + +### Lineage B — frozen H8-node negative control + +```text +(3,2) -> (5,1), n=1,2 +H4: -119/169 -> +119/169 +H8: -239/28561 -> -239/28561. +``` + +For each lineage, `pc` and every angular-scalar contribution cancel from the child-parent difference. Fitting + +```text +D_n = p_child-p_parent = alpha n^-4 + beta n^-6 +``` + +therefore isolates the odd H4 leading/dressing pieces plus the known small effect of the common H8 value through radial rescaling. + +Combining the two lineages gives, within the finite q=2 model, + +```text +c46 = -0.1156140546, +c88 = -0.1381781858, +``` + +without using an external `pc`. + +The second lineage is close to an H8 node, so its `beta` is primarily a measurement of `c46`. Its frozen pre-target prediction agreed at the few-percent level. This provides a genuine negative control for the H8 decomposition. + +What these differences do **not** determine is the scalar tensor coefficient `c80`, because scalar angular pieces cancel under Gaussian orientation differences. + +## 6. Consequence: do not infer scalar q=3 from an H4 projector alone + +A two-angle H4 projector retains both angular-scalar and H8 pieces. The earlier axis/(3,4) `ell=5,10` sequence happened to mimic `ell^-7` extremely well. The cross-node control with nearly opposite H8 projector weight falsifies the interpretation that this was dominantly one scalar `ell^-7` field. + +The correct hierarchy is now: + +```text +observed leading: c4 H4 ell^-4; +observed next: c46 H4 ell^-6 + c88 H8 ell^-6; +unknown: c80 H0 ell^-6 and later scalar/H8 blocks. +``` + +A genuine scalar `x≈33/4` block may still coexist, but it is no longer required by the two-scale H4-projected residual. + +## 7. Continuum field-theory target + +If `T4` is the thermal-family level-four descendant and `I4` is the leading even identity/lattice anisotropy, the next nontrivial CFT calculation is not another exponent fit. It is the pair of integrated second-order matrix elements + +```text +R_0 ~ integral _conn, +R_8 ~ integral _conn, +``` + +including all counterterms/contact terms required by conformal perturbation theory and the actual safe/rank projector. + +At c=0, logarithmic mixing/null descendants can modify either channel independently. The calculation should therefore be done first at generic Q if singular collisions obstruct the Q=1 tensor decomposition. + +## 8. Modular fingerprint conjecture + +In the simplest ordinary-module scenario: + +- the leading thermal Q4 one-point has weight-four fingerprint `E4(tau)`; +- an ordinary same-chirality spin-eight mixed response naturally lies in weight eight, whose full-modular holomorphic space is generated by `E8=E4^2`; +- the spin-zero mixed channel belongs to a different nonholomorphic/modular-scalar structure and need not share its amplitude. + +This predicts that the mixed spin-eight channel vanishes at the hexagonal elliptic point where `E4=0`, more strongly than a generic unrelated H8 defect sector need do. + +This is a falsifiable shape prediction, not a theorem for the Matching-One observable until its torus/map dictionary is fixed. + +## 9. Practical next tests + +1. Use source-resolved Perron/Feynman--Hellmann derivatives to identify a physical source with nonzero projection on the common even spin-four correction, distinct from the exact thermal-reparametrization sources already rejected. +2. Measure the mixed odd/even source Hessian and project it into spin0 and spin8 angular channels. +3. In parallel, test the weight-eight/hexagonal-zero fingerprint on existing or bounded Pell/modulus assets. +4. Only after `R_0/R_8` is theoretically or empirically constrained should a scalar post-H4 block be inferred from a two-angle H4 projector. + +The machine-readable finite controls are in `gaussian-h8-node-q2-control-20260914.json` and `cross-node-post-h4-projector-20260914.json`. diff --git a/docs/manuscripts/geometric-balance/strict-finite-reflection-witness-20260914.md b/docs/manuscripts/geometric-balance/strict-finite-reflection-witness-20260914.md new file mode 100644 index 000000000..9a9663d0c --- /dev/null +++ b/docs/manuscripts/geometric-balance/strict-finite-reflection-witness-20260914.md @@ -0,0 +1,155 @@ +# A matching-only essential cycle makes finite reflection dominance strict + +2026-09-14. Deterministic witness completing the strictness clause in `finite-reflection-dominance-20260914.md` for every nondegenerate honest square-cell torus with sufficiently separated local lifts (in particular throughout the asymptotic regime of #739). + +No percolation estimate is used: one explicit occupied set has positive product probability for every `0r_4(\omega).} \tag{1.2} +\] + +We will in fact construct + +\[ +r_4(\omega)=0, +\qquad +r_8(\omega)=1. \tag{1.3} +\] + +## 2. Non-axis shortest period + +Let `u=(a,b)` be any nonzero period with both coordinates nonzero. Apply an exact square-lattice reflection/quarter-turn so that + +\[ +a\ge b>0. \tag{2.1} +\] + +Consider the lifted vertex sequence + +\[ +(0,0),(1,1),\ldots,(b,b),(b+1,b),\ldots,(a,b)=u. \tag{2.2} +\] + +Project its distinct vertices modulo the period lattice, identifying only the final endpoint `u` with the starting vertex `0`. For a shortest period in an honest quotient, no two other vertices in the monotone rectangle can be period translates: their Euclidean difference is strictly shorter than `|u|`. + +In the matching graph, consecutive diagonal steps followed by horizontal steps form a closed path whose lift displacement is exactly `u`. Hence the occupied set has nonzero ambient homology. + +In the NN graph, every diagonal step is absent. The diagonal vertices before `(b,b)` are isolated from one another in NN connectivity, while the horizontal tail from `(b,b)` back to the identified endpoint `u=0` is only a path. There is no NN closed walk with nonzero gain. Monotonicity of the lifted coordinates rules out an incidental NN chord; such a chord would require two listed vertices to differ by one axial step, which occurs only along the declared horizontal tail. + +Thus (1.3) holds. + +The same argument works for arbitrary sign patterns by reflection. + +## 3. Axis shortest period + +Suppose instead + +\[ +u=(a,0), \tag{3.1} +\] + +with a nondegenerate circumference. Use the lifted matching path + +\[ +(0,0)\to(1,1)\to(2,0)\to(3,0)\to\cdots\to(a,0)=u. \tag{3.2} +\] + +The first two edges are matching diagonals with displacements `(1,1)` and `(1,-1)`; the remaining edges are horizontal NN edges. The total lift displacement is `(a,0)=u`, so after quotienting this is an essential matching cycle. + +In the NN graph, the two diagonal connections are absent. The occupied horizontal tail from `(2,0)` through `(a,0)=0` is a path with one missing link at the detour; `(1,1)` does not close it. Hence the NN ambient rank is zero. + +A vertical axis period is the quarter-turned version. + +For the very shortest quotients one must retain the repository's existing lifted-parallel-edge convention or use the separate finite oracle. In the #739 asymptotic honest-torus scope, where the systole tends to infinity, no local degeneracy enters. + +## 4. Strict finite endpoint inequality + +Fix `00. \tag{4.1} +\] + +Therefore the graph-inclusion event containment is strict: + +\[ +\boxed{P_p^{8}(r=0)1/2.} \tag{5.2} +\] + +Likewise the fair birth-mixture CDF obeys + +\[ +F(p)+F(1-p)<1 \tag{5.3} +\] + +throughout the interior, except that quantile inequalities can become equal at levels affected by finite jumps only under the usual generalized-inverse convention. For the continuous-label birth law on a nondegenerate finite torus the distribution is continuous, and + +\[ +Q(u)+Q(1-u)>1 \tag{5.4} +\] + +whenever the two corresponding quantiles lie in the strict interior regime. + +The universal non-strict statement remains the safe formulation for any tiny quotient outside the honest-cell scope. + +## 6. Interpretation + +The negativity of `M(1/2)` seen in the exact `L=3,4` censuses is therefore not an accidental small-size feature. It is forced by a simple structural fact: + +> the matching graph admits essential cycles on vertex sets whose NN graph has no essential cycle. + +This finite topological asymmetry is the zero-scale precursor of the strict inverse-correlation mass gap `kappa_8(p)0.} \tag{3.3} +\] + +By homogeneity, for every `x in R^2`, + +\[ +\tau_{8,p}(x)+\eta_I|x|\le\tau_{4,p}(x). \tag{3.4} +\] + +Let `B_2` be the Euclidean unit disk. Since support functions add under Minkowski sum, + +\[ +h_{K_{8,p}+\eta_I B_2}(x) +=h_{K_{8,p}}(x)+\eta_I|x|. \tag{3.5} +\] + +Equations (1.2), (3.4) and the support-function characterization of convex-body inclusion give + +\[ +\boxed{ +K_{8,p}+\eta_I B_2\subset K_{4,p} +\qquad(p\in I).} \tag{3.6} +\] + +This is stronger than strict inclusion: the smaller matching body can be thickened by a fixed positive Euclidean radius, uniformly over the whole compact parameter interval, and still remain inside the NN body. + +## 4. Equivalent support and gauge statements + +Equation (3.6) is equivalent to the uniform support gap + +\[ +\boxed{ +h_{K_{4,p}}(e)-h_{K_{8,p}}(e)\ge\eta_I +\quad(e\in S^1,p\in I).} \tag{4.1} +\] + +If `B_{G,p}={x:tau_{G,p}(x)<=1}` is the primal correlation-norm unit ball, then the easier matching model has the larger primal ball, + +\[ +B_{4,p}\subsetneq B_{8,p}. \tag{4.2} +\] + +The dual-body buffer (3.6) is usually the cleaner quantitative statement for first-exit certification, because the finite certificates approximate `K_{G,p}` directly through exponential tilts. + +## 5. Finite first-exit certification consequence + +Let `C^{(G)}_S(p)={t:B^{(G)}_S(t;p)<1}` be a rigorous first-exit certified region. The large-box exhaustion theorem says that for every compact subset of `K_{G,p}^circ`, sufficiently large boxes certify it. + +Fix `I` and choose any `epsilon in (0,eta_I/4)`. Then + +\[ +K_{8,p}+ (\eta_I-\epsilon)B_2 +\Subset K_{4,p} \tag{5.1} +\] + +uniformly for `p in I` after an arbitrarily small inward shrink if one wants compact containment in the interior. + +Thus a numerical/certified Wulff computation has a strong same-parameter cross-graph control: + +1. certify a compact inner approximation to `K_{8,p}`; +2. thicken it by any radius safely below `eta_I` once a rigorous lower bound on the support gap is available; +3. the result must still lie inside the true NN body and should eventually be certifiable by large enough NN first-exit boxes. + +Conversely, a claimed pair of certified bodies that violates `K8 subset K4` has an adjacency, orientation, support-function or outward-rounding error before it indicates physics. + +## 6. Enhancement sandwich gives a second nested-body relation + +The local enhancement argument actually gives, on a compact interval `I`, a positive site-density sprinkling `delta_I` such that + +\[ +\tau_{8,p}(x)\le\tau_{4,p+\delta_I}(x) \tag{6.1} +\] + +for all directions. Therefore + +\[ +\boxed{K_{8,p}\subset K_{4,p+\delta_I}.} \tag{6.2} +\] + +Since `p -> tau_{4,p}` is strictly decreasing, the NN bodies shrink strictly as `p` increases, and + +\[ +K_{4,p+\delta_I}\subsetneq K_{4,p}. \tag{6.3} +\] + +So the same-parameter strict gap can be viewed as the convex-body sandwich + +\[ +\boxed{ +K_{8,p}\subset K_{4,p+\delta_I}\subsetneq K_{4,p}.} \tag{6.4} +\] + +This is the geometric version of “full matching enhancement beats a positive ordinary-site sprinkling.” + +## 7. Dilute asymptotic interpretation + +The dilute directional results on this branch show that the support gap is smallest, to leading order, near the coordinate axes. + +Axially, + +\[ +\tau_{4,p}(e_1)-\tau_{8,p}(e_1) +=\log3+O(p) \tag{7.1} +\] + +as `p downarrow 0`. + +For a genuinely tilted fixed rational direction the coefficient of `log(1/p)` is already strictly larger for NN than matching, so the gap diverges as `p downarrow0`. + +This suggests, and the explicit fixed-rational formulas verify directionwise, that the compact-interval buffer remains macroscopic in the dilute regime rather than collapsing there. + +A uniform asymptotic statement `eta_p -> log3` would additionally require controlling the minimising direction uniformly as `p->0`; it is not asserted here. + +## 8. Claim boundary + +The Minkowski-buffer conclusion is an exact convex consequence of: (i) all-direction strict mass gap; and (ii) continuity of the directional norms on compact subcritical parameter-direction sets. No differentiability in direction or in `p`, no OZ amplitude, and no numerical Wulff construction is required. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/structural-consequences-20260914.md b/docs/manuscripts/geometric-balance/structural-consequences-20260914.md new file mode 100644 index 000000000..959756e10 --- /dev/null +++ b/docs/manuscripts/geometric-balance/structural-consequences-20260914.md @@ -0,0 +1,333 @@ +# Structural consequences from the geometric-balance programme — 2026-09-14 + +Status: owner-authorized continuation from PR #739 head `907a9d94`. This note consolidates deterministic consequences, consequences that additionally use the existing author-level #739 probability arguments, and clearly marked conjectural interfaces. It does **not** upgrade the parent manuscript to independent referee acceptance or certify literature priority. + +The point of this note is to reduce, not enlarge, the open claim surface. Several questions that were being treated as separate analyses collapse once the finite 4/8 topology, the component-intensity point process, and the directional mass are used together. + +## 1. Persistent 4/8 birth reflection + +Let `v_1,...,v_N` be any strict ordering of the sites of an honest square-cell torus and let `S_k={v_1,...,v_k}`. Let `K_1^4,K_2^4` be the first indices at which the NN occupied ambient rank reaches one and two. On the matching graph occupy the same sites in reverse order and define `K_1^8,K_2^8` analogously. + +The configurationwise digital-Alexander identity + +\[ +r_4(S)+r_8(S^c)=2 +\] + +implies the pathwise identities + +\[ +\boxed{K_1^8=N+1-K_2^4,\qquad K_2^8=N+1-K_1^4.} +\] + +Proof: put `R_j=V\setminus S_{N-j}`. Then `r_8(R_j)=2-r_4(S_{N-j})`. Immediately before `K_2^4` the NN rank is at most one and at `K_2^4` it is two, so the reverse matching process first reaches rank at least one at `N+1-K_2^4`. The other identity is the rank-zero/positive version. A direct rank jump `0 -> 2` makes the two identities coincide rather than invalidating them. + +For continuous labels `U_v` and reflected labels `V_v=1-U_v`, this becomes + +\[ +\boxed{T_1^8(V)=1-T_2^4(U),\qquad T_2^8(V)=1-T_1^4(U).} +\] + +At every complementary rank-one pair the black and white ambient homology lines are the same. The existing digital-Alexander proof gives the image relation `C=A^perp`; in the two-dimensional symplectic space `H_1(T^2;Q)`, a one-dimensional isotropic line equals its symplectic orthogonal. + +Finite controls committed with this note exhaust all 512 configurations and all `9!` site orders on `3 x 3`, and all 65,536 configurations plus 20,000 fixed-seed random site orders on `4 x 4`. They test rank duality, the winding-component count identity, rank-one line matching, and the birth-index reflection. These controls are not the proof. + +## 2. Dual-even and dual-odd birth coordinates + +Define, configuration by configuration, + +\[ +G=T_2-T_1,\qquad C=\frac{T_1+T_2-1}{2}. +\] + +Under the 4/8 reflection `(T_1,T_2) -> (1-T_2,1-T_1)`, one has + +\[ +\boxed{G\mapsto G,\qquad C\mapsto-C.} +\] + +Thus `G` is the digital-complement-even coordinate measuring separation of the two births, whereas `C` is the odd coordinate measuring their common displacement from `1/2`. + +The two coordinates have exact integral representations which require no quantile fit: + +\[ +E T_1=\int_0^1P_0(p)\,dp, +\qquad +E T_2=\int_0^1[1-P_2(p)]\,dp, +\] + +hence + +\[ +\boxed{E G=\int_0^1P_1(p)\,dp,} +\qquad +\boxed{E C=-\frac12\int_0^1M(p)\,dp.} +\] + +If `K_1,K_2` are the two birth occupation indices in a uniform random permutation, then conditional order-statistic means give + +\[ +E G=\frac{E(K_2-K_1)}{N+1}, +\qquad +E C=\frac{E[K_1+K_2-(N+1)]}{2(N+1)}. +\] + +Existing rank-birth archives can therefore recover both coordinates without rescanning `p`. + +If an independent fair selector chooses one of the two births, write it as a sign `S=+1/-1`. Then + +\[ +T=\frac12+C+\frac S2G, +\] + +so exactly + +\[ +\boxed{\operatorname{Var}(T)=\operatorname{Var}(C)+\frac14E[G^2].} +\] + +There is no `Cov(C,G)` term because the selector is independent and centered. This gives a clean decomposition of mixture broadness into configuration-to-configuration centre motion and within-configuration birth separation. + +## 3. Exact reduction of same-parameter black/white winding counts + +Let `W_4` be the number of black NN components with nonzero ambient homology and `W_8` the corresponding white matching count on the same labelled finite torus. The wrapping-component classification already used in the branch implies exactly one of + +\[ +(W_4,W_8)=(0,1),\quad (1,0),\quad (K,K),\ K\ge1. +\] + +Consequently the full same-parameter joint probability generating function has the exact form + +\[ +\boxed{E[s^{W_4}t^{W_8}] +=P_2(p)s+P_0(p)t+P_1(p)H_p(st),} +\] + +where `H_p(z)=E[z^K\mid r=1]`. + +Thus a common critical-window model does **not** require an arbitrary two-dimensional joint count law. It is completely specified by the two rank endpoint probabilities and one positive-integer law in the rank-one sector. Independent or generically correlated bivariate Poisson ansatzes contain redundant degrees of freedom and usually violate the exact support. + +Combining this finite support statement with the existing lower-window result `W_4 => Poi(lambda)` and `P(r=2)->0` gives the constrained same-parameter limit + +\[ +P[(W_4,W_8)=(0,1)]\to e^{-\lambda}, +\] + +\[ +P[(W_4,W_8)=(k,k)]\to e^{-\lambda}\lambda^k/k!,\qquad k\ge1. +\] + +The upper-window statement is the colour-reflected version. This is compatible with the exact obstruction to independent black/white Poisson counts at one common parameter, and distinct from the asymptotic independence of the two separated birth windows proved in the branch. + +At the first-birth median `lambda=log 2`. Conditional on winding having occurred, the limiting probability of at least two black winding components is + +\[ +P(K\ge2\mid K\ge1)=0.3068528194\ldots. +\] + +Therefore an event-conditioned winding configuration cannot be treated as containing a unique `birth cluster` by default. Component-Palm sampling is the correct microscopic protocol when a single component shape is required. + +## 4. Alternating barriers and the reciprocal mean white span + +Fix `p infinity` but `log lambda_w=o(w)` preserves the vanishing absolute Poisson error. After scaling vertical coordinates by `nu_w`, the local barrier process tends to unit-rate Poisson. Together with `nu_w L(B_i)->0`, this gives the natural next lemma + +\[ +\nu_w L(W_i)\Rightarrow Exp(1). +\] + +Sampling an interval by a uniform stationary vertical location instead of component Palm length-biases the exponential gap and gives `Gamma(2,1)`. The two observed laws are therefore different sampling protocols for one gap process, not separate shape mechanisms. + +## 5. Marked-Poisson transport: do morphology once at component Palm + +The anchor proof in `poisson-birth-windows.md` already works with local indicators. Any finite-valued mark that is measurable inside the same localization window can be attached to an anchor without changing the dependency graph. Decomposing marks into finitely many types gives a vector of locally dependent indicators and the same Chen--Stein estimates apply componentwise and jointly. + +Therefore the efficient architecture for span/core/occupancy/boundary investigations is: + +1. prove or measure the law of one complete winding component under component Palm; +2. localize/quantize the desired mark if necessary; +3. lift that marked Palm law to the long torus using the existing marked-Poisson machinery. + +There is no reason to resimulate the whole exponentially long torus separately for each morphology observable. + +This also clarifies the #762 operational core issue. The mark must be a precisely defined local functional of the full component. A `bridge-deleted winding connected part` and the union of nonzero-winding vertex-biconnected blocks are not generally the same object; contractible loops attached at an articulation provide a deterministic counterexample. The core definition must be fixed before any conditional shape result is interpreted. + +## 6. Complementary Palm score identity + +For a complete component `C` at site probability `p`, with `n(C)=|C|` and `b(C)` the number of **distinct external boundary sites**, the activity is + +\[ +q_p(C)=p^{n(C)}(1-p)^{b(C)}. +\] + +Differentiating the log activity gives the exact component-Palm score + +\[ +S_p(C)=\frac{n(C)}p-\frac{b(C)}{1-p}. +\] + +The exact complementary intensity identity `nu_w^4(p)=nu_w^8(1-p)` therefore gives, wherever finite differentiation is justified (and exactly at every fixed finite transfer representation), + +\[ +\boxed{ +E_{4,p}\!\left[\frac n p-\frac{b_4}{1-p}\right] ++E_{8,1-p}\!\left[\frac n{1-p}-\frac{b_8}{p}\right]=0.} +\] + +If the alternating-barrier result above is used on the white giant essential component at fixed black-subcritical `p`, then `E L_{white}\asymp1/nu_w` and connectedness with vertical step at most one implies `E n_{white}\ge E L_{white}\gg w`. The black score is only `O(w)` under its ordinary component Palm law. Consequently the white score must cancel internally to leading order, giving the testable prediction + +\[ +\boxed{\frac{E b_{white}}{E n_{white}}\to\frac{p}{1-p}.} +\] + +This boundary/volume ratio is independent of the Brownian-range conjecture and is a useful checksum for future #762-type joint `(L,K,B)` work. + +## 7. Convex structure of the loop--branch rate candidate + +The current #758 candidate Palm-span rate is + +\[ +I_p(A)=\min_{0\le r\le A} +\{\tau_p(1,2r)-\kappa+\kappa(A-r)\}, +\qquad \kappa=\tau_p(1,0)=\tau_p(0,1). +\] + +This section analyzes the candidate only; it does not prove that the actual SITE Palm LDP equals it. + +Put + +\[ +\phi(r)=\tau_p(1,2r),\qquad + g(r)=\phi(r)-\kappa-\kappa r. +\] + +Then + +\[ +I_p(A)=\kappa A+\min_{0\le r\le A}g(r). +\] + +If the directional norm is strictly convex, `g` is strictly convex, `g'(0)=-kappa<0` by reflection symmetry, and norm bounds imply `g(r)->infinity`. Hence there is a unique finite minimizer `r_*>0`, independent of `A`, and + +\[ +\boxed{ +I_p(A)= +\begin{cases} +\tau_p(1,2A)-\kappa,&0\le A\le r_*,\\ +\kappa A+c_*,&A\ge r_*, +\end{cases}} +\] + +with `c_*=g(r_*)`. When differentiable, + +\[ +2\,\partial_y\tau_p(1,2r_*)=\kappa. +\] + +Square symmetry gives `0kappa`. + +Thus the candidate mechanism has a mandatory geometric transition: for small target span all extra height is supplied by bulging the closed essential loop; after `r_*` the optimal bulge saturates and every further unit of span is paid by a linear branch of slope `kappa`. If an eventual measured/rigorous large-`A` rate has slope strictly below `kappa`, the candidate topology is falsified by a cheaper network mechanism. + +## 8. Diffusion coefficient from Wulff curvature: a three-way consistency relation + +Assume the directional norm is twice differentiable at the horizontal direction and write + +\[ +\tau_p(1,y)=\kappa+\frac12a_p y^2+O(y^4), +\qquad a_p=\partial_{yy}\tau_p(1,0)>0. +\] + +On the first branch of the candidate rate, + +\[ +I_p(A)=\tau_p(1,2A)-\kappa +=2a_pA^2+O(A^4). +\] + +If the complete-component Brownian-bridge range picture is also valid, +`L/sqrt(D_p w) => R_BB`, and the Brownian bridge range tail satisfies + +\[ +-\log P(R_{BB}>r)=2r^2+O(\log r). +\] + +In the moderate-deviation overlap `A->0`, `A sqrt(w)->infinity`, the Brownian prediction is + +\[ +-\frac1w\log P(L\gtrsim Aw)\sim\frac{2A^2}{D_p}. +\] + +Matching the two descriptions forces + +\[ +\boxed{D_p^{-1}=\partial_{yy}\tau_p(1,0).} +\] + +If the direction mass is written in angular form `tau(r cos theta,r sin theta)=r kappa(theta)`, reflection gives `kappa'(0)=0` and a direct expansion gives + +\[ +\boxed{D_p^{-1}=\kappa_p(0)+\kappa_p''(0).} +\] + +This is not yet a theorem for the actual complete SITE component. It is a sharp three-way consistency condition connecting: (i) the #758 small-`A` loop-deformation LDP, (ii) the closed Brownian/HK transverse limit, and (iii) the directional mass/Wulff curvature targeted by #766. Proving any two in an overlapping regime determines and tests the third. + +## 9. Directional competition in exponentially elongated tori + +Let `Lambda` have shortest Euclidean period `u`, `|u|=ell`, area `N`, and transverse height `h=N/ell`. For any nonparallel `v in Lambda`, the determinant condition gives `|det(u,v)|>=N`, so `|v|>=h`. + +For a square-symmetric norm `tau_p`, with `kappa=tau_p(1,0)`, convexity and symmetry give + +\[ +\kappa\|z\|_\infty\le\tau_p(z)\le\kappa\|z\|_1. +\] + +Therefore + +\[ +\frac{\tau_p(v)}{\tau_p(u)}\ge\frac{h}{2\ell}. +\] + +Whenever `h/ell -> infinity` (in particular under fixed positive `log h/ell`), all nonparallel homology classes are automatically separated in correlation cost. The only parallel primitive minimizers are `+/-u`. This removes a generic multi-direction competition from the default #765 proof architecture; remaining work is the actual SITE directional connection/seam estimate and parameter inversion. + +## 10. Claim boundary + +The following are finite/deterministic consequences once the already-committed digital-Alexander and wrapping-component classifications are accepted: persistent birth reflection, the `(rank,K)` support reduction, the dual-even/odd integral identities, and the convex algebra of the #758 candidate functional. + +The reciprocal mean white span additionally uses the existing #739 author-level fixed-subcritical component-intensity exponent and component-volume control. The exponential white-span law additionally uses the process Poisson/Palm passage. The `D^{-1}` curvature relation is a consistency implication until the SITE LDP/Brownian overlap is proved. None of these statements identifies a continuum field or changes the original-U source contract. diff --git a/docs/manuscripts/geometric-balance/supercritical-white-slab-bulk-20260914.md b/docs/manuscripts/geometric-balance/supercritical-white-slab-bulk-20260914.md new file mode 100644 index 000000000..6dcd5708b --- /dev/null +++ b/docs/manuscripts/geometric-balance/supercritical-white-slab-bulk-20260914.md @@ -0,0 +1,158 @@ +# Bulk law inside the giant complementary white component + +2026-09-14. Continuation of the alternating-barrier / Poisson-tessellation analysis. This note contains one exact infinite-volume identity and a concrete conditional law for the huge white matching component between rare black winding barriers. + +## 1. An exact infinite-cluster boundary-density identity + +Consider independent site percolation with open probability `q` on any locally finite graph, in a regime with a positive probability of an infinite open cluster. Let + +\[ +\theta(q)=P_q(0\hbox{ belongs to an infinite open cluster}) +\] + +and + +\[ +\beta(q)=P_q(0\hbox{ is closed and has an open neighbour connected to infinity}). +\] + +Then exactly + +\[ +\boxed{\beta(q)=\frac{1-q}{q}\,\theta(q).} +\] + +**Proof.** Freeze the states of all sites except the origin. If the origin is open and lies in an infinite cluster, delete the origin. The infinite cluster minus one finite-degree vertex has at least one infinite connected component: otherwise it would be a finite union of finite neighbour-components plus the origin and hence finite. Thus in the frozen outside configuration some neighbour of the origin is connected to infinity without using the origin. Conversely, if such a neighbour exists, opening the origin puts it in an infinite cluster. Therefore the two events have the **same outside configurations** and differ only in whether the origin is open or closed. Their product-measure weights have ratio `(1-q)/q`. End of proof. + +No uniqueness or one-endedness of the infinite cluster is required for this identity. + +For the white matching graph at white density `q=1-p`, write `theta_8(q)` for the infinite-cluster density. The density of black sites forming the external vertex boundary of that infinite white cluster is therefore + +\[ +\boxed{\beta_8(q)=\frac{p}{1-p}\theta_8(q).} +\] + +This independently explains the boundary/volume ratio forced by the exact complementary component-Palm score in `structural-consequences-20260914.md`. + +## 2. Geometry supplied by the black-barrier process + +Fix black NN density `pp_c(G8)`. Let `nu_w=nu_w^4(p)` be the black winding-component row intensity. The preceding notes give: + +- black essential component thickness is `O(w)` under component Palm; +- black anchors, after vertical rescaling by `nu_w`, converge locally to a unit-rate Poisson process; +- between consecutive black essential components there is exactly one white matching essential component; +- its span `L_w` satisfies + \[ + \nu_wL_w\Rightarrow Exp(1),\qquad \nu_wE L_w\to1. + \] + +Since `nu_w` is exponentially small in `w`, the intervening white component occupies a slab whose vertical length is exponentially larger than both the width and the subcritical black boundary layers. + +## 3. Bulk-density conjecture with an exact target + +Let `N_w` be the number of white sites in the intervening white essential component and `B_w` the number of **distinct black external boundary sites** of that component. + +The natural supercritical slab statement is + +\[ +\boxed{ +\frac{N_w}{wL_w}\xrightarrow{P}\theta_8(q), +\qquad +\frac{B_w}{wL_w}\xrightarrow{P}\beta_8(q) +=\frac{p}{1-p}\theta_8(q).} +\] + +This is not yet proved here. Its target constants, however, are not free amplitudes: they are the ordinary infinite-volume supercritical cluster density and the exact one-site-flip boundary density above. + +A proof should use a bulk/boundary decomposition. Remove from the white interval boundary layers whose thickness grows faster than the supercritical mixing/correlation scale but remains `o(L_w)`. In the remaining slab, local events should be asymptotically unaffected by the two rare black barriers and by the conditioning that no additional black winding barrier occurs. Spatial ergodicity/mixing then gives the two densities above. The already established rare-event Poisson description should quantify why the no-extra-barrier conditioning has vanishing effect on a fixed or slowly growing bulk window. + +This is a cleaner target than attempting to infer `N_w` or `B_w` from small-width component tables. + +## 4. Joint macroscopic law if the bulk-density statement holds + +Combine the bulk law with the exponential span law. If `E~Exp(1)`, then jointly + +\[ +\boxed{ +\left( +\nu_wL_w, +\frac{\nu_wN_w}{w\theta_8(q)}, +\frac{\nu_wB_w}{w\beta_8(q)} +\right) +\Rightarrow(E,E,E).} +\] + +Equivalently, + +\[ +\frac{B_w}{N_w}\xrightarrow{P}\frac{p}{1-p}. +\] + +Thus almost all macroscopic randomness of `(L,N,B)` on the **white giant-component side** comes from the Poisson gap length. Internal density fluctuations are lower order. + +This predicts asymptotic correlations + +\[ +Corr(L_w,N_w)\to1, +\qquad +Corr(L_w,B_w)\to1, +\qquad +Corr(N_w,B_w)\to1, +\] + +provided the second moments are uniformly integrable at the stated scaling. + +That behaviour is deliberately different from the fixed-subcritical **black** complete-component conjecture `J` in `span-resolvent-frontier.md`, where Brownian transverse range and additive occupancy/boundary marks may become asymptotically independent after their own centering. The complementary pair therefore supplies an internal positive/negative control: + +- rare black component: width-`sqrt(w)` diffusive shape with `O(w)` additive marks; +- huge white complementary component: length-`1/nu_w` slab whose additive marks are dominated by the same exponential gap. + +## 5. A sharper fluctuation conjecture + +Condition on the white interval length `L_w`. Standard supercritical mixing suggests + +\[ +N_w-\theta_8(q)wL_w=O_P(\sqrt{wL_w}), +\] + +with a similar joint CLT for `B_w`. Since `L_w` is of order `1/nu_w`, + +\[ +\frac{\sqrt{wL_w}}{wL_w} +=O_P\left(\sqrt{\frac{\nu_w}{w}}\right), +\] + +which is exponentially smaller than the order-one randomness of `nu_wL_w`. + +If a slab CLT is proved, then after subtracting the random gap-length contribution one should see a two-dimensional Gaussian bulk fluctuation: + +\[ +\frac1{\sqrt{wL_w}} +\begin{pmatrix} +N_w-\theta_8wL_w\\ +B_w-\beta_8wL_w +\end{pmatrix} +\Longrightarrow N(0,\Sigma_{bulk}(q)). +\] + +This is a secondary target; the law of large numbers already supplies the main geometric test. + +## 6. Practical consequence for morphology work + +A future #762-style joint `(L,K,B)` measurement on the complementary white side has parameter-free checks before any Brownian or OZ interpretation: + +\[ +N/(wL)\to\theta_8(1-p), +\qquad +B/N\to p/(1-p), +\qquad +\nu_wL\Rightarrow Exp(1). +\] + +The second target is especially cheap because it needs no numerical value of `theta_8`. + +Failure of `B/N -> p/(1-p)` at widths where black thickness is already negligible would localize a definition/sampling problem (wrong external-boundary convention, wrong component Palm, or white component not being the intended complementary giant) before it is interpreted as a failure of a continuum shape theory. + +## 7. Claim boundary + +The one-site-flip identity for `beta/theta` is exact. The black Poisson-tessellation and reciprocal-span statements use the author-level #739 process theorem. The white bulk law and conditional slab CLT are conjectural proof targets. No new sampling is required to formulate them, and no unknown amplitude is introduced. diff --git a/docs/manuscripts/geometric-balance/tagged-cut-records-and-future-clock-20260914.md b/docs/manuscripts/geometric-balance/tagged-cut-records-and-future-clock-20260914.md new file mode 100644 index 000000000..d84f6f944 --- /dev/null +++ b/docs/manuscripts/geometric-balance/tagged-cut-records-and-future-clock-20260914.md @@ -0,0 +1,168 @@ +# 一个白簇的谱系：对数参数时间里的记录过程、两时刻孔洞律与逆向跳跃 + +2026-09-14。接 `monotone-cut-mark-filtration-20260914.md`。本文第1–6节是显式Poisson切分/标记模型的精确计算；前文已把这个模型从两种实际固定宽度site稀疏极限导出。第7节提出固定亚临界、大宽度的更远推广，并严格列出仍需的统一条件。 + +## 1. 不再每个参数重新选一个簇：固定一个空间位置跟随其子孙 + +在R×R_+放置两份独立Poisson测度：cuts强度 dz d tau，holes强度 r dz d tau。时钟tau时，只保留完成时间<=tau的点。cuts把R切成区间；每段继承其中所有已有孔洞。 + +固定空间原点0。它所在的白区间记为I_tau=(-D_-(tau),D_+(tau))。跟随这一个空间位置，而不是每次从全部簇重新做component Palm。D_-与D_+是独立且同分布的下记录过程。 + +在任意固定tau： + + tau D_- , tau D_+ 独立 Exp(1), + Y_tau=tau |I_tau| ~ Gamma(2,1), + H_tau | Y_tau ~ Poi(r Y_tau). (1.1) + +固定宏观区间总长度有限时，切分生成元为 + + L F(partition)=sum_I |I| integral_0^1 + [F(split I at its u-fraction)-F(partition)] du. (1.2) + +孔洞在每段内以rate r|I|均匀加入。既有孔洞在分裂后按真实位置继承，不在两个孩子中分别重采样。零黑屏障时，若空间是圆周而非直线，白色对象是一个cross；第一次cut将它变成rank-one区间，不能遗漏这个拓扑初态。 + +Uniform binary self-similar fragmentation及Poisson记录是已有概率对象[BG,G]；这里的物理接口是前文的site模板完成时间，后面的联合孔洞公式直接计算，不把一般fragmentation框架当成新发明。 + +## 2. 在对数时钟里，典型尺度成为平稳过程 + +令s=log tau，并设单侧X_s=tau D_+(tau)。两次新cut之间D_+不变，X因此按dX/ds=X增长。新cut在(0,D_+)内到达的log-clock速率为X；它的位置在这个区间均匀，所以X跳到UX，U~Uniform(0,1)。生成元为 + + A f(x)=x f'(x)+x integral_0^1 [f(ux)-f(x)] du. (2.1) + +Exp(1)是其平稳分布。特别地，对每个整数n>=1， + + A x^n=n x^n - n/(n+1) x^(n+1), + +在Exp(1)下的均值严格为0。不能因为二阶相关是指数衰减就将此过程叫Gaussian OU：它具有正的Exp边缘和向下的乘法跳跃。 + +对两个时刻tau_2=k tau_1，k>=1，给定X_1=x，令Z~Exp(k-1)，则 + + X_2=k min(x,Z), (2.2) + +其中k=1时按连续极限X_2=x。条件分布在kx处有原子exp[-(k-1)x]；其余是(0,kx)上的截断指数。这是确切转移核，不是只给一个时间相关函数。 + +## 3. 多少次宏观切分？对数时钟上的Poisson计数 + +单侧第一次新cut之前的等待log-time为S。起点X~Exp(1)，条件X=x时 + + P(S>s|X=x)=exp[-x(e^s-1)]. + +混合后P(S>s)=e^-s。更强地，设Y为cut前的X，则(S,Y)联合密度分解为e^-s·y e^-y；cut后UY重新具有Exp(1)分布并独立于S。由未使用Poisson点的独立性逐步迭代，得到单侧记录时间在s轴上是率1 Poisson过程。两侧独立，所以原点白区间的切分次数满足 + + N_split(tau_1,tau_2) ~ Poi(2 log(tau_2/tau_1)). (3.1) + +这也是Gnedin [G, Prop.2]中平面Poisson过程的一阶记录时间定理的直接实例。原文已给时间强度1/t，不把此一般事实宣称首次发现。 + +特别地，原点所在的粗区间在两时刻间没有被进一步切分的概率是 + + (tau_1/tau_2)^2. (3.2) + +传回前文实际site低密度耦合，若黑色基准概率p_2/p_1=t_2/t_1固定，则tau_2/tau_1=(p_2/p_1)^b。因此宏观谱系保持概率趋向 + + (p_1/p_2)^(2b): + w4 matching黑侧的指数为8；w8 NN黑侧的指数为16. (3.3) + +这里“保持”指没有新essential黑屏障切开这个宏观白区间。微观白点仍大量关闭，不能称为同一完整站点集合不变。这是稀疏极限下的参数耦合律，不是平方格临界动力学指数。 + +## 4. 完整两时刻变换：孔洞也有精确的时间记忆 + +把较早时钟归一到1，较晚时钟为k>=1。令Y_1,Y_k为两时刻的归一化长度，H_1,H_k为原点区间的内部孔洞数。对于s,t>=0以及z,v∈[0,1]，设 + + A=1+s+r(1-z), + B=k t+r(1-v)(z+k-1). + +则精确地 + + E[exp(-sY_1-tY_k) z^H_1 v^H_k] + =[(A+k-1)/(A(A+k-1+B))]^2. (4.1) + +证明只需一侧。早期边界距离x~Exp(1)，后期距离y=min(x,Z)，Z~Exp(k-1)。旧孔洞在[0,y]上被两次计数，在(y,x]上只被第一次计数；新增孔洞只贡献第二次。因此其条件指数为 + + -r x(1-z)-r y(1-v)(z+k-1). + +再乘长度Laplace权重，按Z>=x和Z_Exp = _Exp. + +本轮核对m,n=0..6的49个有理数等式。这个反演连接解释了为什么forward图景是切分和向下记录，reverse图景却是移入和衰减。二者共享Exp平稳边缘，但显然不是同一个可逆动力学。 + +## 6. 一个特别重要的取样区别 + +固定参数下，component Palm的间隔长度为Exp；固定空间位置所在区间为Gamma(2)。追踪同一位置的谱系只适用于后者。若每次改变p后重新从所有component中均匀抽一簇，两次样本没有天然的祖先关系，不能拿(3.3)或(4.1)作检验。 + +实际稀疏site实验可保留一个固定站点。它在[t0,T]内染黑或落入微小孔洞的概率趋0；在这小概率异常事件上单列状态，不用悄悄改选一个白站点。长度/孔洞/新屏障形成参数都必须使用同一份uniform标签。 + +## 7. 更远的方向：固定亚临界p的热窗口是否也由同一个切分过程控制？ + +这不是将固定宽度epsilon↓0交换成w→infinity。提出一个独立的多参数接口。 + +固定NN黑色p0 Lambda(x), (7.1) + +Lambda连续严格增加；并在有界x窗口上具有： + +(a) 黑色完整簇和锚点可在多项式高度内统一局部化，错误在O(1/nu_0)行上仍趋0； +(b) 多项式邻域中两个不交essential证据的总概率趋0，包括较低p时是两个簇、较高p时合并的情形； +(c) 最高参数下的一个完整黑簇，在整个参数窗口中至多携带一条粗尺度essential谱系。源锚点的移动在nu_0尺度消失。 + +在这三个条件下，给最高参数的局部完整簇标上“最早获得essential绕行的x”。对标记分箱应用相同的局部依赖Poisson界，得到时空cuts强度dz dLambda(x)。这是一条条件定理：真正的附加内容是同一参数区间上的标记/合并控制，不是再引用每个单独p的Poisson边缘。 + +如果已有质量导数/强度比能使p_w(x)=p0+x/(v w)、Lambda(x)=e^x，那么log clock就是x，上述预测变为 + + 宏观谱系保持概率 = e^-2(x2-x1), + tagged归一化长度自相关 = e^-(x2-x1), + 新分裂次数 -> Poi(2(x2-x1)). (7.2) + +这会把以前的单次出生Gumbel曲线提升成整个参数过滤的Markov切分结构。这里没有把(a)–(c)写成已检查完毕的实际固定p定理。 + +**可行推导。** 在最高参数下截掉w²以上大黑簇；用共同局部块的两条不交绕行证据排除多谱系；随后逐块标记出生x。主要风险不是标签不独立（原始标签独立），而是“一个最终component含两个较早component”的计数在缩放后是否确实消失，以及对x变化的强度归一化是否一致。 + +**可行计算。** 比单参数均值更有区分力的是两个/三个p的共同标签输出：较早屏障是否保留、是否合并、最早新增屏障位置，和tagged白区间的两时刻Laplace变换。可以用三个颜色层的共同转移，或正确加权的稀有条件采样。先复用已有w4/6/8，不默认新增宽度；不要把分别独立抽取的单参数表当作联合数据。 + +另一个独立方向是空间慢变列场。前文已经给出cuts与holes不同的局部速率。如果环境本身在宏观上随机，几何寿命将由随机累计hazard控制；此时非指数间隔可能来自环境而非新的临界机制。应该先指定环境律，再讨论Cox/随机时钟的后果。 + +## 来源与执行边界 + +[G] Alexander Gnedin, *Corners and Records of the Poisson Process in Quadrant*, arXiv:0709.1285v1；最终发表于ECP13(2008),187–193。已读§2–3、Prop.2，PDF印刷第3页已截图核对。其M^(1)过程就是本文单側边界，1/t记录强度是已有结论。https://arxiv.org/pdf/0709.1285 + +[BG] Bertoin–Gnedin, *Asymptotic laws for nonconservative self-similar fragmentations*, arXiv:math/0402227。本轮仅核对作者摘要中x^alpha分裂率的框架，不把它当作site映射定理或本文所有公式的来源。https://arxiv.org/abs/math/0402227 + +新有限结果：6196个w4最小支撑配置；610个共同标签两参数概率等式；216个两时刻变换/条件积分等式；49个时间反演多项式等式；15项本地测试。无Monte Carlo或新大宽度计算。完整仓库测试未运行。 diff --git a/docs/manuscripts/geometric-balance/thermal-null-ward-root-mechanism-20260914.md b/docs/manuscripts/geometric-balance/thermal-null-ward-root-mechanism-20260914.md new file mode 100644 index 000000000..73868e4d7 --- /dev/null +++ b/docs/manuscripts/geometric-balance/thermal-null-ward-root-mechanism-20260914.md @@ -0,0 +1,149 @@ +# 热零向量、E4 形状律与 Matching-root：一个可失败的机制 + +2026-09-14。服务 #768/#802。主要读取快照为 #771 `8e1282f6d4397f03c80a2883b917c2d31a4b93a3`；写入前分支已继续推进，不覆盖其他文件。本篇不再增加一套宽度阶梯。 + +**核心新目标**：把“thermal spin4”从尺寸幂和旋转标签，推进到可计算的一点响应与整个环面形状函数。以下 Virasoro/Ward 计算是明确条件下的推导；实际 square-site 根的模函数规律仍是猜想。标准 Ward/Zhu 技术和热通道的 level4 选择已有文献，不重新计作通用方法发现。 + +## 1. 热候选的具体代表，而非 spin/momentum 排除 + +取 c=0，热主场 epsilon 的 h=bar h=5/8。局部平坦坐标中的二级奇异向量为 + + chi=(L_-2-(2/3)L_-1^2)epsilon. + +当前 `verify-768-momentum-exclusion-20260914.md` §2.1 仍把局部场当成满足 `[P,phi]=(2pi/L)(h-hbar)phi` 的平移动量本征算符，遗漏了插入位置导数。[GL](2.3)明确给出 `[L0,V(v,z)]=(z partial_z+h)V(v,z)`。空间零模不因局部 spin 非零就自动消失。这里直接计算热模块，不只用 identity-family I3 作反例。 + +定义 + + Q epsilon=[L_-4+(20/11)L_-3 L_-1-(160/561)L_-1^4]epsilon, + U4=-(11/29)[Q epsilon-(80/33)L_-2 chi]. + +有理 PBW 计算给出 L1 chi=L2 chi=0，L1 Q epsilon=(80/11)L_-1 chi，L1 U4=0。因此 U4 是全 Verma 空间里的准初级代表。在商掉 null 后裔与总导数后，它的类恰为 L_-4 epsilon；level4 的五维空间中，被商子空间秩4，L_-4 不在其中。 + +更有用的是不商 null 时的精确关系： + + U4=L_-4 epsilon+(80/87)L_-2 chi+L_-1 V3. (1) + +V3 是一个三级态。故“null 是否影响本观察”可用一个具体振幅检验，而不是先替整个 c=0 理论贴模块标签。 + +## 2. 圆柱零模确实允许非零 + +若 `=C_m z^-h`，平面 Ward 给 + + /C_m=m+h, + /C_m=h(h+1). + +在 C_m 非零、该三点通道 null 解耦时，m+h-(2/3)h(h+1)=0，得到 m=5/96。这与热—磁性融合相容；该融合选择已有 [JPS] §4.2。 + +周长2pi圆柱、z=exp(t)中，正规化应力张量插入是 + + /C_m + =m-c/24+h exp(t)/(exp(t)-1)^2. + +Laurent 展开 `exp(t)/(exp(t)-1)^2=t^-2-1/12+t^2/240+...` 因而给 + + =(m-c/24-h/12)=0, + =(h/240)=/384. (2) + +null 后裔解耦时 `=/384`。恢复周长 L，实的两手征和系数为 pi^4/(12 L^4)。这并不保证实际格点 C_m 非零；它已经排除了“平移普遍杀掉该 spin4 零模”的错误推理。真空、磁性外态与 rank 投影 trace 仍须区分。 + +## 3. Torus Ward 给出完整 E4 因子 + +取复周期 u,v，tau=v/u，Im(tau)>0。设一个线性插入泛函 F 对主场 epsilon 满足普通周期 Ward 恒等式、插入位置平移不变、总导数期望为零，且无额外 seam/contact 项。普通 CFT trace 满足相应标准关系；**rank/同调投影是否满足，是实际模型的待证接口**。 + +定义 + + G4(u,v)=sum_(lambda in Zu+Zv,lambda!=0) lambda^-4 + =(pi^4/45)u^-4 E4(tau), + E4(tau)=1+240 sum_(n>=1) sigma_3(n) exp(2pi i n tau). + +Weierstrass 函数展开为 `wp(z)=z^-2+3G4 z^2+...`。单主场 Ward 恒等式使 F(T(z)epsilon) 的位置依赖为 h wp(z)F(epsilon) 加一个与 z 无关的项。取 z^2 系数： + + F(L_-4 epsilon)=3hG4 F(epsilon). (3) + +无需除以可能为0的 F(epsilon)。同一式可从 [GL](2.10) 的 Zhu 递推得到。如果 F(L_-2 chi)=0，由(1) + + F(U4)=(pi^4/24)u^-4 E4(tau)F(epsilon), + F(U4+bar U4)=(pi^4/12)Re[u^-4 E4(tau)]F(epsilon). (4) + +**全模块 null 解耦只是充分条件，不是必要条件：目标一点振幅 F(L_-2 chi)=0 就够。** 若其不为0，精确差额是 + + F(U4)-3hG4 F(epsilon)=(80/87)F(L_-2 chi). (5) + +对于普通未正规化手征 trace f、u=1，右侧可写为 `(80/87)(2pi i)^4 D_(h+2)D_h f`，其中 D_h=q_mod partial_q_mod-h E2/12。广义对数 trace 或投影接缝异常应另列，不能机械复制普通主场式。 + +当 null 在 trace 中解耦，D_h f=0，故 f=C eta_D(tau)^(5/4)，其首项为 + + q_mod^(5/96)[1-(5/4)q_mod-(35/32)q_mod^2+(45/128)q_mod^3+...]. + +这里 eta_D 是 Dedekind eta，q_mod 不是占据概率。本轮精确核对了乘积展开与微分递推。[JPS] 已讨论热通道二级后裔消失、四级后裔先贡献；它同时指出 Q=1 的无条件能量一点函数为0。**不能把该零函数当成 rank 条件热响应，更不能凭它取消本任务的源映射。** + +## 4. 形状、方向和周期基是同一个协变量 + +周期换基 `u'=u(c tau+d), tau'=(a tau+b)/(c tau+d)` 下，E4 的权重4变换给 + + u'^-4 E4(tau')=u^-4 E4(tau). + +若 u=ell exp(i theta)，几何因子为 + + ell^-4 Re[exp(-4i theta)E4(tau)]. (6) + +一般斜模参数中不能随意更换相位符号。固定面积 N=|u|^2 Im(tau) 时，N^2 倍因子为 `(Im tau)^2 Re[exp(-4i theta)E4(tau)]`。 + +它包含：长圆柱 E4(i infinity)=1；方形 E4(i)=1.45576289226870932246242200359886929...；长宽比2矩形 E4(2i)=1.00083698843473765919291512747422264...；六角 rho=exp(i pi/3) 上 E4(rho)=0，且 E4'(rho)=-(2pi i/3)E6(rho)不为0。 + +既有 double-null 方案已经有角向/六角零点。本篇增加的是零点之间的完整形状律，而非再增加一个零检验。 + +## 5. Actual-site 猜想：一个系数控制整个根形状 + +所需条件明确为： + +H1：实际 rank 扇区具有上述临界 Ward 接口，使用同一个热场正规化； +H2：leading matching-odd correction 是同一微观系数乘 U4+bar U4，且目标 null 异常消失，其他同阶 scalar/module/seam 项已排除或分开； +H3：根局部非退化，余项对固定模参数紧集一致，并另外控制 finite-aspect 到 cylinder 的对应。 + +这时每个 Z_r 的 first-order insertion 都满足 `partial_g4 Z_r=C4(u,v)partial_t Z_r`，总 Z 同样满足，故正规化 P_r 也同样满足。将热度量、符号和微观系数收进一个未知常数 A，预言 + + p*_Lambda-pc=A Re[u^-4 E4(tau)]+o(|u|^-4). (7) + +**(7)是 square-site 机制猜想，不是(3)证明的格点定理。** A未计算，每个形状不能另拟合一个A。一个后裔可在临界一点响应上等价于热移动，却不等于全理论的冗余算符；不能对临界恒等式直接再微分，忽略二次热插入与接触项。 + +因此特别预言，同一周长L的方形环面根与无穷长轴向圆柱 charge 根，若共同A非零，则 + + (p*_square(L)-pc)/(p*_cylinder(L)-pc) -> E4(i). (8) + +这把根指数、角向spin4、六角零点和此前的切向盲点，连接成同一个可失败机制。 + +## 6. 复用已返回数据，不生产新宽度 + +团队L5/L6整数rank-by-K表，配仓库已有同宽圆柱根，给： + +|L|方形环面根|已有圆柱根|用pc_ref形成的位移比|相对E4(i)偏差| +|---|---|---|---|---| +|5|0.591988256518333844610968680211928879|0.5922358232050258|1.48520835204458070455|+2.0226824%| +|6|0.592395070817704237693855807642505434|0.592507356205638|1.47041447230994514445|+1.0064537%| + +pc_ref=0.59274605079210 是已有参考值，不是严格pc区间。圆柱根也是已有打印值，不能把位移比的全部显示位数当认证精度。每个k的三rank和等于binom(L^2,k)已复核，但这不认证实际rank分类。未重新枚举2^25或2^36，未重跑Perron。 + +两个小尺寸趋近目标不是渐近证明，也不唯一识别thermal模块；这是同一旧数据的新比较，没有拟合振幅/指数来选目标。参考误差灵敏度为 `R'(c)=(a-b)/(b-c)^2`，当前约-951、-1971。今后若已有三个同面积不同形状的根，可用根差之比同时消去pc与A；不为此自动开启形状阶梯。 + +## 7. 现在只追真实接口 + +#768：计算 rank 投影下应力张量运输是否产生seam/contact项，以及 F(L_-2 chi) 是否消失。可只证明一个实际扇区/源的一点关系，不要求先解决整个对数模块。零范、奇异与在指定关联函数中解耦不是同一句话。 + +#802：先修正上一轮已指出的正规化与 H_W=N-2K+H_B 冗余，保留两压力与各自导数。复用现有 finite-aspect 根检验(7)，一个不符合E4的法向/形状响应比更多同轴根更能改变判断。非E4残差可能是null或投影异常，也可能是同阶scalar项，不能直接命名。 + +#780：新 `intrinsic-birth-clock-construction-20260914.md` 已用最终簇出生CDF beta 构造单调时钟；本轮吸收该进展，不重新购买时钟存在性或更多Poisson核矩。压力时钟是另一实现，不是第二套独立生产。 + +本轮只加本文件、独立控制脚本、测试和结果。没有修改原始U、STATUS或团队文件，未启动机器。12项局部测试核对PBW、null异常、模变换、eta系数和原始表算术；高精度模函数/求根不是区间认证，未运行全仓CI。 + +## 来源 + +[GL] Gaberdiel–Lang, arXiv:0810.0106，PDF印刷页3–4、6，(2.3),(2.10),(2.14),(3.4)–(3.7)：标准torus Ward/Zhu和null微分方程。只取实际读取的递推，不将一般有理性假设移植到渗流。https://arxiv.org/abs/0810.0106 + +[JPS] Javerzat–Picco–Santachiara, arXiv:1907.11041v2，§2.2、4.2，(4.11)–(4.18)：热/磁性融合、二级null及四级后裔，Q1无条件能量零与连通函数的奇异正规化。其Potts/FK连通不等于本项目rank投影。https://arxiv.org/html/1907.11041v2 + +[HS] He–Sun, arXiv:2004.07486v2，§2.1(4),(5)：未变形CFT的周期Ward恒等式。这里不引入T Tbar变形模型。https://arxiv.org/html/2004.07486v2 + +内部输入：主读SHA上的L5/L6 rank-sector-C JSON，fixed-width-charge-spectrum-derivatives-w4-w8 JSON，double-null、two-observable-response、topological-clapeyron及intrinsic-birth-clock四份说明。完整输入路径和blob见结果JSON。 + +复现：`python scripts/thermal_ward_root_control.py --output results/research-dispatch/thermal-null-ward-root-20260914.json`；`python -m unittest discover -s tests -p 'test_thermal_ward_root_control.py' -v`。依赖mpmath，精确代数用标准库Fraction。 diff --git a/docs/manuscripts/geometric-balance/thermal-window-clock-and-effective-range-20260914.md b/docs/manuscripts/geometric-balance/thermal-window-clock-and-effective-range-20260914.md new file mode 100644 index 000000000..e7f6fa412 --- /dev/null +++ b/docs/manuscripts/geometric-balance/thermal-window-clock-and-effective-range-20260914.md @@ -0,0 +1,169 @@ +# 强度时钟的一致误差与有效范围：在固定宽度精确引擎上的核验 + +2026-09-14。续接 #780 与 `thermal-window-no-merger-20260914.md`（作者推导）。 +本文件是**核验 + 缺口攻击**，不是新渐近定理。所有数字由 `bridge780b` 在 +`DevEnvC_TVVfoB`（16 vCPU，Python 3.9.9 + numpy 2.0.2）上跑出；本机不做计算。 + +## 0. 一句话结论 + +对 `#780` 的并合界推导：**五条关键步骤全部成立**（其中两条须显式标注输入/前因子过计数）， +**三项「未宣称」项目没有被偷偷使用**；对 **Q1**：强度时钟的一致误差**可以算出来**， +在固定宽度精确引擎上它等于 + +``` +E_w(X) = sup_{|x|<=X} | nu_w(p_0+x/(v_w w))/nu_w(p_0) - x | + = |kappa_w''(p_0)| X^2 / (2 kappa_w'(p_0)^2 w) * (1+o(1)) （v_w := -kappa_w'(p_0)） + = O(X^2/w) ， +``` + +系数逐 `(w,p_0)` 可计算，并被**共同标签** Monte Carlo 独立验证（差 <1 s.e. 当 `|x|<=0.15`）。 +**缺的不是误差公式，而是两件输入**：(i) `kappa_w'`,`kappa_w''` 的收敛（w<=8 未认证）； +(ii) 固定位置协议需要的额外条件（见 §4）。 + +## 1. 有限检查的独立复现（任务 A1） + +见 `results/research-dispatch/thermal-window-clock-20260914.json` 的 +`A1_finite_check_independent_reproduction`。要点： + +* 自写脚本、**不 import 包内脚本**；两个独立绕行判据（提升势 BFS / 万有覆盖探索） + 在全部 512 个 `3x3` 自由纵向 mask 上逐点相等。 +* 复现 `512 / 19683 / 7 / 3`、显式并合 `(455,463)`、三组精确概率与 BK 界。 +* **`721/125000000` 的我自己的推导**：三色权重 `(1/5)^e(1/20)^a(3/4)^c`，`e+a+c=9`，有 + `(1/5)^e(1/20)^a(3/4)^c * 20^9 = 4^e 15^c`（`a` 恰好约掉），故等价于整数恒等式 + `sum_{有并合} 4^e 15^c = 721*4096 = 2953216`（我独立枚举得 2953216 ✓）； + 另两组：`sum 2^{3c}3^e = 217*729 = 158193` ✓、`sum 2^{e+c} = 1216` ✓。 +* 极小宽度上「并合概率 = 阶乘对期望 = 谱系损失期望」三者相等（每次并合恰为 + 一个 late essential 片含 2 个 early 片、损失 1）。**仅极小宽度；未外推。** + +## 2. 五条关键步骤的判定（任务 A2） + +| # | 步骤 | 判定 | +|---|---|---| +| 1 | 中间参数的并合 ⟹ 最终簇含两个顶点不交的绕行见证 | **成立**（确定性蕴含；需「同标签单调」+「essential = 非平凡绕行」）。充分非等价，正文已声明。 | +| 2 | `Q_w <= wH^2 p e^{-(w-1)kappa(p)}` 及其 `p` 依赖 | **成立**；`p` 依赖自洽（见下）。两处须标注。 | +| 3 | BK 用在「占据绕行见证」上 | **成立**（site BK 给上界；事件单调；只在平面提升上用；不对非单调的簇锚点用）。 | +| 4 | 一致体积尾：周期标签 vs 平面标签 | **成立**（发现树单射 ⟹ `|C_cyl|<=|C_plane|`；[AV] Thm 3 为引用输入）。 | +| 5 | `m x Q_w^2` 的指数账 | **闭合**（`-> -kappa(p_0)<0`；尾项超指数压住；因子 2 本质）。 | + +### 2.1 第 2 步：`p` 只以 `p^1` 出现，且这是唯一自洽的写法 + +* `w` 因子 = 起始列在 `Z/wZ` 的 `w` 种；`H^2` = 两端点纵坐标（**并集界，故意松**）； + `p^1` = 平面连接要求端点占据（`t_p <= p`）；`e^{-(w-1)kappa(p)}` = `w` 个站点的代价。 +* **不是 `p^w`**：被界的是两点函数 `t_p`（内部已求和）。固定 `p_0` 下 `kappa(p_0) < log(1/p_0)`， + 故 `e^{-kappa w} >> p^w`，用 `p^w` 会**低估**。稀疏极限 `kappa(p) ~ log(1/p)` 时两者才合并。 + 实测 `kappa_w(p_0)`：`p_0=0.25` 时 1.253（w=8，`log(1/p_0)=1.386`）； + `p_0=0.40` 时 0.717（`log(1/p_0)=0.916`）；`p_0=0.50` 时 0.496（`0.693`）。⇒ 差距随 `p_0 -> p_c` 扩大。 +* **须标注 (i)**：`wH^2` 是并集界计数。实测真值 `Q_w(H;p) ~~ H nu_w`（`Q/(H nu_w) in [0.62,0.985]`）， + 故 `界/Q` 从 64（w=4）涨到 455（w=8），`ln(界/Q) ~ 2.85 ln w`。**指数相同、前因子多算 `~w^{2.9}`**： + 对 `(4.2)` 无影响，对 w=4/6/8 的任何数值代入有害（正文已声明）。 +* **须标注 (ii)**：`(2.1)` 的 `kappa` 与 `(4.1)` 的 `kappa` 是**同一个**——这是一个识别性输入， + 本文件只做内部一致性检验（§3.4 的闭合边距）。 + +## 3. Q1：强度时钟的一致误差 + +**引擎**：自写的分量退休行转移（前沿配分 + 整数提升增益 + 每活分量一个绕行旗标）。 +校验：`(w,h)=(2,4),(3,3),(3,4),(4,3),(4,4)` 全部 mask 的转移奖励与独立万有覆盖 BFS 计数 +**0 处不符**；`nu_2,nu_3` 闭式（引用）差 `<=1.4e-16`；状态数 `6/14/38/102/282/786/2214`（w=2..8）与既有产物一致。 + +### 3.1 主结果 + +`F_w(p)=log nu_w(p)`，`v_w := -kappa_w'(p_0) = F_w'(p_0)/w`，`phi_w(x)=F_w(p_0+x/(v_w w))-F_w(p_0)`。 +则 `phi_w(x) - x ~~ (F_w''/(2F_w'^2)) x^2`，即 + +``` +E_w(X) = |F_w''(p_0)| X^2 /(2 F_w'(p_0)^2) = |kappa_w''(p_0)| X^2 / (2 kappa_w'(p_0)^2 w) +``` + +| `p_0` | w | `E_w(1)` 实测 | 上式预测 | `E_w(1)*w` | 可达 `x` 窗口 | +|---|---|---|---|---|---| +| 0.25 | 4 | 0.146 | 0.125 | 0.58 | 1.98 | +| 0.25 | 6 | 0.0796 | 0.0726 | 0.478 | 3.19 | +| 0.25 | 8 | 0.0553 | 0.0518 | 0.442 | 4.41 | +| 0.40 | 4 | 0.261 | 0.249 | 1.04 | 1.09 | +| 0.40 | 6 | 0.130 | 0.127 | 0.781 | 1.86 | +| 0.40 | 8 | 0.0844 | 0.0833 | 0.675 | 2.62 | +| 0.50 | 8 | 0.332（窗口 0.95） | 0.318 | — | 0.95 | + +* `E_w(0.5)/E_w(1) ~~ 1/4.1` ⇒ **`X^2` 律**；`E_w(1)*w` 近常数 ⇒ 支持 **`1/w`**（尚未完全稳，见 §3.3）。 +* **可达窗口** `Xmax = F_w'(p_0)*dmax`（`dmax` 为到 `p_c` 的安全余量）：`p_0=0.25` 时 1.4→4.4； + `p_0=0.50` 时仅 0.21→0.95。⇒ 深亚临界时时钟窗口够用，越靠 `p_c` 越塌缩。 + +### 3.2 共同标签核对（不是独立边缘） + +单批 `U` 标签、`w x 1200` 圆柱（横向周期、纵向自由）、每宽度 4092 样本。 +`nu_hat` 与精确值差 0.5–1%（`O(1/L)` 边界偏差，符合预期）。用同一批标签造 `phi_w(x)`： + +| `p_0=0.40` | `delta=+0.01` 的 `phi-x` | `+0.02` | `+0.04` | MC s.e. | +|---|---|---|---|---| +| w=4 | −0.0018 | −0.0070 | −0.0320 | 0.003 | +| w=5 | −0.0008 | −0.0103 | −0.0441 | 0.005 | +| w=6 | −0.0020 | −0.0125 | −0.0447 | 0.006 | + +反解曲率系数 `C_w` = 0.249 / 0.170 / 0.122，与预测 0.2487 / 0.1699 / 0.1273 吻合。 +⇒ **共同标签的实现与精确仿射时钟一致，且 `−C_w x^2` 的形状被复现。** + +### 3.3 缺口 1：时钟的「率」不能从 w<=8 认证 + +| `p_0` | `v_w`（w=3..8） | `v_inf` (1/w) | `v_inf` (1/w^2) | 差 | +|---|---|---|---|---| +| 0.25 | 3.867, 4.128, 4.304, 4.430, 4.521, 4.589 | 5.012 | 5.132 | 2.4% | +| 0.40 | 1.990, 2.278, 2.462, 2.585, 2.671, 2.734 | 3.181 | 3.211 | 1.0% | +| 0.50 | 0.967, 1.190, 1.346, 1.462, 1.554, 1.629 | 2.000 | 2.199 | 10% | + +⇒ 要把 `Lambda` 归一到指定形状（如 `e^x`）需**先证** `kappa_w - kappa = c/w + o(1/w)` +（= 局部对数斜率收敛）。正文点名「需另证」的正是这一项；本文件只给 `v_w` 与其漂移。 +另：`E_w=O(X^2/w)` 需要 `kappa_w',kappa_w'' -> kappa',kappa''`，而 `(4.2)` 只需 `kappa` 连续。 + +### 3.4 缺口 2（Q3 的可计算切片）：闭合边距的有效范围 + +`2 kappa_w(p_+) - kappa_w(p_0)`（取 `p_+ = p_0 + C/w`）： + +* `C=1`：全部为正，w=8 时 +0.325 / +0.203 / +0.357（`p_0=0.25/0.40/0.50`）。 +* `C=2`、`p_0=0.25`、`w=5..8`：**为负**（w=8 时 −0.260），因为此时 `p_+ = 0.50` 而 + `kappa_w(0.50)=0.496 < kappa_w(0.25)/2 = 0.626`。 + +⇒ 正文的充分条件 `2kappa(p_+) > kappa(p_0)` **不是自动的**，它给出一个具体的参数上界： +`delta p` 一旦大到让 `kappa` 掉一半以上，双见证项就不再被压住。这是**有效范围边界的一个切片**， +**不是**相变线（正文亦如此声明）。 + +## 4. 固定位置协议：所有者点名要修的那一处 + +**必须写成**：固定 `p_0>0` 时，固定微观点以概率 `p_0` 是黑的；即使白，也可能在**有限白簇**里。 +全部数字按**固定行/固定微观点**统计，**没有在异常时换点**： + +| `p_0`（白侧参数） | P(站点黑) | P(白且绕行) | P(白且有限) | 固定行上 P(无绕行白簇) | P(>=2 个绕行白簇覆盖该行) | +|---|---|---|---|---|---| +| 0.25 (0.75) | 0.250 | 0.741 (MC) | 0.007 | 0.015→0.002 (w=4→6) | 0.0022→0.0042 | +| 0.40 (0.60) | 0.400 | 0.478 (MC) | 0.115 | **0.249→0.221 (w=4→6)** | 0.0027→0.0112 | +| 0.50 (0.50) | 0.500 | 0.057 (w=8) | 0.443 (w=8) | 0.877 (w=8) | 0（单侧定义） | + +* `p_0=0.25`（白 0.75，远离 `p_c`）：协议可用，异常 0.2–0.5%。 +* `p_0=0.40`（白 0.60，**紧贴** `p_c≈0.592746`）：**不在渐近区**，`P(无)=0.22–0.25`， + 且异常在 `w=4..6` 反而**变大**。 +* `p_0 > 1-p_c ≈ 0.4073`：白侧次临界，固定行上通常**没有**绕行白簇（`p_0=0.5`、w=8 时 0.877）。 + ⇒ §6 的「取覆盖该行的唯一完整白色 essential 分量」**必须另列条件**。 + +**并且该异常不是一个单侧 Markov 观测量**：给出显式见证（w=5，17 站点，自由纵向）—— +第 2 行整行环（分量 A，带向下尾巴）与分量 B（第 4–8 行经 wrap 边闭合成绕行圈）**同时覆盖第 4 行**。 +我的转移矩阵在任意行上只有 0/1 个「**已经**绕过的分量」（`P(u>=2)=0` 精确，w<=8）， +而按「整簇是否绕行」定义 `P(u>=2)>0`。**两个定义在有限 w 下不同**； +想精确算 §6 的异常概率需要**双向**转移矩阵，或改用整簇定义。 + +## 5. 不能宣称什么 + +1. 不能说并合不存在：小宽度并合非零（`721/125000000` 复现）。`B_w/nu_- -> 0` 是**要证的目标**； + 本文件的 MC 只给三个点的回归（`p_±=0.25/0.29` 与 `0.40/0.44`），**三点支撑不了渐近断言**。 +2. `E_w(X)=O(X^2/w)` 是**有限宽度精确量 + 有限 w 数值**，不是已证的极限条款（见 §3.3）。 +3. 仿射热时钟与 `Lambda=e^x` **没有**实现；只给了 `v_w` 与漂移。 +4. `(2.1)` 的界**数值上不紧**（差 `~w^{2.9}`）。 +5. 固定行协议**没有修好**：`p_0=0.40` 下异常不随 w 下降（3 点），`p_0>1-p_c` 时协议本身失效。 +6. **引用未复算**：`[AV]` 体积尾/两点函数界、`(4.1)` 的种子论证、`nu_2/nu_3` 闭式、`p_c≈0.592746`。 + +## 6. 脚本与产物 + +云端 `/workspace/genealogy/`： +`bridge780b_work/a1_merger_control_independent.py`、`b1_clock_and_interfaces.py`、 +`b2_common_label_mc.py`、`b3b_witness_search.py`； +`out/a1-independent.json`、`b1-validate.json`、`b1-clock.json`、`b1-qw.json`、`b1-fixedrow.json`、 +`b2-mc.json`（`p_0=0.40`）、`b2-mc-p025.json`、`b3b-witness.json`。 +结构化摘要见 `results/research-dispatch/thermal-window-clock-20260914.json`。 diff --git a/docs/manuscripts/geometric-balance/thermal-window-no-merger-20260914.md b/docs/manuscripts/geometric-balance/thermal-window-no-merger-20260914.md new file mode 100644 index 000000000..e4930547e --- /dev/null +++ b/docs/manuscripts/geometric-balance/thermal-window-no-merger-20260914.md @@ -0,0 +1,147 @@ +# 共同标签热窗口：双绕行见证界与出生标记过程 + +2026-09-14。续接 #780，读取的基础分支为 #771 `6192f7df2e16627a9e5d2bef17761d7af5ea8eed`。这是本轮给出的作者推导，不从有限枚举推断宽度极限。 + +## 1. 对象与结论 + +轴向无限圆柱 C_w=(Z/wZ)×Z，w≥3，黑色 NN 独立 SITE。所有参数共用 U_v~Uniform(0,1)，p 时占据当且仅当 U_v≤p。固定 p0∈(0,pc)，ν0=ν_w(p0) 是每纵向行的完整 essential 黑簇密度。 + +若 p_w^-≤p_w^+ 且两者均趋向 p0，在长度 T/ν0 的窗口内，整个 [p_w^-,p_w^+] 上两个已经 essential 的不同黑簇发生并合的概率趋零。下文在平面质量连续性、site 体积指数尾与有限连接种子输入下给出 + +`limsup_w w^-1 log Pr(有一次并合) ≤ -κ(p0)<0`。 + +不需要 OZ 前因子、κ 的可微性或先证明仿射热时间。固定 p0 不能在本证明中改成 p0(w)→pc；移动方向和标签刷新也不在范围内。 + +## 2. 完整周长的局部概率上界 + +令 E_{w,H}(p) 表示自由纵向边界的 H 行圆柱带内有水平 essential 占据闭路，Q_w(H;p)=Pr(E_{w,H}(p))。 + +平面两点函数 t_p(z)=Pr_p(0↔z)，s_p=t_p/p。条件拼接点占据后，Harris 给 s_p(x+y)≥s_p(x)s_p(y)。轴向质量定义及反射给 + +`t_p(n,d)²/p ≤ t_p(2n,0) ≤ p exp(-2κ(p)n)`， + +故 t_p(n,d)≤p exp(-κ(p)n)。 + +提升一条圆柱绕行闭路，跟随到横向跨度首次达到 w-1，截取连接横向两端的子路径。它位于连续 w 列、原 H 行内。保留提升的边，切掉额外周期接缝边后，支撑可单射到平面。起始列模 w 有 w 种，两端纵坐标各 H 种。因此 + +`Q_w(H;p) ≤ w H² p exp[-(w-1)κ(p)]`. (2.1) + +这里只比较一段支撑已经单射的路径，不把整条商空间闭路的两个部分当成独立平面连接。H 可以远大于 w。 + +两个顶点不交的 essential 占据见证属于 E∘E，site BK 给 + +`Pr(E_{w,H}∘E_{w,H}) ≤ Q_w(H;p)²`. (2.2) + +完整簇锚点事件带空置边界、不是单调事件；BK 施加在占据见证上，而不是锚点上。 + +## 3. 一次检查最后参数，控制所有中间参数 + +若两个此前不同的 essential 分量在某参数并合，它们分别含有顶点不交的 essential 闭路。单调性保证两条闭路在最终 p_w^+ 仍存在，并处于同一最终簇。因此 + +`某时刻发生并合 => 最终某簇含两个顶点不交的 essential 见证`. (3.1) + +这是坏事件的包围，不是等价：最终双圈簇可能从来没有在当前区间并合两条既有谱系。 + +按确定次序探索圆柱根簇。首次查询一个圆柱站点时，沿发现树父边选择一个尚未查询的平面提升读取标签。不同圆柱站点有不同 residue，因此其提升不相同；再次遇到旧站点不查询新副本。发现树嵌入真正独立的平面簇。这保留顶点数，不保留所有周期闭边。 + +取固定 p_b κ(p0)`. (4.1) + +这个接口不必依赖 Poisson 结论：上界由 (2.1)、跨度截断和 (3.2) 得到。下界从平面两点函数极限选一个足够长但固定的有限盒连接种子，使单位长度代价≤κ+ε。沿周向连续拼接，最后用有界路径接回初点的 w-平移。各局部盒最终都单射入圆柱，重叠种子用 Harris，不用独立性。得到固定带高 D 内的 essential 种子事件，概率≥exp[-(κ+2ε)w]。 + +排除概率≤CwD exp(-cH) 的大体积簇后，这个事件必有一个完整 essential 簇锚点在 D+2H 行内。Campbell 计数给 + +`(D+2H+1)ν ≥ exp[-(κ+2ε)w]-CwD exp(-cH)`。 + +不要强制关闭整条长度 w 的外边界，那会错误增加指数代价。 + +将 m=ceil(T/ν0)、H=w² 代入 (3.3)，使用 κ 在 p0 的连续性，得到 + +`limsup w^-1 log Pr(宏观窗口有并合) ≤ -κ(p0)`. (4.2) + +常数与起始宽度没有数值优化；这不意味着 w=4、6、8 的并合已可忽略。 + +### 原工单的阶乘并合密度 + +最终簇 C 含 n_C 条不同早期 essential 谱系，定义 + +`B_w=E sum_{C:min_y C=0} binom(n_C,2)`。 + +每个早期 essential 至少 w 点。跨度≤H、最低行为0的所有最终簇共有至多 wH 个站点，所以 sum n_C≤H。用 (2.2)，小簇贡献≤(H²/2)Q_w(H;p_w^+)²。大簇用 n_C≤|C|/w 和 (3.2)，得到 + +`B_w ≤ (H²/2)Q_w(H;p_w^+)²+(C/w)exp(-c'H)`. (4.3) + +早期参数趋向 p0 时，用固定 p_aκ(p_a) 仍能获得指数小上界；该充分条件不是已经得到的并合相变线。 + +## 5. 带明示时钟条件的标记 Poisson 结论 + +设 x∈[a,b]，p_w(x) 单调连续，sup_x|p_w(x)-p0|→0，并且 + +`ν_w(p_w(x))/ν0 -> Λ(x)` 一致，Λ 正、连续、严格增加。 + +在最高参数 b，将每个最终完整 essential 簇标为其内部第一次出现 essential 子簇的 x；已经在 a 时存在的簇标为 a。最低行及其最小列确定唯一锚点。 + +截断 |C|≤H=w² 后，锚点、完整簇和出生标记均由上下 H+1 行标签决定。对有限空间/参数分箱，索引为“候选站点×标记箱”的 Bernoulli 指示；同站点不同箱互斥。不同成功锚点属于不同完整簇，因此局部双事件仍由两个不交 essential 见证控制。依赖邻域 O(wH)，在长度 O(1/ν0) 内，AGG 的 b1、b2 均≤poly(w) exp[-2κ(p_w(b))(w-1)]/ν0→0；邻域外标签独立，b3=0。截断误差≤poly(w)exp(-cw²)/ν0。 + +识别强度时，某参数 x 的每个早期分量唯一归属于一个最终簇；n 条早期分量造成计数损失 (n-1)_+≤binom(n,2)。平移协变的锚点质量搬运和 (4.3) 给 + +`0≤ν_w(p_w(x))-β_w(x)≤B_w(x,b)`， + +其中 β_w(x) 为“出生标记≤x”的最终簇密度。有限窗口锚点移动≤H，缩放后 Hν0→0。因此 β_w(x)/ν0→Λ(x)。 + +由 [AGG, Theorem 2] 的过程近似和连续 Λ 的分箱细化，极限强度测度是 + +`dz [Λ(a)δ_a(dx)+1_{(a,b]}dΛ(x)]`. (5.1) + +a处原子编码初始屏障，不是说它们实际同时出生。以后参数箱的增量独立；它们来自共同标签，不是独立重采样的边缘。 + +这给局部标记点过程与有限组非跳时刻的切分观测收敛。不附带宣布所有路径拓扑或高阶矩。实现 Λ=e^x 的仿射 p_w(x)=p0+x/(vw) 仍需局部对数强度斜率；本节不把这个条件藏起来。 + +## 6. 固定位置的正确含义 + +当前 p0 固定时，一个固定微观站点为白色或属于 essential 白簇的概率不趋于1。因此不得直接照搬稀疏 p→0 对照里“始终追踪该站点白簇”的约定。 + +应选择固定纵向行，读取其跨度覆盖该行的唯一 essential 白簇；若零个/多个则单列异常。使用已有交替拓扑与 ν E L_black→0 控制异常。等价地在极限中读最近两道黑屏障之间的区间。 + +这属于 fixed-location 的 Gamma(2) 协议。每完整簇一次仍是 Exp 协议。若坚持固定微观站点，必须保留其非essential事件和相应条件分布，不在它变黑后偷偷换站点。 + +## 7. 算力后果与有限检查 + +新证明不需要直接模拟指数长窗口来等待一次并合。#780的现有作业可保留一个真正的双参数祖先映射或有限并合见证；后续机器更适合估计尚未控制的强度时钟和模型误差，而不是不断重复“未观察到并合”。 + +有限并合确实存在：C3×{0,1,2}中早期上下两行全黑、中间全空，后来中间任一点打开，就合并两个早期 essential 分量。 + +独立脚本 `essential_lineage_merger_control.py` 用提升BFS与位移并查集核对512个物理配置，穷尽3^9=19683个嵌套早晚对，逐项检查祖先归属、损失≤阶乘并合数、并合蕴含两个不交见证和三组有理权重。发现7个并合嵌套对；p_-=1/5,p_+=1/4时有限带并合概率为721/125000000。第三组p_+=3/5只作有限代数检查，不被纳入亚临界渐近结论。这些是有限接口，不估计无限 B_w 或 ν。 + +## 8. 下一层猜想与边界 + +猜想：固定亚临界窗口的领先谱系只通过强度时钟依赖微观细节，首个过程修正由局部双绕行连接和锚点移动决定。其修正系数与统一误差尚未求得。 + +若 p0(w)→pc，κ和体积尾常数退化，需要联合控制 wκ、Hν及截断误差；不能只因强度比有限就宣称纯切分。双见证竞争是一个明确失败方向，而非已证的转变线。 + +本记录不计算匹配根的两个稀有扇区相对权重，不识别L^-4场，不改变#275原始源合同。#768/#802继续是另一条主攻。 + +## 来源 + +[AV] Antunović–Veselić, arXiv:0707.1089v3，§2 Theorem3及§3 Definition7/Fundamental Tools；本轮读取site定义、体积尾、Harris/BK原文。https://arxiv.org/html/0707.1089v3 + +[AGG] Arratia–Goldstein–Gordon, Ann. Probab.17 (1989),9–25，§2 Theorem2。读取并截图核对PDF印刷页10–11的过程误差界。通用Poisson工具不是当前site谱系的现成定理。https://dornsife.usc.edu/larry-goldstein/wp-content/uploads/sites/221/2023/06/AGG-1.pdf + +内部输入：exponential-birth-centres.md（质量与有限种子）；root-response-and-bridge-targets-20260914.md（B_w接口）；#780；黑白区间映射另用black-white-gap-law.md的交替输入。没有用Gaussian-sheet、Kingman树及其更多矩来代替本次桥接。 diff --git a/docs/manuscripts/geometric-balance/topological-action-spectrum-beyond-witness-count-20260914.md b/docs/manuscripts/geometric-balance/topological-action-spectrum-beyond-witness-count-20260914.md new file mode 100644 index 000000000..50846d936 --- /dev/null +++ b/docs/manuscripts/geometric-balance/topological-action-spectrum-beyond-witness-count-20260914.md @@ -0,0 +1,196 @@ +# Beyond witness counting: an incremental topological action spectrum + +Date: 2026-09-14 + +Status: correction to the earlier single-fugacity synthesis. Existing BK bounds and Wulff/network results retain their original status. The new proposal is to organize rare topology by constrained incremental actions, not by assuming integer powers of one universal fugacity. + +## 1. Why the one-fugacity picture was too strong + +The earlier shorthand + +```text +epsilon_w = exp[-w kappa(p)] +``` + +is useful for the baseline probability of one horizontal essential connection/component. It became too strong when it was used to suggest that every additional topology-changing effect costs exactly one more factor of `epsilon_w`. + +There are at least three distinct mechanisms: + +```text +merger of two earlier essential lineages, +wrap-sensitive correction to cylinder mass, +spectral tunnelling producing the #800 slow doublet. +``` + +Only the first currently has a direct disjoint-witness probability bound. The second is an irreducible-decoration perturbation problem; the third is a visible operator/eigenvalue splitting. They need not share an action. + +## 2. Incremental action definition + +Let `B_w` be a baseline topological event with + +```text +P(B_w)=exp[-w a_B(p)+o(w)]. +``` + +For an additional defect/event `E_w`, define when meaningful + +```text +sigma_E(p) + = liminf_(w->inf) + -1/w log [ P(E_w intersect B_w) / P(B_w) ]. +``` + +This is the incremental large-deviation cost conditioned on the baseline topology. + +For spectral perturbations that are not literally event probabilities, use the analogous exponential rate of the matrix element/eigenvalue splitting and keep it separately typed. + +## 3. What #780 really proves + +A merger of two already-essential lineages implies two vertex-disjoint essential occupied witnesses in the final configuration. The one-witness probability has exponent `kappa`; site BK gives an absolute two-witness upper probability with exponent at least `2 kappa`. + +Since the baseline component intensity has exponent `kappa`, the rigorous consequence is an incremental lower bound + +```text +sigma_merger >= kappa +``` + +in the exponential-rate sense used by the proof. + +Nothing in the argument proves + +```text +sigma_merger = kappa. +``` + +The merger can be strictly rarer because of connectivity/anchor constraints or because the optimal two-witness geometry costs more than two independent cheapest crossings. + +## 4. #760 and #800 require distinct actions + +### Periodic mass locality + +Pure periodization preserves the zero Fourier mode, so `gamma_w-kappa` comes from an irreducible piece that sees a periodic image. Define its leading action + +```text +sigma_wrap(p). +``` + +Then generically one expects + +```text +gamma_w-kappa + = poly/subexp(w) * exp[-w sigma_wrap(p)] +``` + +if a unique leading defect exists. `sigma_wrap=kappa` is one candidate, not the definition. + +### Slow doublet + +For the projected slow-pole splitting define + +```text +sigma_split(p) + = liminf -w^-1 log Delta gamma_w. +``` + +The observed `Delta gamma/(w nu)=O(1)` at NN `p=1/4` suggests + +```text +sigma_split ~= kappa +``` + +but this remains a diagnostic until the tagged operator eigenvectors identify the actual tunnelling mechanism. + +Opposite residues indicate an approximately antisymmetric source/readout, not by themselves a one-witness probability event. + +## 5. Wulff-network actions are the natural replacement + +Subcritical 2D connectivity already comes with a directional norm `tau_p`. A constrained topology is naturally assigned a network cost by minimizing sums of `tau_p` over a graph/skeleton compatible with that topology. + +Therefore propose: + +```text +sigma_T(p) + = minimum Wulff/network excess cost for defect topology T + relative to the baseline winding object. +``` + +This makes #758's loop/branch variational programme part of the same language: its rate is not generally an integer multiple of `kappa`, because branches can share geometry and the optimum can change topology. + +A witness-count argument gives useful inequalities on `sigma_T`; it need not give the exact minimizer. + +## 6. Near-critical conjecture: universal action ratios, not integer powers + +If rotational restoration holds near criticality so that + +```text +tau_p(v)/kappa(p) -> |v|_2, +``` + +then every fixed finite network topology has a Euclidean variational constant `c_T`. This motivates + +```text +boxed: +sigma_T(p)/kappa(p) -> c_T +``` + +for the relevant class of defects as `p->pc` from the subcritical side. + +Then with + +```text +s=w kappa(p), +``` + +the massive tail is organized schematically by + +```text +sum_T exp[-c_T s] P_T(s), +``` + +not necessarily by integer powers `exp[-j s]`. + +This is a stronger but safer unification: the same mass sets the unit of action, while topology determines the coefficient `c_T`. + +## 7. What remains true about `w kappa` + +The #780 merger upper bound implies that + +```text +w kappa -> infinity +``` + +is sufficient to suppress macro-merger errors in the fixed-subcritical fragmentation picture. + +It does **not** prove that every other defect becomes small with exactly `exp[-w kappa]`, nor that finite `s=w kappa` alone determines the whole process. + +At finite near-critical `s`, a split--merge/hard-core process remains a plausible continuum target, but its rates may involve several `c_T` values. + +## 8. Minimal high-information tests + +1. `#800`: identify the two slow eigenvectors and the matrix element coupling them; only then decide whether the tunnelling skeleton has action `kappa`. +2. `#760`: characterize the minimal wrap-sensitive irreducible piece before fitting an exponential rate. +3. `#780`: no need to simulate deeper fixed-subcritical no-merger; existing theorem already gives suppression. A future finite-`s` calculation should measure one merger observable and compare it with a declared network action. +4. `#758`: use the variational skeleton to compute candidate `c_T` values which can be compared with spectral/process rates. + +## 9. Claim boundary + +Retained exact/author-level inputs: + +- baseline winding/component exponential mass `kappa`; +- BK double-witness upper bound for merger; +- zero-Fourier periodization identity; +- projected slow-doublet numerical diagnostics; +- existing directional/Wulff network definitions. + +New conjecture: + +- each topological/spectral defect has its own incremental action `sigma_T`; +- near criticality, ratios `sigma_T/kappa` approach geometry/topology constants `c_T` when isotropic Wulff scaling applies. + +Superseded simplification: + +```text +one extra witness => exactly one universal factor exp[-w kappa] +``` + +is no longer used as a general law. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/topological-clapeyron-response-20260914.md b/docs/manuscripts/geometric-balance/topological-clapeyron-response-20260914.md new file mode 100644 index 000000000..7a662d58b --- /dev/null +++ b/docs/manuscripts/geometric-balance/topological-clapeyron-response-20260914.md @@ -0,0 +1,290 @@ +# Topological Clapeyron response: root motion as a difference of quasi-stationary source scores + +Date: 2026-09-14 + +Status: exact finite-state Perron/Feynman--Hellmann calculus once a physical source is represented in the two safe transfer operators. Large-width/CFT interpretations are separate. + +## 1. The matching root is a coexistence line between two topological void phases + +At fixed circumference `w`, let + +```text +I4(p,g) = -log lambda4(p,g), +I8(q,g) = -log lambda8(q,g), +q=1-p, +``` + +be the Perron free energies of the NN-safe and complementary-matching-safe transfer operators in the presence of a microscopic source `g`. + +The source must be physically specified on each graph/colour representation; the symbols `g` on the two sides mean the declared matched perturbation, not an arbitrary matrix entry. + +Define the charge free-energy difference + +```text +Theta_w(p,g)=I4(p,g)-I8(1-p,g). +``` + +The fixed-width charge-coexistence root `p_w(g)` satisfies + +```text +Theta_w(p_w(g),g)=0. +``` + +Thus the matching root is literally a coexistence line between two quasi-stationary topological void phases. + +## 2. Perron/Doob source scores + +Let a safe transfer entry carry microscopic weight `W_G(i,M,j;p,g)`. Normalize positive Perron vectors by `l^T r=1` and define the one-step Doob/Perron law + +```text +Q_G(i,M,j) + = l_i W_G(i,M,j) r_j / lambda_G. +``` + +For a source derivative define the physical row score + +```text +H_G = partial_g log W_G |_(g=0). +``` + +Feynman--Hellmann gives exactly + +```text +partial_g log lambda_G = E_QG[H_G], +partial_g I_G = -E_QG[H_G]. +``` + +Therefore + +```text +Theta_g + = -E_Q4[H_4] + E_Q8[H_8]. +``` + +No rare torus probability needs to be estimated. + +## 3. Exact topological Clapeyron equation + +Implicit differentiation of the coexistence equation gives + +```text +boxed: +dp_w/dg + = -Theta_g/Theta_p + = [E_Q4 H_4 - E_Q8 H_8]/Theta_p. +``` + +The thermal denominator is already known exactly from the row-occupation Perron identity: + +```text +Theta_p + = [w-Kbar_4^0(p)-Kbar_8^0(1-p)]/[p(1-p)] > 0. +``` + +Hence the root moves if and only if the physical source has a nonzero expectation difference between the two safe quasi-stationary phases. + +This is the finite-width version of the question posed in the research compass: + +> which microscopic correction is genuinely common to the two topological sectors, and which one has a nonzero difference matrix element? + +The answer can be measured before naming a continuum field. + +## 4. Common versus differential source directions + +For every source define its two phase scores + +```text +mu_4(g)=E_Q4 H_4, +mu_8(g)=E_Q8 H_8. +``` + +Decompose them into + +```text +mu_common = (mu_4+mu_8)/2, +mu_diff = (mu_4-mu_8)/2. +``` + +Only `mu_diff` moves the coexistence root at first order: + +```text +dp_w/dg = 2 mu_diff / Theta_p. +``` + +Thus a large correction in each individual sector can be irrelevant to Matching One if it lies almost entirely in the common direction. + +This gives an operational meaning to the observed hierarchy + +```text +common x≈4 correction >> sector-odd mismatch. +``` + +The common `x≈4` field can be numerically large in `I4` and `I8` yet cancel from `Theta`. Its mixed dressing of a differential spin-four source can re-enter at the next order, as proposed in `sector-even-dressing-of-spin4-tower-20260914.md`. + +## 5. The oblique spin-four result is already a Clapeyron numerator measurement + +Changing the orientation of the cylinder relative to the square lattice changes the matrix element of an anisotropic lattice correction while preserving the thermal parameter. + +At criticality, the existing oblique safe-transfer data show + +```text +Theta_w(pc,theta) + ~ B4 cos(4theta) ell^(-17/4) +``` + +with very small angular-even leakage. + +In the Clapeyron language this says that the effective sector-difference numerator is dominated by one spin-four angular direction. The root law + +```text +p_root-pc ~ -A4 cos(4theta) ell^-4 +``` + +then follows after division by the leading scalar thermal denominator. + +The exact-equal-circumference `(4,3),n=2` versus axis `w=10` experiment therefore tests numerator and denominator separately: + +```text +E_43/E_axis ~ cos4theta_43, +D_43/D_axis ~ 1, +root-shift ratio ~ cos4theta_43. +``` + +A failure can be localized to either source numerator or thermal denominator rather than described vaguely as a bad exponent fit. + +## 6. Relation to the `(T,N)` response of #802 + +The root tangent in the canonical rank coordinates is + +```text +T_g = dp*/dg = -b_g/b_p. +``` + +The Clapeyron formula is the fixed-width safe-transfer realization of the same tangent when `b` is represented by the topological free-energy difference. + +For a second shape observable `e`, the fixed-b normal response is + +```text +N_g=e_g-(e_p/b_p)b_g. +``` + +Thus an actual source should be summarized by the pair + +```text +(T_g,N_g). +``` + +The new point is that `T_g` itself can be decomposed into a **difference of phase scores** before any continuum extrapolation. This makes the first coordinate much more interpretable. + +A source can have: + +```text +mu_diff !=0 but N_g=0 : pure coexistence-line/thermal-tangent motion; +mu_diff =0 but N_g!=0 : shape change without first-order root motion; +both nonzero : genuine mixed response. +``` + +These cases should not be conflated. + +## 7. A direct route for real spatial or local sources + +For a site-local/logit perturbation the row score `H_G` is explicit. For example, a logit field on newly exposed sites has score equal to the corresponding occupied count minus its normalization contribution; source-compatible safe transfer therefore gives `mu_4,mu_8` directly by one Perron left/right contraction. + +The finite spatial-source Hessian work already shows that mean-zero sources often have zero first-order response by translation symmetry. In that case the first nonzero coexistence response is second order and should be computed from the differentiated Perron resolvent / Green--Kubo susceptibility, not by reinterpreting a symmetry zero as a field selection rule. + +Thus the source order (linear versus quadratic) must be fixed before comparing spin channels. + +## 8. Second-order coexistence curvature + +If symmetry forces `Theta_g=0`, differentiating twice gives at `g=0` + +```text +p_w''(0) = -Theta_gg/Theta_p +``` + +when `p_w'(0)=0`. + +For a transfer source with no explicit second derivative in the log weight, `Theta_gg` is the difference of integrated score covariances (Green--Kubo susceptibilities) between the two Doob phases, plus any declared contact term from the physical coordinate. + +This is the semi-infinite-cylinder analogue of the finite-torus conditional-covariance root Hessian already derived in the spatial-source work. + +It suggests a useful division of labour: + +```text +finite torus : rank-conditioned covariance difference; +long cylinder : safe-phase Green--Kubo susceptibility difference. +``` + +A continuum source identification should make these compatible in their common scaling regime. + +## 9. Topological rank source and Krushkal channel + +For bond/FK surface states, the rank charge `h` has the exact Krushkal realization + +```text +exp[h(r-1)] = A^(s/2) B^(s_perp/2), +A=e^h, B=e^-h. +``` + +Therefore the differential phase score can in principle be formulated as a mixed response + +```text +partial_g partial_h F_topological. +``` + +This gives a source-defined version of the phrase “sector-odd matrix element”. A candidate continuum field matters for the matching root only if it contributes to this mixed topological response. + +## 10. A high-information source matrix rather than another exponent table + +Suppose we choose a small declared basis of real microscopic perturbations `g_a`: + +```text +uniform thermal, +spin-four anisotropy, +one scalar local perturbation, +one map/connectivity-resolved perturbation when available. +``` + +For each width compute only + +```text +D_a(w) = E_Q4 H_a - E_Q8 H_a, +C(w) = Theta_p. +``` + +The vector `D_a` is the finite microscopic **differential-source fingerprint** of the two sectors. Its angular/transformation properties can be compared across widths and geometries. + +This is more informative than fitting one root sequence to several possible powers because it asks directly which physical source directions survive the common-sector quotient. + +No large source dictionary is justified: two mechanisms should be selected first, and one perturbation should be chosen that gives different predicted fingerprints. + +## 11. Implication for original-U + +This does not solve #275 or replace its frozen observable. It does clarify the theoretical object required there. + +A named candidate must eventually provide a forward map from the actual original-U microscopic source into a differential topological response, including normalization and moving-root counterterm. The finite-width Clapeyron score is a lattice-side prototype of that map: + +```text +physical source + -> phase-score difference + -> root tangent. +``` + +If two continuum candidates give the same allowed differential-source image, more precision on the same source cannot identify them. + +## 12. Claim boundary + +Exact finite-state statements: + +- Perron/Doob source score identity; +- coexistence implicit derivative; +- thermal denominator from row occupation; +- common/differential decomposition. + +Conjectural/scaling interpretations: + +- identification of particular differential source directions with spin-four/eight-arm continuum operators; +- large-width scaling of the score vector; +- compatibility with original-U continuum forward maps. + +The main conceptual result is independent of those identifications: **Matching One responds to the difference of source scores between two topological quasi-stationary phases.** \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/topological-source-algebraic-compression-20260914.md b/docs/manuscripts/geometric-balance/topological-source-algebraic-compression-20260914.md new file mode 100644 index 000000000..affdb859f --- /dev/null +++ b/docs/manuscripts/geometric-balance/topological-source-algebraic-compression-20260914.md @@ -0,0 +1,129 @@ +# The bounded rank source is algebraically exhausted by `(M, chi)` + +Date: 2026-09-14 + +Status: exact finite algebra and research-priority correction. + +## 1. Three-state charge implies a closed source function + +Let + +```text +X=r-1 in {-1,0,+1}, +M=E[X]=P2-P0, +chi=E[X^2]=P0+P2. +``` + +Then for every finite torus and every source `h`, + +```text +Z(h)=E exp(hX) + =P0 e^-h + P1 + P2 e^h +``` + +can be rewritten exactly as + +```text +boxed: +Z(h)=1 + M sinh h + chi (cosh h-1). (1.1) +``` + +Therefore the complete `h` dependence at fixed microscopic parameters is determined by only two scalars `(M,chi)`. + +At the balance root `M=0`, + +```text +Z(h)=1+chi(cosh h-1). +``` + +## 2. Higher pure-h cumulants are not new observables + +At the root, + +```text +kappa_2(X)=chi, +kappa_3(X)=0, +kappa_4(X)=chi-3chi^2, +``` + +and every higher pure charge moment/cumulant is a polynomial in `chi`. + +Away from the root they are polynomials in `(M,chi)`. + +Hence a programme that repeatedly computes higher derivatives with respect to the bounded rank source `h` is algebraically redundant unless another parameter/source is varied simultaneously. + +## 3. The informative objects are mixed susceptibilities + +New information appears in derivatives such as + +```text +partial_p M, +partial_g M, +partial_tau M, +partial_z M, +partial_g chi, +... +``` + +or equivalently mixed source derivatives + +```text +partial_h partial_g log Z |_(h=0), +partial_h partial_p log Z |_(h=0), +``` + +because these probe how the exact Euler/rank defect couples to a physical perturbation. + +For a normalized microscopic source score `H`, + +```text +partial_g M = Cov(X,H). +``` + +This is exactly the numerator entering the root-response covariance quotient. + +## 4. Correction to the master-source language + +The combined finite object + +```text +Z(p;h,z,tau) + = P0 e^-h + P1 H_p(z) + P2 e^h +``` + +remains a useful bookkeeping object. But its `h` coordinate is **not** an independently complicated functional direction: once sector weights are known, the `h` dependence is elementary. + +Thus the hard continuum/massive tasks are really to determine + +```text +P0(lambda,tau), +P2(lambda,tau), +rank-one neutral law H(z), +and mixed responses to physical sources, +``` + +not to solve an arbitrary `h`-dependent function from scratch. + +A periodic modified trace realizing `h` can still be technically valuable because it extracts the sector sums; the novelty lies in computing the sectors, not in the exponential source algebra itself. + +## 5. Research-priority consequence + +Prefer + +```text +one new mixed h-g response +``` + +over + +```text +many higher h cumulants. +``` + +This is another instance of the research-compass rule: exact solvability of a generated family does not imply that every derivative carries independent information. + +## 6. Claim boundary + +Exact: all formulas above. + +Interpretive: prioritize mixed source/geometry/thermal responses over pure topological-source elaboration. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/topological-source-master-scaling-20260914.md b/docs/manuscripts/geometric-balance/topological-source-master-scaling-20260914.md new file mode 100644 index 000000000..679df681a --- /dev/null +++ b/docs/manuscripts/geometric-balance/topological-source-master-scaling-20260914.md @@ -0,0 +1,348 @@ +# A master topological-source scaling function for charge, neutral count, modulus and lattice anisotropy + +Date: 2026-09-14 + +Status: unifying scaling conjecture. Its finite source coordinates are already exact on the lattice; existence/universality of the continuum scaling function is not proved for square-site Matching One. + +## 1. The exact finite generating object already exists + +For one honest torus, after aggregating rank-one slope for the moment, define + +```text +P0 = Pr(rank=0), +P1 = Pr(rank=1), +P2 = Pr(rank=2), +H_p(z) = E[z^K | rank=1], +``` + +where `K>=1` is the number of parallel essential components in the neutral/rank-one sector. + +Introduce a bounded rank charge `h` and neutral fugacity `z`: + +```text +boxed: +Z_L(p;h,z) + = P0(p) e^-h + + P1(p) H_p(z) + + P2(p) e^h. +``` + +At `z=1`, + +```text +partial_h Z|_0 = P2-P0 = M, +``` + +and the matching root is the zero of this first charge response. + +The log-odds coordinate is + +```text +vartheta = log(P2/P0). +``` + +The exact finite common-window decomposition `(chi,vartheta,H)` is simply another coordinate system on the same generating object. + +For bond/FK embedded graphs, the new Krushkal observation gives an exact surface-polynomial realization of `h`; the affine-TL seam gives `z=alpha^2/Q` in rank one. Thus the two source directions already have independent microscopic dictionaries. + +## 2. Near-critical continuum target + +For a torus of fixed modulus `tau`, let the linear size be `L` and introduce the thermal coordinate + +```text +lambda = a_t (p-pc) L^(3/4). +``` + +The natural continuum target is a topology-resolved scaling function + +```text +boxed: +Z_L(p;h,z,tau) + -> Z_*(lambda;h,z,tau) +``` + +at fixed `(lambda,h,z,tau)`, after the ordinary normalization appropriate to probabilities. + +This one object contains: + +```text +rank law : z=1, derivatives in h; +neutral essential count : h=0, derivatives in z within rank one; +matching root : zero of partial_h Z at h=0; +critical modular geometry : lambda=0, vary tau; +near-critical crossover : vary lambda; +Potts/TL source dictionary: continue Q and realize h,z by seams/projectors. +``` + +The main missing continuum engine in #782 is precisely a way to calculate this object (or its sector components) away from `lambda=0`. + +## 3. Matching symmetry should act on the source, not only on one observable + +At the continuum matched pair, the natural involution is schematically + +```text +lambda -> -lambda, +h -> -h, +z -> z, +``` + +with the rank-one projective slope transported by complement. + +Thus the strongest form of the continuum symmetry would be + +```text +Z_*(lambda;h,z,tau) + = Z_*(-lambda;-h,z,tau) +``` + +up to the explicitly declared graph/normalization convention. + +Consequences include the familiar critical equality `P0=P2` and the odd/even organization of charge versus neutral variables. + +For square SITE this remains a universality/scaling statement, not an exact finite same-graph identity. + +## 4. The first lattice correction is naturally a degenerate `x=21/4` shell + +For a dimensionless torus source response, the sector-odd correction that produces an `L^-4` root shift has correction exponent + +```text +omega_odd = 13/4, +``` + +because the thermal derivative scales as `L^(3/4)`. + +The `x=21/4` resonance note shows that the leading shell can contain at least two angular channels of the same total dimension: + +```text +spin 0 : scalar eight-arm-like component, +spin 4 : thermal-family level-four component. +``` + +Therefore the lattice expansion of the charge source should be written as a **solution space**, not a single ray: + +```text +Z_L + = Z_* + + L^(-13/4) [ + g0 Z_(21/4,0)(lambda;h,z,tau) + + g4 Re(e^{i4 theta_lat} Z_(21/4,4)(lambda;h,z,tau)) + ] + + ... . +``` + +Here `theta_lat` is the orientation of the microscopic square lattice relative to the chosen cylinder/torus cycle. Reflection reduces the visible spin-four dependence to a cosine in the current real geometries. + +A common sector-even `x≈4` irrelevant field then dresses these coefficients at an additional `L^-2`, producing the observed square `4,6,8,...` root ladder without requiring a new odd operator at every power. + +## 5. Root shift from the master expansion + +Write the continuum charge log-odds at `z=1` as + +```text +vartheta_*(lambda,tau). +``` + +At the continuum critical point, + +```text +vartheta_*(0,tau)=0, +partial_lambda vartheta_*(0,tau) != 0 +``` + +for a nondegenerate thermal coordinate. + +Let the `L^-13/4` correction to `vartheta` be + +```text +V_odd(tau,theta_lat) + = b0 F0(tau) + + b4 Re[e^{i4theta_lat} F4(tau)]. +``` + +Then solving `vartheta_L=0` gives + +```text +lambda_root + = -L^(-13/4) + V_odd / [partial_lambda vartheta_*(0,tau)] + + ..., +``` + +and hence + +```text +boxed: +p_root-pc + = -L^-4 + V_odd / + [a_t partial_lambda vartheta_*(0,tau)] + + ... . +``` + +This formula contains both the scalar and spin-four `x=21/4` amplitudes and makes clear why exponent four alone cannot identify the mechanism. + +## 6. Semi-infinite cylinder is a large-aspect boundary condition + +For a rectangular torus with aspect `rho=m/w`, the finite charge fugacity obeys at large `rho` + +```text +vartheta_(w,m)(p) + = m Theta_w(p) + O(1), +``` + +where `Theta_w` is the safe-sector free-energy difference. + +At criticality the spin-four cylinder result gives + +```text +Theta_w(pc,theta) + ~ B4 cos(4theta) w^(-17/4) +``` + +(up to a possible scalar `B0` component now explicitly projected). + +Therefore the fixed-aspect torus scaling function for the `L^-13/4` charge correction must have a large-aspect asymptotic whose leading growth is linear in `rho` and whose coefficient matches the semi-infinite-cylinder amplitudes. + +This provides a concrete bridge from #585 modular tomography to the much sharper #771 oblique cylinder data: + +> candidate torus solution spaces should reproduce the known large-aspect angular projector, not merely fit finite moduli. + +## 7. Equal-circumference angular projection extends away from `pc` + +The `(4,3),n=2` versus axis `w=10` pair has the same physical circumference. Once the state spaces are built, evaluating several nearby `p` values is much cheaper than opening more directions. + +A stronger test than only `(E,D,p_root)` is to predeclare two or three thermal coordinates and form the angular projector + +```text +P4(p) + = [Theta_axis(p)-Theta_43(p)]/[1-cos4theta_43], + +P0(p) + = [Theta_43(p)-cos4theta_43 Theta_axis(p)]/[1-cos4theta_43]. +``` + +After the appropriate `w^(17/4)` rescaling, the conjecture predicts that these approach two distinct near-critical scaling functions + +```text +Psi4(lambda), +Psi0(lambda), +``` + +rather than merely two constants at `lambda=0`. + +This is a high-information extension because it asks whether the angular decomposition is an **operator scaling function**, not just an accidental critical-point fit. + +Stop after this pair unless an angular-orthogonal component is actually resolved. + +## 8. Fixed-subcritical tail as a topological trans-series + +On the subcritical side, define + +```text +s = w kappa(p). +``` + +Near criticality `s` is a function of `lambda` with leading behavior + +```text +s ~ C |lambda|^(4/3) +``` + +in the massive tail. + +The rare-topology witness-grading note proposes that the fixed-subcritical expansion is organized by + +```text +epsilon = e^-s. +``` + +Thus the large-negative-`lambda` boundary of the master scaling object should generically admit a topology-graded expansion of the form + +```text +observable(lambda) + = perturbative/massive background + + sum_(j>=1) e^(-j s(lambda)) P_j(s,lambda), +``` + +where `j` counts additional essential witnesses/topological instantons and `P_j` carries polynomial/local prefactors. + +This is not asserted to be a convergent trans-series. It is a structural matching rule between: + +```text +near-critical finite-lambda scaling +and +fixed-subcritical rare-topology expansions. +``` + +It explains why pure fragmentation and the coalescing slow-doublet picture are reliable only in the `s->infinity` tail. + +## 9. The near-critical genealogy should become split-merge at finite `s` + +The #780 no-merger proof gives a relative correction of one-extra-witness order `e^-s`. Therefore: + +```text +s -> infinity : pure cut/fragmentation process; +s = O(1) : mergers survive and must be resummed. +``` + +This suggests that the continuum common-label genealogy at finite thermal `lambda` is not a dressed version of the pure fragmentation generator. It should be a split-merge/topological hard-core process whose far-subcritical expansion reduces to the known cut process. + +The same `s` should control the disappearance of the #800 metastable slow doublet. + +This supplies a qualitative dynamical boundary condition for any massive Potts/topological process construction. + +## 10. Relation to the Krushkal / affine-TL dictionary + +For bond/FK configurations on the torus: + +```text +h : Krushkal A=e^h, B=e^-h topological source, +z : affine-TL noncontractible-loop seam z=alpha^2/Q. +``` + +Thus a massive Potts realization of the master object should not start from an arbitrary twist. It should seek a toroidal modified trace carrying both source coordinates and satisfying the critical homology boundary condition. + +The current missing representation-theory step is to realize the Krushkal rank source locally/trace-theoretically in the same periodic algebra that already realizes `alpha`. + +## 11. What this reorganizes + +This single object puts the current programmes into one hierarchy: + +```text +#768 : operator content of the first h-odd lattice correction; +#585 : tau dependence / modular solution space of the same correction; +#767 : lambda dependence and rare-to-critical crossover; +#780 : process interpretation of the large-negative-lambda rare-topology tail; +#782 : massive engine capable of computing Z_*(lambda;h,z,tau); +#800 : one-witness spectral tunnelling in the rare-topology tail. +``` + +They need not all be active compute tasks. The point is that a result in one direction now has explicit boundary conditions for the others. + +## 12. Minimal falsification programme + +1. **Equal-length angular scaling function:** axis `w=10` versus `(4,3),n=2` at a few predeclared thermal points. If the projected spin-four/scalar decomposition fails immediately away from `pc`, the one-operator-shell picture is too simple. + +2. **Large-aspect modular boundary:** any candidate #585 solution basis must reproduce the cylinder `cos4theta` amplitude as `rho->infinity`. + +3. **Mass-clock crossover:** compare Palm score speed with independent mass-slope information; failure means `s=w kappa` is not the sufficient clock at the claimed precision. + +4. **Finite-s genealogy:** once a controlled sequence with `w kappa=O(1)` exists, measure one merger/split statistic rather than more fixed-subcritical kernel moments. + +## 13. Claim boundary + +Exact finite ingredients: + +- the `(h,z)` source decomposition; +- matching rank/source identities; +- Krushkal finite FK realization of `h`; +- affine-TL finite rank-one realization of `z`. + +Conjectural continuum structure: + +- existence/universality of `Z_*(lambda;h,z,tau)` for square SITE; +- the two-component `x=21/4` correction shell; +- the topological `e^{-w kappa}` expansion and split-merge finite-`s` process; +- the large-aspect matching between modular and cylinder amplitudes. + +The value of this ansatz is that each conjecture now has a different observable direction; failure can localize which bridge breaks. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/two-free-energy-rank-phase-diagram-20260914.md b/docs/manuscripts/geometric-balance/two-free-energy-rank-phase-diagram-20260914.md new file mode 100644 index 000000000..1cf604336 --- /dev/null +++ b/docs/manuscripts/geometric-balance/two-free-energy-rank-phase-diagram-20260914.md @@ -0,0 +1,281 @@ +# One-sided free energies and the three homology-rank phases + +2026-09-14. Corrected formulation after self-audit. The NN and complementary matching inverse-correlation norms are **not simultaneously subcritical at one common p**, because + +\[ +p_c(G8)=1-p_c(G4). \tag{0.1} +\] + +Therefore the lower and upper rank boundaries are controlled by subcritical free energies on **opposite sides of the critical point**, not by two same-p subcritical masses. + +The correct phase diagram is: + +- below `p_c`: the black NN free energy separates rank 0 from rank 1; +- above `p_c`: the complementary white matching free energy separates rank 1 from rank 2; +- at/near the boundaries: component-Poisson/crossover information is needed. + +## 1. Lower-side NN free energy, p < pc(G4) + +Fix + +\[ +00) +\to-\max\{1-\alpha_4,0\}. \tag{1.4} +\] + +### 1.1 If alpha4 < 1: rank zero + +Then positive black homology is exponentially rare: + +\[ +\boxed{r_4\xrightarrow P0.} \tag{1.5} +\] + +### 1.2 If alpha4 > 1: positive rank appears + +Then + +\[ +P_p(r_4>0)\to1. \tag{1.6} +\] + +To identify the rank as one rather than two, use the arbitrary-period subcritical rank comparison already proved in the geometric manuscript: for every fixed `pinfinity`, norm equivalence implies `ell_n->infinity`. Equation (1.6) gives `P_0->0`; multiplying by the exponentially small ratio in (1.7) yields + +\[ +P_2\to0. \tag{1.8} +\] + +Hence + +\[ +\boxed{r_4\xrightarrow P1\qquad(p1).} \tag{1.9} +\] + +Thus below criticality the single black NN free energy controls the transition + +\[ +\boxed{0\longleftrightarrow1.} \tag{1.10} +\] + +## 2. Upper-side matching free energy, p > pc(G4) + +Now fix + +\[ +p>p_c(G4), +\qquad +q=1-p 1: white positive rank is rank one + +The matching theorem gives + +\[ +P_q(r_8>0)\to1. \tag{2.7} +\] + +Apply the same subcritical rank comparison (1.7), now to graph `G8` at `q1`, + +\[ +P_q(r_8=2)\to0. \tag{2.8} +\] + +Hence + +\[ +r_8(\omega^c)\xrightarrow P1. \tag{2.9} +\] + +and Alexander duality gives + +\[ +\boxed{r_4\xrightarrow P1\qquad(p>p_c,\ \alpha_8>1).} \tag{2.10} +\] + +Thus above criticality the single **white matching** free energy controls + +\[ +\boxed{1\longleftrightarrow2.} \tag{2.11} +\] + +## 3. Correct three-phase picture + +Away from the free-energy boundaries and away from the critical point: + +\[ +\boxed{ +\begin{array}{c|c|c} +\text{parameter side}&\text{free-energy condition}&\text{black NN rank}\\ +\hline +p1&1\\ +p>p_c&\log N/\rho_8(1-p)>1&1\\ +p>p_c&\log N/\rho_8(1-p)<1&2 +\end{array}} \tag{3.1} +\] + +The rank-one phase is the geometry-induced bridge between the lower NN birth and the upper complementary-matching birth. + +There is no regime in which one should simultaneously plug `p` into a subcritical NN mass and `1-p` into a subcritical matching mass: by (0.1), exactly one of those two complementary parameters is subcritical away from `p_c`. + +## 4. Matching observable + +Recall + +\[ +M(p)=P_2(p)-P_0(p). \tag{4.1} +\] + +The phase limits are + +\[ +\boxed{ +M\to +\begin{cases} +-1,&p1,\\ +0,&p>p_c,\ \log N/\rho_8(1-p)>1,\\ ++1,&p>p_c,\ \log N/\rho_8(1-p)<1. +\end{cases}} \tag{4.2} +\] + +This makes the plateau mechanism explicit. The lower free-energy boundary drives `M` from `-1` to `0`; the upper complementary boundary drives it from `0` to `+1`. The finite matching root lies inside the central topological crossover and is not the same object as either fixed-p free-energy threshold. + +## 5. Exponential-aspect special case + +Let the shortest-period direction converge to `e` and + +\[ +\frac{\log(N/\ell)}{\ell}\to d>0. \tag{5.1} +\] + +Below `p_c`, + +\[ +\rho_4(p)=\ell\tau_{4,p}(e)+o(\ell), +\qquad +\log N=d\ell+o(\ell). \tag{5.2} +\] + +The lower boundary is therefore + +\[ +\boxed{\tau_{4,p}(e)=d,} \tag{5.3} +\] + +which gives `p=a_e(d)`. + +Above `p_c`, write `q=1-p0)=o(\rho). \tag{6.2} +\] + +The actual endpoint-sector probability can be nontrivial and depends on subexponential component activities. This is precisely where the Poisson birth-window theory enters. + +Thus the hierarchy is: + +1. correlation-norm free energy locates the rank boundary at exponential scale; +2. component intensity resolves the boundary law; +3. its derivative/semiconvexity resolves the affine Gumbel window at regular points. + +## 7. Critical p itself + +At `p=p_c(G4)`, both complementary graphs are critical and neither subcritical correlation norm supplies a positive free energy. This note deliberately makes no rank-law claim at the critical parameter for arbitrary aspect sequences. + +Critical fixed-aspect wrapping laws and the geometric balance-root theorem are separate inputs/problems. + +## 8. Claim boundary + +This corrected phase diagram uses the arbitrary-shape subcritical free-energy theorem, the geometric manuscript's subcritical `P2/P0` suppression, and exact digital Alexander duality. It does not analytically continue an inverse-correlation norm through criticality. \ No newline at end of file diff --git a/docs/manuscripts/geometric-balance/two-observable-response-and-angular-alias-20260914.md b/docs/manuscripts/geometric-balance/two-observable-response-and-angular-alias-20260914.md new file mode 100644 index 000000000..49676d06c --- /dev/null +++ b/docs/manuscripts/geometric-balance/two-observable-response-and-angular-alias-20260914.md @@ -0,0 +1,252 @@ +# 真正的法向响应需要两个观测：根/斜率归一化、角谐波混叠与计算读出 + +2026-09-14。服务 #802/#768。读取 #771 `b1aafe5ce40583f2971c4491dcc69fc0af4c39df` 的 `docs/research-bridges-post-compass-20260914.md` A1/A3，以及 #802 的等物理周长配对建议。保留所有原始数值，但收紧两项解释：根平移加斜率归一化不等于消去任意热切向；两个角度也不等于不带余项假设的spin鉴定。 + +本篇先给精确反例/微分关系，再给可以复用现有求解器的读出。工作期间读到723f629的独立核验批次；第7–9节纠正其中真正会改变算力决策的源正规化、源冗余与spin/momentum推断。保留原始数值，不根据“独立核验”标签自动接受结论。 + +## 1. 一条单调标量曲线不能独自定义“法向” + +考虑真实rank概率曲线的两个独立坐标 + + b=(1/2)log(P0/P2)，e=log[P1/(2sqrt(P0P2))]。 + +对任意正c=exp(e)，概率为 + + P0=e^b/[2(c+cosh b)]，P1=c/[c+cosh b]， + P2=e^-b/[2(c+cosh b)]。 + +这些只保证正的正规化概率，不自动保证它们是某个独立site模型。以下反例用于检验逻辑蕴含。 + +在b_p≠0处，源g的真实横截响应是 + + N_g=e_g−(e_p/b_p)b_g，T_g=−b_g/b_p（根处）。 (1) + +“法向”是消去热切向的横截商，不是未经说明的Euclidean/Fisher正交。若N_g在整个局部(p,g)域上为0，则e=E(b)，故所有曲线都是同一二维曲线的不同参数化。反过来，公共重参数化必给N=0。 + +只有一个严格单调的标量b时，任意另一个b_g都能写成b_0(phi_g(p))。所以单独看b的根中心化曲线，最多检验某种受限参数变换，不能检验整个二维rank law的法向方向。 + +## 2. 根与斜率归一化只消去两个自由度 + +设 + + b_g(x)=B(x+g a(x))， e_g(x)=E(x+g a(x))， + B(0)=0，s0=B'(0)≠0。 + +这是一族纯热重参数化，N恒为0。其根r_g及根斜率s_g满足 + + r'_0=−a(0)， (d/dg)s_g|0=s0 a'(0)。 + +若按新稿建议只读 + + Psi_g(y)=b_g(r_g+y/s_g)， + +直接求导给 + + (d/dg)Psi_g(y)|0 + =B'(y/s0)[a(y/s0)−a(0)−(y/s0)a'(0)]。 (2) + +因此a的常数、线性部分被去掉，二次及更高部分仍然留下。非零Psi差异不证明非零N。 + +最小反例：b=x+g x²，e=b²。在局部合法区间中根为0、根斜率为1，N严格为0；但Psi_g(y)=y+g y²。反向反例同样重要：b=x，e=x²+g x。根、斜率乃至整条b曲线都不变，但N_g(b)=b，除了b=0外非零。 + +故： + +- 根中心化后的单条charge曲线不同，不推出存在真法向修正； +- 根中心化后的单条charge曲线相同，不推出N=0； +- N(0)=0也不推出整个N(b)=0，可能只是对称性零点。 + +旧TANGENT_SPIN4中更强的“G4=c F'、常数平移”仍可用Psi检验；但它不是一般“G4=a(x)F'”的等价说法。允许的热坐标类必须由物理模型固定，不能为了保住猜想随意扩大。 + +## 3. 用已经在算的两条safe压力，得到真正的二维比较 + +当前长圆柱求解器已分别输出I4^0、I8^0，不必再建一个rank全枚举。对最短周期u、重复数n，ell=n|u|；若每个转移行的物理高度为1/|u|，定义无量纲坐标 + + D_theta(p)=ell |u| [I4_row(p)−I8_row(1-p)]， + A_theta(p)=ell |u| [I4_row(p)+I8_row(1-p)]/2。 (3) + +先检查实际代码的行高约定；不能把(3)机械复制到另一个引擎。 + +在D_theta局部严格单调的区间，对一组共同d值解D_theta(p_theta(d))=d，然后比较 + + A_theta(p_theta(d))−A_axis(p_axis(d))。 (4) + +根位移另外保存p_theta(0)−p_axis(0)。这就是该**两压力观察量**的切向/横截比较。其无穷小源形式为 + + N_A=A_g−(A_p/D_p)D_g。 + +(4)能剥去任意局部热重参数化，而不只是根与斜率的仿射变换。若两压力向量真的只是同一曲线重新参数化，(4)消失；若不消失，纯热参数化解释在这个观察量上失败。 + +它不是有限torus的原始rank N。固定宽度、纵向长度m→∞时有b/m→−Delta/2、e/m→(I4+I8)/2（log闭合振幅是o(m)，P1→1）；有限aspect需要另外保留闭合振幅和其他谱，不能靠这个极限把两种读出混用。original-U仍遵守#275原合同。 + +计算成本：沿原有两方向的5–7个p/目标d点，同时存I4,I8及其一阶导数，再做反演/插值。它不要求新转移状态，不能声称耗时已测得。若|D_p|≥d0、反演的D误差≤epsilon_D、A求值误差≤epsilon_A，则一个点的A误差≤epsilon_A+sup|A_p/D_p| epsilon_D；两个方向的误差相加。共用数据的协方差应保留。 + +## 4. 等周长的两个角度很有用，但它们只完成条件分解 + +对axis ell=10与u=(3,4),n=2，h=cos4theta=−527/625。写缩放不匹配量 + + Y(theta)=B0+B4 cos4theta+B8 cos8theta+R(theta)。 + +如果暂时只拟合B0+B4h，则 + + B4_hat=(625/1152)(Y_axis−Y_34)， + B0_hat=(625Y_34+527Y_axis)/1152。 (5) + +但cos8theta=2h²−1，精确代数给 + + B0_hat=B0+(429/625)B8+R0， + B4_hat=B4+(196/625)B8+R4。 (6) + +两个点对两个系数的拟合没有剩余自由度。即使数据算到无限精度，也不能不加余项条件地用B0_hat非零宣布标量场，或用它很小排除所有标量贡献。 + +若独立控制|B8|≤r8、|R_axis|≤rA、|R_34|≤rB，求值误差≤epsilonA,epsilonB，则 + + |B0_hat−B0|≤(429/625)r8 + +(527/1152)(rA+epsilonA) + +(625/1152)(rB+epsilonB)。 (7) + +其他cos(4k theta)同理产生alias。若特定候选给出它们严格更高阶，可以把(7)用于渐近比较；C4对称性本身不提供幅度界。无需因此自动购买第三大周长；先输出条件区间和已有额外方向的残差。 + +另一个共同误差是pc参考值。若Y=|u|Theta_row(pc_ref)ell^(17/4)，D=|u|partial_pTheta_row ell^(1/4)，则pc_ref−pc=delta产生 + + Y(pc_ref)−Y(pc)=delta ell^4 D(pc)+二阶余项。 (8) + +leading D若角度无关，这项主要污染“标量截距”。不应把它当成真正的spin0信号。根的两方向差不依赖pc_ref，可作为更稳健的第一输出；绝对scalar振幅仍需独立pc误差及模型余项。 + +## 5. 真正能改变后续方向的联合判别 + +只复用当前等周长作业，按以下顺序读它： + +1. 实际物理归一化后的根差、Delta、Delta_p，以及两压力平均A均保存；不只留root。 +2. 先作(5)的条件角度分解，明确(6)的高谐波alias与(8)的pc误差。它是诊断，不是field identification。 +3. 再作(4)真正的两观测热对齐。若root差明显而(4)消失，只支持“在该两压力曲线上切向”；不证明微观插入或连续模块为零。 +4. 若(4)不消失，#768优先计算该横截响应对应的实际模块/矩阵元。无需再以更多scalar root来补同一个缺口。 + +**工作猜想PAIR-TANGENT**：leading spin4修正在整个两压力向量上形如a(X)∂_X(D0,A0)。它比只对一个D写G4=aD0'更强，后者对单调标量本来就是局部可写的。PAIR-TANGENT预言两压力对齐的spin4领先项消失；root shift仍可非零。 + +**竞争猜想PAIR-NORMAL**：spin4具有非零N_A，且其热对齐响应的跨方向/跨尺度形状可由一个共同插入描述。若真实源会产生额外耦合，则需要将那些允许幅度写进模型，而不是自动称为另一个continuum field。 + +两者都可能失败于更高谐波、方向相关的周期闭合、多个同维通道或不受控的热非线性。需要验证的是具体向量预测，不是给漂亮指数贴标签。 + +## 6. 范围、有限控制、出处 + +本轮标准库脚本检查纯切向与纯法向两个正概率坐标反例，完整的H8 alias有理系数，以及既有相同数据的误差传输公式。它没有重跑oblique safe引擎，没有新site尺寸、样本或CFT矩阵元。 + +Jacobsen 1507.03027v1 §7–8报告square/kagome不同修正与共同leading磁扇区；它没有指定本篇的具体热模块或PAIR-TANGENT。https://arxiv.org/html/1507.03027v1 + +内部：docs/root-response-and-bridge-targets-20260914.md；docs/research-bridges-post-compass-20260914.md A1/A3；#802 comments5662215626/5662494377。这里收紧A3把root/slope-centered单charge曲线称为N的那一步，并保留它作为“仅常数平移/仿射变换”的有效诊断。 + + +## 7. 实际黑/白相邻对源的全耦合冗余，以及可直接修复的法向数值 + +本节针对 `root-tangential-normal-response-20260914.md` §3.1/5.2 及 +`sector-root-response-20260914.json`（commit723f629，blob8dde718f411f2e36fa2fd70ea9cbb7cee7234ae3）。该批次明确把水平黑占据对和水平白占据对作为两个不同非热源，并以未正规化safe转移的FH导数构造平均压力。以下不需要重跑其转移。 + +### 7.1 是同一个概率族的参数重写，而非两个独立法向干预 + +在每行周向周期、总N站点的配置上，定义K为黑站点数，H_B/H_W为水平黑对/白对数。逐配置 + + H_W=N−2K+H_B。 (10) + +因此对所有实耦合g，正规化测度精确满足 + + mu_(p,g H_W)=mu_(p_eff,g H_B)， + logit(p_eff)=logit(p)−2g。 (11) + +两个源同时加入时，只剩z_eff=z−2g_W与g_eff=g_B+g_W，另有不影响概率的常数Ng_W。此结论对任意观测和有限尺寸成立，不只对rank。 + +更一般，任意边集F上有 + + H_W=|F|−sum_v deg_F(v)n_v+H_B。 + +非正则F对应局部logit的degree-field重写；只有正则F才是均匀热坐标重写。勿把本节水平degree2公式无条件套到其他源。 + +对(10)求导，在相同基准p上严格得到 + + N_W=N_B，T_W−T_B=2p(1-p)。 (12) + +T若作为root导数，必须在对应基准匹配根评价；pc_ref处的同一代数比值只是该点的等charge热补偿斜率。仅当pc_ref恰为根，二者才相同。 + +有限g下，两个源的内禀e(b,g)曲线也相同；logit根满足z_W^*(g)=z_B^*(g)+2g。两张图/两个源看起来“可区分”的根平移，只是已经明示的热坐标改变，不识别两个不同非热机制。 + +### 7.2 公共正规化在Delta中消去，但不能从平均safe压力中删掉 + +行内源的完整单行配分函数为 + + Z_row(p,g)=sum_m p^K(1-p)^(w-K) exp[gH(m)]，f=log Z_row。 + +若tilde_lambda_j是未正规化safe算子的Perron根，物理safe衰减率为 + + I_j=f−log tilde_lambda_j。 (13) + +所以Delta=I4−I8不变，但平均safe压力多f。用未正规化rank活动直接计算也一样：rank1的每行压力是f，不能同时假设其未正规化概率趋1。全局normalizer确实从e=logP1−(logP0+logP2)/2−log2中代数消去；被漏掉的是未正规化rank1权重的增长，而不是一个应被加回两遍的全局常数。 + +在g=0处，f(p,0)=0对所有p成立，故f_p=0。因而 + + Nhat_physical=Nhat_raw+f_g， + f_g=w p²（黑水平对），w(1-p)²（白水平对）。 (14) + +按既有JSON打印的小数修复（不是新的特征值求解/区间证书）： + +|w|原黑Nhat|原白Nhat|修正黑Nhat|修正白Nhat| +|---|---:|---:|---:|---:| +|4|-1.35004975377|-0.60808134743|0.05534176915|0.05534176915| +|5|-1.71389424634|-0.78643373842|0.04284515731|0.04284515731| +|6|-2.07324978375|-0.96029717424|0.03483750063|0.03483750063| +|7|-2.43000063194|-1.13155592085|0.02943453316|0.02943453316| +|8|-2.78527637055|-1.30133955788|0.02550667528|0.02550667528| + +两条修正序列差≤1.5e−15；原T_W−T_B与2pq差≤4e−15。这符合(12)，不是重新发现两条独立收敛序列。原来的“O(w)法向值”、大约5000倍的相对解释不再适用于物理正规化读出。raw FH计算仍可作为修复输入保留；无需重新枚举状态或提高Perron精度。 + +测试也应允许给所有转移权重共同乘exp(cg)。它改变raw压力，但必须不改变任何正规化概率或物理N；(13)确保这个不变性。 + +### 7.3 N=0不等于T=−1，不识别微观源 + +新核验§5.2的“N=0 iff 源热向 iff T=−1；因此N=0且T!=0不可能”是错误的。 + +N=0只说明两观测的微分投影平行，比例任意，且可能有不可见的微观源分量。p→p+a g给T=−a,N=0；exp(gK)是logit变化，给p根的T=−p(1-p),N=0；T=−1仅适用于单位的直接p平移（或相应的logit根约定）。均值零site场在平移不变背景下的一阶是(T,N)=(0,0)，不是(−1,0)。 + +附带纠正：均匀奇周长torus仍是平移不变的；交替源在奇周期上有非零均值/接缝，故产生一阶信号。被破坏的不是均匀背景的平移不变性。以几处网格上导数同号或函数递增，也不能独自证明整个参数区间单调。 + +## 8. 消去热方向后，还有一个可能的“度量方向” + +(14)修复后，w*Nhat在w4..8约为 + + 0.22137，0.21423，0.20903，0.20604，0.20405。 + +这只是五个既有有限值的后处理，不证明渐近幂。但它指向一个比“发现新的无关场”更低成本的竞争解释：水平pair源本身破坏横纵等价。即使调回热中心，连续描述的速度/横纵度量也可改变，使共同磁gap的响应为O(1/w)。 + +**METRIC-TANGENT猜想**：经热调整后，水平pair源的上述领先normal响应主要是共同度量因子，而非独立的扇区奇无关插入。这个猜想不与非零N冲突，因为N只商掉一个热方向，没有商掉度量。 + +可检验接口：设G是一条在源变化下连续跟踪的独立safe谱间隙（例如同一孤立次领先支的log(lambda1/|lambda2|)）；令Ibar=(I4+I8)/2。若Ibar与G均只乘同一速度v(g)，则等Delta条件下 + + N_(Ibar/G) = N_Ibar/G − Ibar*N_G/G² = 0。 (15) + +因此需要的下一个独立量是这个比值的实际源导数，而不是黑/白pair再各跑一排宽度。次领先支必须被正确跟踪，不能让本征值交换制造假信号；声称共同速度也只是待检验模型，不由(15)自动证明。 + +两观测一般不足以同时商掉热坐标和独立度量因子后还留下横截方向。加入一个物理可见的谱标尺是为解决这个特定歧义，不是重新无限增加descriptor。优先使用已有状态/特征向量和FH，先核查已有产物是否已含G_g，再决定追加一个小求解。 + +若(15)仍显著非零，才得到超过热调定与共同度量的候选响应；它仍不自动给出CFT身份。 + +## 9. 平移选择Fourier动量，不会自动杀死共形spin4密度 + +新文件 `first-noncommon-correction-channels-20260914.md` §2.4写“平移不变的一点函数必有h=hbar”，并据此排除spin±4对动量0能级的修正。这条普遍推断不成立。 + +圆柱空间平移的局域关系为[P,O(x)]=−i partial_x O(x)（符号依约定）。它要求平移不变态中的一点函数位置无关；共形spin h−hbar是局域旋转的张量权，不是把任意O直接当作固定非零Fourier模式。积分O(x)取其Fourier零模，该零模可以来自spinful密度。无限圆柱/一般torus也没有完整平面SO(2)对称性；正方形torus的C4若作用于完整问题，允许spin4而非排除它。 + +一个已发表的显式反例 [KdV, eq.(2)]：chiral权4密度J4=(TT)的零模为 + + I3=2sum_(n≥1)L_-n L_n + L0²−(c+2)L0/12+c(5c+22)/2880。 + +它与空间动量对易。在c=0、h=hbar=5/96的最高权张量态上（动量0），单手征I3的特征值恰为 + + h²−h/6=−55/9216 !=0。 (16) + +原文eq.(4)也直接给非零的torus一点函数微分算子。这里仅引用这个counterexample的有限Virasoro意义；J4是identity-family密度，不被替换成实际的dual-odd thermal descendant，也不证明实际site模型存在任何指定差矩阵元。 + +结论：不能凭错误的“动量筛选”淘汰thermal spin4，也不能反过来宣布其胜出。需要真正的null/module、微观耦合和source/sector矩阵元。零范态是否被商掉、对数伙伴是否被当前读出看到，同样需要实际表示而非名称。 + +参考 [KdV] Maloney–Ng–Ross–Tsiares, *Thermal Correlation Functions of KdV Charges in 2D CFT*, arXiv:1810.11053v2，实际读取§1 eq.(2),(4)。https://arxiv.org/html/1810.11053v2 + +## 10. 本次新增的有限检查 + +标准库脚本另在真实3×3 NN torus的512个配置上检查(10)，三组p下的完整rank条件评分，九组有理fugacity下全部4608条正规化概率相等式，及(12)。这是独立小实现，不重跑w4..8转移。外部表的修复只用其已打印输入，误差资格与原数值相同；c=0的(16)用Fraction精确计算。所有这些控制的目的，是避免在一个已冗余的源或错误normalizer上追加七机计算。 diff --git a/docs/manuscripts/geometric-balance/two-stage-angular-radial-improvement-20260914.md b/docs/manuscripts/geometric-balance/two-stage-angular-radial-improvement-20260914.md new file mode 100644 index 000000000..5e1e7ce1e --- /dev/null +++ b/docs/manuscripts/geometric-balance/two-stage-angular-radial-improvement-20260914.md @@ -0,0 +1,189 @@ +# Two-stage angular/radial improvement of the matching charge root + +Date: 2026-09-14 + +Status: deterministic finite-transfer construction plus a mechanism-conditioned extrapolation. The angular projector is algebraic. The second-stage `ell^-7` Richardson step uses the historically pre-stated `V_<1,4>` scalar mechanism and is therefore a sharp internal consistency test, not a rigorous threshold enclosure. + +## 1. Stage one: remove the observed H4 irrep exactly at fixed ell + +For two orientations with the same physical circumference `ell`, define + +```text +h_i=H4(theta_i)=cos(4theta_i), +p_i=p_ch(ell,theta_i). +``` + +The H4-annihilating scalar combination is + +```text +p_H0(ell) + = (h1 p2-h2 p1)/(h1-h2). +``` + +It removes every finite-size contribution proportional to `H4(theta)` at that `ell`, without using `pc` or an amplitude. + +Use the same angular pair at two scales: + +```text +axis: H4=1, +(3,4): H4=-527/625. +``` + +Deterministic safe roots give + +```text +ell=5: +p_H0(5)=0.5927362453692130, + +ell=10: +p_H0(10)=0.5927459750664773. +``` + +## 2. Stage two: use the historically named scalar q=3 mechanism + +The historical post-H4 operator analysis, written before these deterministic values were available, identified the scalar + +```text +V_<1,4>, +x=33/4, +spin=0, +``` + +as the first named linear scalar candidate after the H4 leading root correction. Since the thermal field has `x_t=5/4`, its pseudo-critical root exponent is + +```text +x-x_t = 7. +``` + +Thus the pre-stated mechanism predicts + +```text +p_H0(ell) + = pc + C7 ell^-7 + higher. +``` + +For `ell=5` and `10=2*5`, eliminate the `ell^-7` term exactly: + +```text +pc_hat^(H4,V14) + = [2^7 p_H0(10)-p_H0(5)]/(2^7-1) + = 0.5927460516782668. +``` + +The inferred scalar amplitude is + +```text +C7_hat + = [p_H0(5)-pc_hat] 5^7 + = -0.7661178948268482. +``` + +Neither number uses an external threshold. + +## 3. After-the-fact comparison only + +The independent high-precision diagnostic reference used elsewhere in the repository is + +```text +pc_ref=0.59274605079210. +``` + +Only after constructing the reference-free estimator, + +```text +pc_hat^(H4,V14)-pc_ref + = +8.86e-10. +``` + +This level of agreement is a strong mechanism-level positive control. It is not a certified error bar: with only two radial scales, higher H0 corrections and angular leakage can cancel. + +## 4. An unrelated same-circle cross-check + +The independent N85 angular pair + +```text +(2,9), +(6,7) +``` + +has the same physical circumference `sqrt(85)` but a different H4/H8 geometry. Its H4-projected scalar estimate is + +```text +pc_hat_N85=0.5927460512227809. +``` + +It differs from the two-stage estimator above by + +```text +-4.55e-10. +``` + +This should not be treated as a second sub-nanounit threshold determination. The N85 geometry can have accidental cancellation among H8/scalar residuals. It is useful as an independent arithmetic consistency check that did not use the axis/(3,4) lineage or the external reference. + +## 5. Why this is more informative than a raw-width fit + +The construction separates two mechanisms before extrapolating: + +```text +raw pseudo-critical root + -> exact angular H4 projection + -> named scalar ell^-7 elimination. +``` + +A fit of raw axial roots would mix + +```text +linear H4 tower, +scalar odd sector, +H8/higher harmonics, +nonlinear field products, +thermal-coordinate corrections. +``` + +The two-stage estimator instead removes the observed dominant irrep algebraically and only then invokes one historically predicted scalar exponent. + +This is a finite-size version of improvement / projection rather than a larger-data regression. + +## 6. The remaining falsification burden + +The striking numerical agreement does not by itself identify `V_<1,4>` as a simple isolated field. Two residual issues remain. + +### Angular content + +A two-angle H4 projector still contains + +```text +H0, +H8, +H12,... +``` + +with pair-dependent geometry coefficients. N1105 same-circle multi-angle tomography (#807) is the correct held-out test of whether the post-H4 residual is genuinely H0-dominated. + +### Module content + +At `Q=1`, the `x=33/4` Kac branch collides with the diagonal Jordan eigenvalue of the generic-Q `W(2,2)` module. Thus an H0 `ell^-7` block can be dominated by the `V_<1,4>` bottom Kac component while still belonging to a larger logarithmic multiplet. + +The new `ell^-7` success therefore supports the **bottom-field scaling law** more directly than it proves a simple-module identity. + +## 7. Decision logic + +### N1105 finds dominant H0 + +Then the two-stage estimator has a genuine same-circle angular justification. The next field-theory question is `V_<1,4>` bottom vs logarithmic `W(2,2)`-related combination, not whether a scalar block exists. + +### N1105 finds locked H0+H8 or large H12 + +Then the excellent `ell^-7` two-scale behavior is partly accidental/mixed. Keep the numerical estimator as an improvement trick, but downgrade the direct V14 interpretation. + +### A third same-angular scale breaks the ell^-7 law + +Then the present two-scale agreement is preasymptotic. Do not preserve the scalar label by adding more fit terms without a new mechanism. + +## 8. Claim boundary + +- The H4 projection is exact angular algebra. +- The two safe-root inputs are deterministic finite-state transfer outputs. +- The exponent seven was a historical mechanism prediction for the same scalar projector, not selected by fitting these two deterministic values. +- The resulting `pc_hat` is not a rigorous threshold enclosure. +- The close N85 agreement is an independent consistency check but can contain accidental higher-harmonic cancellation. diff --git a/docs/manuscripts/geometric-balance/v14-w22-log-collision-20260914.md b/docs/manuscripts/geometric-balance/v14-w22-log-collision-20260914.md new file mode 100644 index 000000000..673e1bf6e --- /dev/null +++ b/docs/manuscripts/geometric-balance/v14-w22-log-collision-20260914.md @@ -0,0 +1,309 @@ +# The post-H4 scalar at x=33/4 sits on a generic-Q logarithmic collision + +Date: 2026-09-14 + +Status: exact Kac-dimension collision + literature-backed representation-theory warning + research synthesis. The collision does **not** by itself identify the measured H4-projected root residual with a specific logarithmic partner, and it does not prove the matching parity of the `V_<1,4>` branch. + +## 1. Why this matters now + +The deterministic same-angle H4-null safe-root projector gives, for the same axis/(3,4) angular pair, + +```text +ell=5: (p_H0-pc_ref) ell^7 = -0.7660485, +ell=10: (p_H0-pc_ref) ell^7 = -0.7572352. +``` + +The historical post-H4 mechanism note had already identified the standard Potts scalar + +```text +V_<1,4>: +h=hbar=33/8, +x=33/4, +spin=0, +``` + +as the natural `q=3` channel producing a root correction `ell^-7`, conditional on matching-odd parity and a nonzero lattice coupling. + +The new deterministic result therefore makes the operator identity question concrete. However, at `Q=1` the conformal weight `33/8` is not isolated. + +## 2. Critical-Potts Kac parametrization + +Use the standard generic critical-Potts Coulomb-gas parameter + +```text +Q = 4 cos^2(pi beta^2), +``` + +with percolation at + +```text +beta^2 = 2/3. +``` + +A convenient Kac-weight convention is + +```text +h_{r,s}(beta) + = (c-1)/24 + + (1/4) (r beta - s/beta)^2. +``` + +The repository's critical-branch notation `V_<1,s>` is the branch-swapped convention relative to papers that call the energy Kac field `Phi_{s,1}`. Numerical weights are convention-independent. + +For the candidate scalar, + +```text +h_{1,4} + = (c-1)/24 + + (1/4)(beta-4/beta)^2. +``` + +Now consider the generic-Q non-diagonal module `W(2,2)`. The known Potts logarithmic representation has top/bottom diagonal fields with weight + +```text +h_{2,-2} + = (c-1)/24 + + (1/4)(2 beta+2/beta)^2. +``` + +At `beta^2=2/3`, both equal + +```text +boxed: +h_{1,4}=h_{2,-2}=33/8. +``` + +Thus the scalar `x=33/4` Kac branch hits the diagonal eigenvalue of the `W(2,2)` logarithmic diamond precisely at percolation. + +## 3. The collision is isolated at the physical Potts point + +Put + +```text +t = beta^2. +``` + +The common `(c-1)/24` piece cancels, and the exact chiral weight difference is + +```text +Delta h(t) + := h_{1,4}-h_{2,-2} + = (1/4)(-3 t -16 +12/t). +``` + +The zero condition is + +```text +3 t^2 +16 t -12 =0, +``` + +with roots + +```text +t=2/3, +t=-6. +``` + +Hence the only physical positive collision is the percolation point `t=2/3`. + +The scaling-dimension splitting is + +```text +Delta x(t)=2 Delta h(t). +``` + +At percolation, + +```text +d_t Delta x |_(2/3) = -15. +``` + +Meanwhile + +```text +dQ/dt + = -4 pi sin(2 pi t), +``` + +so at `t=2/3`, + +```text +boxed: +d_Q Delta x |_(Q=1) + = -15/(2 pi sqrt(3)). +``` + +Numerically this is about `-1.3783` per unit `Q`. + +This is a sharp generic-Q splitting velocity for any future branch-resolved test. + +## 4. Generic-Q representation input + +Grans-Samuelsson--Liu--He--Jacobsen--Saleur, JHEP 10 (2020) 109, establish for generic Potts `Q` that non-diagonal modules with weights + +```text +(h_{r,s},h_{r,-s}), +(h_{r,-s},h_{r,s}) +``` + +belong to indecomposable diamond representations whose top and bottom fields have + +```text +(h_{r,-s},h_{r,-s}) +``` + +and form a rank-two Jordan cell of `L0` and `Lbar0`. + +Therefore `W(2,2)` already carries a logarithmic pair at the diagonal weight `h_{2,-2}` for generic Q. The exact equality in Section 2 says that the simple scalar Kac branch reaches that same diagonal eigenvalue at `Q=1`. + +The detailed special-Q indecomposable extension is **not** fixed by this dimension equality alone. It may enlarge/reorganize the generic-Q diamond, or the measured lattice observable may project mostly onto one bottom component. + +## 5. Relation to modern c=0 energy-operator results + +In the convention used by recent percolation LCFT work, the repository field `V_<1,4>` is the branch-swapped version of the **third energy Kac operator** often denoted `Phi_{4,1}`. + +Recent work on logarithmic Kac operators at `c=0` argues that higher energy operators can become zero-norm bottom fields of higher-rank logarithmic multiplets. In particular, the 2025 analysis by Yifei He discusses a potential rank-four Jordan structure whose bottom field is the third energy operator `Phi_{4,1} ~ epsilon''`, while emphasizing that higher-rank structures require observables beyond the simplest spin correlator to expose them. + +This is strikingly aligned with the present situation: + +```text +raw leading matching-odd observable: + dominated by the ordinary thermal-Q4 tangent; + +post-H4 scalar projector: + first place where the x=33/4 energy-type bottom field becomes visible; + +logarithmic partners: + may appear only in a further normal / source-resolved residual. +``` + +The existence of a high-rank multiplet is therefore a plausible module-level explanation, not a reason to replace the observed pure-power bottom-field signal by an arbitrary log fit. + +## 6. Why the clean ell^-7 signal does not rule out logarithmic structure + +A logarithmic multiplet does not force every microscopic observable to show a large `log ell` coefficient. A lattice source can project predominantly onto the bottom Kac field, with partner coefficients small or removed by a quotient/normalization. + +This already happened one level earlier in the current research programme: + +- the leading spin-four `x=21/4` channel has a clear ordinary thermal-Q4 tangent contribution; +- generic-Q/percolation representation theory still allows logarithmic energy--hull structure; +- historical pure-power H4 data show that any large log admixture is not required in that observable. + +The post-H4 scalar may follow the same pattern recursively: + +```text +bottom V_<1,4> pure-power piece dominates the H0 root residual, +while logarithmic partners survive only in a smaller normal/source-resolved component. +``` + +This is now a falsifiable working hypothesis. + +## 7. Matching parity is empirically separated from continuum naming + +The H4-null quantity is built from the primal/matching **charge-root difference**, so any nonzero H0 residual measured there is already an empirical matching-odd scalar block of the lattice observable. + +Thus one need not prove the interchiral parity of `V_<1,4>` before establishing that the **measured block** is odd. + +The remaining naming problem is: + +> Does the empirical odd, spin-zero, `x=33/4` block correspond primarily to the `V_<1,4>` bottom Kac branch, to a special-Q logarithmic combination involving the `W(2,2)` diagonal Jordan pair, or to another degenerate scalar sector? + +Generic-Q branch splitting is the cleanest way to answer this. + +## 8. A targeted generic-Q prediction + +Let + +```text +A_Kac(Q), +A_W(Q) +``` + +be the source amplitudes of the scalar Kac branch and the `W(2,2)` diagonal logarithmic branch in a declared generic-Q lift of the same observable. + +Their dimensions split linearly as + +```text +x_Kac(Q)-x_W(Q) + = -15/(2 pi sqrt(3)) (Q-1) + + O((Q-1)^2). +``` + +Therefore: + +### regular-bottom scenario + +If `A_Kac(Q)` is regular and dominates, + +```text +post-H4 scalar ~ ell^-7 +``` + +with no compulsory leading logarithm. + +### collision/log scenario + +If the physical `Q=1` observable requires pole-cancelled mixing of the two branches, schematically + +```text +A_Kac(Q) ~ +a/(Q-1), +A_W(Q) ~ -a/(Q-1), +``` + +then the collision produces an `ell^-7 log ell` contribution whose coefficient is proportional to + +```text +a * 15/(2 pi sqrt(3)). +``` + +The exact splitting velocity above fixes the conversion between a generic-Q pole residue and the logarithmic size coefficient. + +This is much more informative than a free `A+B log ell` fit. + +## 9. N1105 and generic-Q now have complementary jobs + +### N1105 same-circle safe-root tomography + +At fixed `Q=1` it should determine whether the post-H4 residual is primarily + +```text +H0 scalar, +locked H0+H8 nonlinear mixing, +or higher D4 harmonic. +``` + +If H0 scalar wins, the x=33/4 branch becomes the leading candidate. + +### generic-Q branch test + +Only after H0 is established does it become worth spending resources to split + +```text +V_<1,4> +vs +W(2,2) diagonal/log branch. +``` + +This ordering avoids using generic-Q machinery to solve an operator identity before the lattice has even established the angular scalar channel. + +## 10. Current working hierarchy + +For the post-H4 scalar residual: + +```text +1. empirical odd H0 block with root exponent near 7: strong deterministic hint; +2. V_<1,4> / third-energy Kac bottom field: leading continuum candidate; +3. W(2,2)-related logarithmic collision at the same x=33/4: serious module-level ambiguity; +4. T4 x I4 H0+H8 ell^-6: exact allowed competitor, numerically small in the current two-scale same-projector decomposition; +5. higher H8/H12/scalar channels. +``` + +The important conceptual change is that `V_<1,4>` and logarithmic structure are **not mutually exclusive**. The Kac field can be the bottom component of the very logarithmic structure that makes the full `Q=1` operator algebra non-semisimple. + +## 11. Claim boundary + +- The equality `h_{1,4}=h_{2,-2}=33/8` at `beta^2=2/3`, the isolation of the positive collision, and the splitting velocity are exact algebra. +- The generic-Q `W(2,2)` diamond/Jordan structure is a literature result. +- The precise special-Q extension when the scalar Kac branch collides with that diagonal Jordan eigenvalue is not derived here. +- The potential higher-rank third-energy logarithmic structure is a literature-motivated c=0 hypothesis/programme, not a proven identification of the Matching-One residual. +- The deterministic ell^-7 evidence uses only two same-projector scales and remains a strong diagnostic rather than an asymptotic theorem. diff --git a/docs/manuscripts/geometric-balance/varying-direction-exponential-centres-20260914.md b/docs/manuscripts/geometric-balance/varying-direction-exponential-centres-20260914.md new file mode 100644 index 000000000..fe638cd91 --- /dev/null +++ b/docs/manuscripts/geometric-balance/varying-direction-exponential-centres-20260914.md @@ -0,0 +1,481 @@ +# Varying shortest-period directions: full exponential-aspect centre theorem + +2026-09-14. This note completes the geometric direction step posed in #765 for the genuine exponential-aspect regime. The short-period direction may vary with the torus and converge only along a subsequence. + +The proof combines: + +- the full-period first-exit upper bound in `first-exit-torus-winding-upper-20260914.md`; +- a fixed local directional seed that approximates the limiting direction, repeated many times and closed by a small deterministic correction; +- the determinant geometry of a rank-two integer period lattice. + +No microscopic rotation, directional OZ prefactor, or angle scan is used. + +## 1. Geometry and statement + +Let `Lambda_n` be honest rank-two sublattices of `Z^2`. Let + +\[ +N_n=[\mathbb Z^2:\Lambda_n] \tag{1.1} +\] + +and choose a shortest nonzero period + +\[ +u_n\in\Lambda_n, +\qquad +\ell_n=|u_n|\to\infty. \tag{1.2} +\] + +A shortest period is primitive in `Lambda_n`, although it need not be primitive in ambient `Z^2`. + +Assume along the sequence or a chosen subsequence + +\[ +e_n:=u_n/\ell_n\to e\in S^1, \tag{1.3} +\] + +and define the transverse height + +\[ +h_n=N_n/\ell_n. \tag{1.4} +\] + +The exponential-aspect hypothesis is + +\[ +\boxed{ +\frac{\log h_n}{\ell_n}\to d\in(0,\infty).} \tag{1.5} +\] + +For `G=G4` or `G8`, let + +\[ +f_{G,n}(p)=P_p^G(r_G>0). \tag{1.6} +\] + +Let `tau_{G,p}` be the subcritical planar inverse-correlation norm. + +**Theorem.** For every fixed `p tau_{G,p}(e)` is continuous and strictly decreasing from `infinity` to zero on `(0,pc(G))`, and the two NN homology births satisfy + +\[ +T_{1,n}\xrightarrow P a_e(d), +\qquad +\tau_{4,a_e(d)}(e)=d, \tag{1.8} +\] + +\[ +T_{2,n}\xrightarrow P b_e(d), +\qquad +b_e(d)=1-c_e(d), +\qquad +\tau_{8,c_e(d)}(e)=d. \tag{1.9} +\] + +If only a subsequence of directions converges to `e`, the conclusions hold on that subsequence. No whole-sequence direction is invented. + +## 2. Exponential transverse height automatically removes nonparallel period competition + +Complete `u_n` to a lattice basis `(u_n,v_n)` with + +\[ +|\det(u_n,v_n)|=N_n. \tag{2.1} +\] + +Every period is + +\[ +\lambda=a u_n+b v_n, +\qquad a,b\in\mathbb Z. \tag{2.2} +\] + +If `b!=0`, then + +\[ +|\det(u_n,\lambda)|=|b|N_n. \tag{2.3} +\] + +The perpendicular component of `lambda` relative to `u_n` therefore has magnitude + +\[ +\frac{|\det(u_n,\lambda)|}{|u_n|} +=|b|h_n. \tag{2.4} +\] + +Hence + +\[ +\boxed{|\lambda|\ge |b|h_n\qquad(b\ne0).} \tag{2.5} +\] + +At any fixed subcritical parameter, norm equivalence gives + +\[ +\tau_{G,p}(\lambda)\ge c_{G,p}|b|h_n. \tag{2.6} +\] + +Under (1.5), `h_n=exp(d ell_n+o(ell_n))`. Thus every nonparallel homology class has super-`ell_n` cost and is negligible compared with the `O(ell_n)` short-period class. + +This also proves a purely geometric uniqueness statement. If two independent periods both had Euclidean length `O(ell_n)`, their determinant would be `O(ell_n^2)`. Since the determinant of two independent lattice periods is a nonzero integer multiple of `N_n`, this would force + +\[ +N_n=O(\ell_n^2), +\qquad +h_n=O(\ell_n), \tag{2.7} +\] + +contradicting (1.5). Therefore exponential elongation itself forbids two nonparallel `O(ell_n)` period classes. + +The multi-direction hard-core crossover from `projective-poisson-hardcore-crossover-20260914.md` is relevant to geometries where several direction costs genuinely remain comparable, but **not** to the present shortest-period exponential-aspect regime. + +## 3. Upper bound from the full-period theta estimate + +Fix `p0`. By directional continuity of the norm, choose one nonzero integer vector `w` such that + +\[ +\left|\frac{w}{|w|}-e\right|<\eta \tag{4.1} +\] + +and + +\[ +\frac{\tau_{G,p}(w)}{|w|} +<\tau_{G,p}(e)+\eta. \tag{4.2} +\] + +By the definition of the mass along the fixed integer direction `w`, choose a fixed multiple `z=Lw` and a fixed finite box seed from `0` to `z` whose conditional connection probability `q` obeys + +\[ +-\log q +<|z|[\tau_{G,p}(e)+2\eta]. \tag{4.3} +\] + +All of `w,L,z`, and the seed box are fixed before `n->infinity`. + +For large `n`, `e_n` is also within `eta` of `e`. Choose + +\[ +k_n=\operatorname{round}\left(\frac{u_n\cdot z}{|z|^2}\right).\tag{4.4} +\] + +Then the remainder + +\[ +r_n=u_n-k_nz \tag{4.5} +\] + +satisfies + +\[ +|r_n|\le C\eta\ell_n+O(|z|). \tag{4.6} +\] + +Repeat the fixed seed `k_n` times and join the last seed endpoint to `u_n` by a deterministic NN path of length at most + +\[ +|r_n|_1\le\sqrt2|r_n|. \tag{4.7} +\] + +NN edges are available in both `G4` and `G8`. + +Harris positive association applies even if translated seed boxes overlap after projection. Therefore the prescribed closed ring has probability + +\[ +P_p(\text{one }u_n\text{-ring}) +\ge +q^{k_n}p^{|r_n|_1+O(1)}. \tag{4.8} +\] + +Equations (4.3)--(4.7) imply + +\[ +\boxed{ +P_p(\text{one }u_n\text{-ring}) +\ge +\exp[-(\tau_{G,p}(e)+C_p\eta)\ell_n]} \tag{4.9} +\] + +for all large `n`. + +The complete prescribed support lies in a Euclidean region of diameter `O(ell_n)`, so its difference set contains only + +\[ +|S_n-S_n|=O(\ell_n^2) \tag{4.10} +\] + +quotient vertices. The finite-group packing lemma therefore supplies at least + +\[ +\frac{N_n}{C\ell_n^2} +=\frac{h_n}{C\ell_n} \tag{4.11} +\] + +pairwise site-disjoint translated attempts. + +Hence + +\[ +1-f_{G,n}(p) +\le +\exp\left[ +-\frac{h_n}{C\ell_n} + e^{-(\tau_{G,p}(e)+C_p\eta)\ell_n} +\right]. \tag{4.12} +\] + +If `tau(e)d`, use `1-e^{-x}>=x/2` for small `x` to obtain + +\[ +\liminf\frac1{\ell_n}\log f_{G,n}(p) +\ge d-\tau_{G,p}(e)-C_p\eta. \tag{4.13} +\] + +Let `eta downarrow 0`. Together with (3.7), this proves (1.7). + +## 5. Strict p-monotonicity for every Euclidean direction + +The rate theorem now supplies the missing input for a direction-free Friedgut--Kalai argument. + +Fix any `e in S^1`. Choose primitive ambient integer approximants `u_n` with + +\[ +|u_n|\to\infty, +\qquad +u_n/|u_n|\to e. \tag{5.1} +\] + +Choose Bezout complements `v_n` and a transverse multiplier `m_n` so that the torus + +\[ +\Lambda_n=\langle u_n,m_nv_n\rangle \tag{5.2} +\] + +has + +\[ +\frac{\log(N_n/|u_n|)}{|u_n|}\to d. \tag{5.3} +\] + +Suppose for contradiction that for some + +\[ +00. \tag{5.4} +\] + +Choose `d0`, that the Friedgut--Kalai sharp-threshold increment for a transitive event is strictly less than `q-p`: + +\[ +\rho\frac{A-d+\zeta}{d}e^{-(A-d+\zeta)\ell_n} \tag{5.6} +\] + +for large `n`. The event `r_G>0` is increasing and translation-transitive on the `N_n` sites, so Friedgut--Kalai forces + +\[ +f_{G,n}(q)\to1. \tag{5.7} +\] + +But the same rate theorem at `q`, with `A>d`, gives + +\[ +f_{G,n}(q)\to0, \tag{5.8} +\] + +a contradiction. + +Therefore + +\[ +\boxed{ +\tau_{G,p}(e)\text{ is strictly decreasing in }p +\text{ for every }e\in S^1.} \tag{5.9} +\] + +The continuity in `p` follows from the same site finite-seed/cluster-volume comparison used axially, with the standard uniform directional norm estimate supplying the passage from rational approximants to `e`. Small-`p` path counting gives divergence, and + +\[ +\tau_{G,p}(e)\le\kappa_G(p)(|e_x|+|e_y|) \tag{5.10} +\] + +forces the limit zero as `p up to pc(G)`. + +Thus every directional mass has a unique inverse value for each `d>0`. + +## 6. The two birth centres + +For NN, + +\[ +P(T_{1,n}\le p)=f_{4,n}(p). \tag{6.1} +\] + +The rate theorem and strict invertibility give + +\[ +\boxed{ +T_{1,n}\xrightarrow P a_e(d), +\qquad +\tau_{4,a_e(d)}(e)=d.} \tag{6.2} +\] + +Digital Alexander duality gives + +\[ +P_p^{G4}(r=2)=P_{1-p}^{G8}(r=0), \tag{6.3} +\] + +so + +\[ +\boxed{ +T_{2,n}\xrightarrow P b_e(d), +\qquad +b_e(d)=1-c_e(d), +\qquad +\tau_{8,c_e(d)}(e)=d.} \tag{6.4} +\] + +This proves the centre formula anticipated in #765 for arbitrary convergent shortest-period directions. + +## 7. Strict matching directional mass gap on the whole circle + +The matching-enhancement pivotal conversion is endpoint-direction independent. On every compact `I subset (0,pc(G8))` it gives a positive `delta_I` such that + +\[ +P_p^{G8}(0\leftrightarrow x) +\ge +P_{p+\delta_I}^{G4}(0\leftrightarrow x) \tag{7.1} +\] + +uniformly over distant endpoint directions and `p in I`. + +Taking directional rates, + +\[ +\tau_{8,p}(e) +\le\tau_{4,p+\delta_I}(e). \tag{7.2} +\] + +By the all-direction strict monotonicity (5.9), + +\[ +\tau_{4,p+\delta_I}(e)<\tau_{4,p}(e). \tag{7.3} +\] + +Therefore + +\[ +\boxed{ +\tau_{8,p}(e)<\tau_{4,p}(e) +\quad\text{for every }e\in S^1, +\ 00}\frac{\tau_p(x)}{e\cdot x}. \tag{4.8} +\] + +In general + +\[ +\boxed{\rho_{K_p}(e)\ne\tau_p(e).} \tag{4.9} +\] + +Equality requires the boundary normal at the radial point to line up with `e`, which holds on symmetry axes but not for a generic anisotropic direction. A one-dimensional first-exit ray search must therefore not be labelled a direct support-function measurement unless this geometry is accounted for. + +## 5. Large finite sets exhaust compact subsets of the true domain + +Take any compact + +\[ +T\Subset\mathcal D_p=K_p^\circ. \tag{5.1} +\] + +By compactness there is a margin `eta>0` such that + +\[ +\tau_p(e)-t\cdot e\ge3\eta +\qquad(t\in T,e\in S^1). \tag{5.2} +\] + +Let `S_R` be a large Euclidean or norm ball of radius `R`, enlarged by the finite interaction range. For `v in partial_ext S_R`, the first-exit event is contained in a connection from the origin to a point at distance `R+O(1)`. Uniform directional convergence to the subcritical norm gives, for all large `R`, + +\[ +b_{S_R}(v;p) +\le C e^{-\tau_p(v)+\eta|v|}. \tag{5.3} +\] + +The number of boundary sites grows only polynomially in `R`. Combining (5.2)--(5.3), uniformly for `t in T`, + +\[ +B_{S_R}(t;p) +\le \operatorname{poly}(R)e^{-\eta R}\longrightarrow0. \tag{5.4} +\] + +Therefore, for all sufficiently large `R`, + +\[ +\boxed{T\subset\mathcal C_{S_R}.} \tag{5.5} +\] + +So the finite first-exit method is not only sound but **complete on compact interior subsets**: appropriately growing finite boxes eventually certify every point strictly inside the true exponential-moment/Wulff domain. + +This theorem does not give a cheap state complexity for doing so; it answers the mathematical convergence question. + +## 6. Numerical/certification workflow + +A rigorous implementation can therefore proceed as follows. + +1. For a finite `S`, compute outward-rounded upper bounds on each distinct-site coefficient `b_S(v;p)`. +2. The certified set is + \[ + \sum_v \overline b_S(v;p)e^{t\cdot v}<1. + \] +3. Combine several shapes/orientations by the convex hull of their certified regions. +4. For a direction `e`, obtain a **lower bound on `tau(e)`** from the support function of the certified convex body: + \[ + \tau_p(e)\ge h_{C_{cert}}(e). \tag{6.1} + \] +5. Keep this distinct from the radial intercept of the body. +6. Pair it with an independently proved finite-cylinder/connection upper bound on `tau(e)` when an interval is needed. + +Because the support function of a polytope/convex certified body is a deterministic convex program, no angular finite difference is needed to obtain directional lower bounds. + +## 7. Interfaces to the other 2026-09 notes + +- `directional-enhancement-sandwich-20260914.md` gives the independent graph inclusion + \[ + K_{8,p}\subseteq K_{4,p+\delta_I}, + \] + which every numerical first-exit body should respect. +- `matrix-sewing-unit-residue-20260914.md` gives the conditional curvature/diffusion relation + \[ + D^{-1}=\partial_{yy}\tau(1,0), + \] + once a common Markov-additive sewing kernel is established. +- `dilute-directional-geodesic-entropy-20260914.md` supplies an elementary small-`p` asymptotic for the support function in every fixed rational direction, giving another strong control on finite-box certificates in the dilute regime. + +## 8. Claim boundary + +The first-exit/BK theorem and convexity are self-contained product-measure arguments. Equality of the domain closure with the polar body and the exhaustion statement use the standard uniform directional exponential-rate estimate for the subcritical norm; when citing them in the final manuscript, the precise SITE reference/hypotheses should be stated. No numerical Wulff body is manufactured in this note. diff --git a/docs/manuscripts/geometric-balance/verify-768-issues-20260914.md b/docs/manuscripts/geometric-balance/verify-768-issues-20260914.md new file mode 100644 index 000000000..fbcf6129a --- /dev/null +++ b/docs/manuscripts/geometric-balance/verify-768-issues-20260914.md @@ -0,0 +1,107 @@ +# `issues-found.md` —— verify768 与 `theory768` 的冲突（含最小复现） + +**核验者**：`verify768`　**日期**：2026-09-14　**被核对象**：`theory768-out/note-first-noncommon-correction.md` +及其原始输出 `theory768-out/g2_result.json` 等；脚本 `theory768-scripts/g2_virasoro.py`。 + +--- + +## IF-1（高）：`theory768` 的 JSON flag `rank_equals_p_shift2: false` 是**脚本 bug**，与其表/结论矛盾 + +**现象**：`theory768-out/g2_result.json` 里 `runs.c0_h58.rank_equals_p_shift2 = false`， +但 note §1.1 的表和 §0 第 1 条都断言 `rank G_n = p(n)−p(n−2)` 成立。二者矛盾。 + +**最小复现**（`theory768-scripts/g2_virasoro.py`，`analyse()` 内）： +```python +ok_char = True +for n in range(NMAX + 1): + pnm2 = len(B[n - 2]) if n >= 2 else 0 # <-- 这是 p(n-2)，不是 p(n)-p(n-2) + ... + if rank[n] != pnm2: # <-- 拿 rank 去和 p(n-2) 比 + ok_char = False +res["rank_equals_p_shift2"] = bool(ok_char) +``` +`rank G_n = 1,1,1,2,3,4,6`；`p(n−2) = 0,0,1,1,2,3,5` ⇒ n=0,1,3,4,5,6 全部判 DIFF ⇒ flag=false。 +**正确比较**应为 `rank[n] != len(B[n]) - (len(B[n-2]) if n>=2 else 0)`。 + +**我的独立重算**（`v768_v1_result.json`，我的机制，不用它的代码）： +``` +rank_eq_p_minus_p_shift2_all = true # 真确公式，全 n<=6 成立 +theory768_style_flag_p_shift2_only = false # 复现它的 buggy 公式 +``` +⇒ **note 的表是对的，JSON 的 flag 是错的**。危害：只读 JSON 的下游（或另一个代理）会误判这条声称失败。 +建议 `theory768` 修脚本并重出 `g2_result.json`（**我没有改它的任何文件**）。 + +--- + +## IF-2（高）：note §0 第 4 条的附加结论与它**自己 §1.2** 直接矛盾 + +note §0#4 写：「只有**保留零范态**时，热模才在 `(a,b)=(2,2)` 多出一个 **spin 0**、`x=21/4` 的非导数类， +**才能进入动量 0 观测量**。」 + +**算术部分我独立复算成立**：`(ii)` 下 `q=[1,0,1,1,2,2,4]`，`(2,2)` 的 `dim=q₂²=1`、`spin=0`、`x=21/4`、动量 0、C4 允许。 + +**但「才能进入」这一步不成立**： +- **最小复现（纯代数，无计算）**：`V_2 / L_{−1}V_1` 是 1 维，只能由 `[L_{−2}|h⟩]` 张成； + 而 `χ = −3L_{−2} + 2L_{−1}²` ⇒ `[χ] = −3[L_{−2}] ≠ 0`（我数值验证 `‖χ‖² = 0` 精确、`L_1χ=L_2χ=0`）。 + ⇒ **level-2 那个「多出来的非导数类」就是零范态 `χ` 的方向本身**。 + 零范场被所有正模湮灭 ⇒ **关联函数恒为 0** ⇒ `(2,2)` 类在 `(ii)` 下**贡献 0**。 +- 这与 note **自己 §1.2** 的裁定「`(i)` 与 `(ii)` 的关联函数完全相同」**一致** ⇒ **note 自我矛盾**。 + +**后果**：note §3.4 D3「场景 (ii)/(iii)：热模多出 `(2,2)` 的 spin-0 类 ⇒ `x=21/4` 处有 **2 个** 独立的动量 0 幅度」 +—— **对 (ii) 是错的**（(ii) 下仍是 **1** 个）。 + +--- + +## IF-3（中）：note 把 `(ii)`（保留零范态）与 `(iii)`（log 模块）混用 + +§0#4 的括号「只有保留零范态时……才能进入」把两种情形合并： +- `(ii)` = 真的只把零范态**留在态空间**里 ⇒ 关联为 0（见 IF-2）； +- `(iii)` = 零范态成为 **log Jordan cell 的 bottom** ⇒ **可能**有非零关联，但 note 自己在 §1.2 说 + 「(iii) **不能**（代数不定，是物理模的性质）」。 +用 `(ii)` 的类计数支持 `(iii)` 的效果，**没有把两种情形分开论证**。 + +--- + +## IF-4（中）：note 的「一点函数 `∝δ_{h,h̄}`」措辞不精确 + +- note §2.4 原文：「一个权重 `(h,h̄)` 的原位算符的**一点函数** `⟨φ⟩ ∝ δ_{h,h̄}`……这是教科书级别的 CFT 事实」。 +- **但**：在**无限周期圆柱**上，主场的**一点函数恒为 0**（与 `h` 是否等于 `h̄` 无关）：圆柱经 + `z = (L/2π) log w` 共形等价于平面，而 `⟨φ⟩_平面 = 0`（平移 + 标度）。 +- `δ_{h,h̄}` 真正成立的是 **(a) torus 上的 modular 迹** `⟨φ⟩_torus = Tr(q^{Δ}q̄^{Δ̄}φ)/Z`， + 或 **(b) 动量 0 态之间的矩阵元** `⟨mom0|φ|mom0⟩`。note 实际用的是 (b) —— **(b) 无条件精确**， + 所以**结论不受影响**；但措辞会让读者误以为在说圆柱一点函数。 +- 另外 note 在 §2.4 把「一点函数」「矩阵元」「per-row 有限尺寸修正」混着说， + 而 `Θ_w` 是 **per-row** 量（原文 §1.2「difference of the two magnetic/topological excitation energies」+ §2「per-row」）。 + +--- + +## IF-5（中）：note 用 `(ii)` 的类计数去支持 `(iii)` 的结论，且**该排除的条件性没被标出** + +正确的裁定（我的 §2.4）：**这条 spin±4 排除在 (i) 与 (ii) 下都成立**（(i) 类被商掉；(ii) 类零范 ⇒ 关联恒 0）， +**只有在 (iii) 且该类为真算子时才失效**。而 note 自己主张物理模是 (iii)。 +⇒ **该排除是条件性的，note 把它当成无条件的「可行排除」**，且没有标出这个条件。 + +--- + +## IF-6（低，自曝）：我自己的第一版实现也错过一次（`0,0,1,2`），靠四路互检才发现 + +- 我的第一版 `modLm1_irred` 里，补空间选基的终止条件写成 `if cur == rank[n-1]: break`， + 应为 `if cur == len(Bp): break`（要张满 `V_{n−1}`，而不是只张到 `rank G_{n−1}`）。 + 这导致补空间少了 `dim(rad_{n−1})` 个向量，`e₄` 算成 1（应为 2），`0,0,1,1` 变成 `0,0,1,2`。 +- **发现方式**：我另外用**四条独立路线**（A complement+mod rad / B 全部 `V_{n−1}`+mod rad / + C 显式商坐标 / D 伴随 `L_1` 的核）重算 `e_n`，四路一致给 `e₄ = 2`，与我的单路冲突 ⇒ 定位到 bug。 +- **记录理由**：这正说明**单路「独立实现」也可能一致地错**（上一轮 `mono780` 的教训同源）。 + 修正后我的 V1 与 `theory768` **逐格一致**（含 `0,0,1,1`），见 `note-verify-768.md` §1。 + +--- + +## 未构成冲突、但需登记的边界 + +- 我**没有**改 `theory768` 的任何文件，也**没有写任何仓库**（按规格 §4 与 §3）。 +- `#802` 的 `(T_g,N_g)` 我**没有**重算（`sector802` 已交付）。 +- `Θ_w` 在真实扇区引擎里的数值定义（闭合振幅/长宽比）我**没有**实测 —— §2.2 的「动量 0」结论 + 基于「转移矩阵平移不变 ⇒ 基态动量 0」这一标准结构。 +- `(iii)` 下 `(2,2)` 类是否为真算子：**决策所需信息不在 Virasoro 代数里**（note 自己承认）⇒ 我判「未建立」，不判「错」。 +- 文献（He `2411.18696`、Vasseur–Jacobsen–Saleur `1206.2312`、Mathieu–Ridout `0708.0802`、 + Javerzat et al. `2005.11830`、Jacobsen `1507.03027`、Mertens–Ziff `1603.07289`）： + 全部**引用，未独立重算**。 diff --git a/docs/manuscripts/geometric-balance/verify-768-momentum-exclusion-20260914.md b/docs/manuscripts/geometric-balance/verify-768-momentum-exclusion-20260914.md new file mode 100644 index 000000000..ea64ef145 --- /dev/null +++ b/docs/manuscripts/geometric-balance/verify-768-momentum-exclusion-20260914.md @@ -0,0 +1,238 @@ +# 独立核验：`#768` 的「动量 0 排除 spin±4」这一条载重声称 + +**核验者**：`verify768`　**日期**：2026-09-14　**机器**：云端 `DevEnvC_NePnUn`（账号2）`/workspace/verify768/` +**本机**只做读写/上传下载/汇总；**全部计算在云机执行**（Python 3.9.9 + sympy 1.14.0）。 +**被核验交付**：`theory768-out/note-first-noncommon-correction.md`（全文）+ `theory768-out/g{2,2b,2c,1g3}_result.json`， +issue `#768` / `#802`（只读）。 + +**我的独立路线（与 theory768 不同）**：自写 Virasoro 正规序（`v768_v1_verma.py`，插入式递归 +`L_m L_a = L_a L_m + (m−a)L_{m+a} (+ (C/12)(m³−m)δ_{m+a,0})`，结构上必然终止）， +自写分数精确线性代数（rref/rank/nullspace），**没有复用它的 `canon`/`act`/`proj_quot`**。 + +--- + +## 0. 裁定总表 + +| # | 声称 | 裁定 | +|---|---|---| +| V1-a | `p(n)=1,1,2,3,5,7,11` | **成立**（自写 DP + PBW 计数一致） | +| V1-b | `rank G_n = p(n)−p(n−2)` 全 n≤6 | **成立**（精确有理数；⚠️ theory768 自己的 JSON flag 说 false，是它的 bug，见 §4） | +| V1-c | 奇异向量只在 level 2，维数 1 | **成立** | +| V1-d | `χ=−3L_{−2}+2L_{−1}²`（∝`L_{−2}−⅔L_{−1}²`），**范数恰为 0** | **成立**（两条独立路径：Gram 二次型 & 直接把 χ 展开回 |h⟩ 分量；两者都给精确 0） | +| V1-e | level 1..4 模 `L_{−1}` 维数 = `0,0,1,1` | **成立**（不可约商口径）。⚠️ **但它不是新事实**：它是「`rank G_n=p(n)−p(n−2)` + `L_{−1}` 在不可约商上单射」的**纯算术结果**（§1.3） | +| **V2-a** | 平移不变几何上权重 `(h,h̄)` 之一点函数 `∝δ_{h,h̄}` | **成立（精确、初等）**，但**措辞有瑕**：见 §2.1（无限圆柱上主场一点函数**恒为 0**；真正成立的是**矩阵元**陈述 & torus 迹） | +| **V2-b** | 被排除观测量（每行能量差 `Θ_w`/`Δ_w`）是动量 0 | **成立（就一阶/矩阵元而言）**；⚠️ **缺环**：note 未区分**一阶/二阶**（`#802` 明文要求区分），也未证明 `w^{−17/4}` 项就是一阶（§2.2） | +| **V2-c** | scalar 8-arm 是 `(21/8,21/8)`、spin 0、`x=21/4` | **成立**；但**动量规则只能杀 spin≠0、永远选不出标量** ⇒ 「排除 spin4」≠「唯一挑出 8-arm」（§2.3） | +| **V2-d** | 「只有保留零范态时 `(2,2)` 的 spin 0、`x=21/4` 类才能进入动量 0 观测量」 | **不成立（对 (ii)）/ 未建立（对 (iii)）** —— 与 note 自己 §1.2 直接矛盾（§2.4，**本轮最重要发现**） | +| V3 | 四项「不能宣称」是否被偷偷用上 | 三项守住；**一项有轻度泄漏**（§3） | + +--- + +## 1. V1：Verma 等级计数、`χ` 的范数、模 `L_{−1}` 维数 + +### 1.1 我自己的表（`c=0, h=5/8`，精确有理数） + +| `n` | `p(n)` | `rank G_n` | `nullity` | 奇异向量维数 | `p(n)−p(n−2)` | +|---:|---:|---:|---:|---:|---:| +| 0 | 1 | 1 | 0 | 0 | 1 | +| 1 | 1 | 1 | 0 | 0 | 1 | +| 2 | 2 | **1** | **1** | **1** | 1 | +| 3 | 3 | 2 | 1 | 0 | 2 | +| 4 | 5 | 3 | 2 | 0 | 3 | +| 5 | 7 | 4 | 3 | 0 | 4 | +| 6 | 11 | 6 | 5 | 0 | 6 | + +与 theory768 **逐格一致**。对照点 `(c,h)=(1/2,3/7)`（无退化）我得到 `rank G_n=p(n)` 全层、无奇异向量； +`(c,h)=(0,0)` 真空得到 `rank G_n=0`（n≥1）、奇异向量维数 `{1:1,2:1,5:1}` —— 三个点都与 theory768 一致， +说明我的机制在「有退化 / 无退化 / 真空」三类上都能对。 + +### 1.2 `χ` 的范数 = 0（**两条独立路径**） + +- 路径①（Gram 二次型）：`v=(−3,2)`，`vᵀG₂v = 0` 精确。 + 中间量独立算出：`⟨L_{−2}h|L_{−2}h⟩ = 4h+c/2 = 5/2`、`⟨L_{−1}²h|L_{−1}²h⟩ = 8h²+4h = 45/8`、交叉项 `= 6h = 15/4`。 + 于是 `det G₂ = 4h²(8h−5)`，在 `h=5/8` 处 = 0 ✓（与 theory768 的因式分解一致；我另外独立手算过 `det G_2` 的通式）。 +- 路径②（把 `χ` 直接约化）：`−3·L_{−2}|h⟩ + 2·L_{−1}²|h⟩` 展开后**就是** `(−3,2)` 这两个 PBW 词， + 再对全空间求和 `Σ c_u c_v G(u,v) = 0` 精确。两条路径一致（`chi_agrees=true`）。 +- `L_1χ = L_2χ = L_3χ = 0` 精确（I 与 III 都验证过）。 +- **裁定：`χ` 是真零范奇异向量。成立。** + +### 1.3 ⚠️ `0,0,1,1` 是不是「只是某个恒等式的重述」？——**是（只有一个非算术输入）** + +设 `d_n = rank G_n`（= 不可约商 level n 的维数）、`e_n = dim(L_{−1}` 在商 level n 的像`)`、`q_n = d_n − e_n`。 +我**分别**用**四路互相独立**的方法算了 `e_n`（`v768_v1b_diag.py`）： + +- **A**：`dim((L_{−1}P_{n−1} + rad_n)/rad_n)`，`P` = `rad_{n−1}` 的补（我第一版的路线） +- **B**：`dim((L_{−1}V_{n−1} + rad_n)/rad_n)`，用**全部** `V_{n−1}`（等价于 theory768 的 `proj_quot` 路线） +- **C**：用**显式商坐标**（rref 主元坐标做代表）算诱导映射的 rank +- **D**：用**伴随** `L_1` 的核：`e_n = d_n − (dim W − dim rad_n)`，`W = {v∈V_n : L_1 v ∈ rad_{n−1}}` + +四路结果（`c=0,h=5/8`）： + +| `n` | 1 | 2 | 3 | 4 | 5 | 6 | +|---|---:|---:|---:|---:|---:|---:| +| `d_n = rank G_n` | 1 | 1 | 1 | 2 | 3 | 4 | +| `e_n`（A=B=C=D） | 1 | 1 | 1 | 2 | 3 | 4 | +| `d_{n−1}` | 1 | 1 | 1 | 2 | 3 | 4 | +| `q_n = d_n − e_n` | 0 | 0 | 1 | 1 | 1 | 2 | + +⇒ `e_n = d_{n−1}` ⇒ `L_{−1}` 在不可约商上**单射**（四路一致，n≤6）。 +另外 n=4 的**显式核向量证书**：诱导映射的核里唯一的向量恰好落在 `rad_3` 内（`u = (−3/2,−3/2,1)`， +`u ∈ rad_3`，其 `L_{−1}u` 范数 0 且与 `V_4` 全空间正交）——即**没有非平凡的核**，与单射一致。 + +**⇒ 结论：`q_n = d_n − d_{n−1}`。代入 `d_n = p(n)−p(n−2)`：** +``` +q_n = p(n) − p(n−2) − p(n−1) + p(n−3) +q_1 = 1−0−1+0 = 0 +q_2 = 2−1−1+0 = 0 +q_3 = 3−1−2+1 = 1 +q_4 = 5−2−3+1 = 1 +``` +**所以 `0,0,1,1` 是「`rank G_n = p(n)−p(n−2)`」+「`L_{−1}` 单射」的纯算术推论。** +我的脚本把这件事单独判了出来：`is_0011_arithmetic_restatement = true`。 +**它不是一个独立于 `rank G_n` 公式的新事实**（上一轮见过的「两条路径代数恒等时把网格大小当独立证据」正是这一类）。 +**非算术的输入只有一个**：`L_{−1}` 在不可约商上单射（`e_n = d_{n−1}`）。 +⇒ 裁定：**`0,0,1,1` 成立；但 note 说「我自己用精确有理数复算」，容易让人以为它独立于 `rank G_n` 公式，实际不是。** + +--- + +## 2. V2（核心）：核「动量 0 排除 spin±4」 + +### 2.1 `δ_{h,h̄}` 的推导与条件 —— **成立，但 note 的措辞把两件事混在一起** + +**推导（精确、初等）**：设紧致方向平移生成元 `P = (2π/L)(L_0 − L̄_0)`。真空满足 `P|0⟩=0` +（周期边界 ⇒ 真空平移不变）。对定权算符 `[P,φ_{h,h̄}] = (2π/L)(h−h̄)φ`。于是 +``` +⟨0|φ|0⟩ = ⟨0|e^{iPa}φe^{−iPa}|0⟩ (平移不变性) +0 = d/da ⟨0|e^{iPa}φe^{−iPa}|0⟩ = i(2π/L)(h−h̄)⟨φ⟩ ⇒ (h−h̄)⟨φ⟩ = 0 +``` +⇒ 动量 0 态之间的矩阵元 `⟨mom0|φ|mom0⟩ = 0` 当 `h≠h̄`。**这是精确的**。 +需要的条件（note 没全列）： +1. **沿紧致方向平移不变** = 周期边界（**开放/自由边界下不成立**）；环面双周期 ✓。 +2. 初末态是**动量 0**（真空或某扇区的动量 0 基态）。 +3. φ 有确定 `(h,h̄)`（平移本征）。 + +⚠️ **措辞之瑕（要记下来）**：note 说「权重 `(h,h̄)` 的算符**一点函数** `∝δ_{h,h̄}`」。 +但在**无限周期圆柱**上，主场的**一点函数恒为 0**（与 h 无关）：圆柱经 `z=(L/2π)log w` 共形等价于平面， +而 `⟨φ⟩_平面 = 0`（平移+标度）⇒ `⟨φ⟩_圆柱 = |dw/dz|^{2h}⟨φ⟩_平面 = 0`。 +`δ_{h,h̄}` 真正成立的地方是 **(a) torus 上的 modular 迹**（`⟨φ⟩_torus = Tr(q^{Δ}q̄^{Δ̄}φ)/Z`，动量守恒 ⇒ 只有 `h=h̄` 项） +或 **(b) 动量 0 态之间的矩阵元**。note 把「一点函数」「矩阵元」「per-row 有限尺寸修正」三件事混着说， +**只有 (b) 是无条件精确的**。这不影响它引出的结论（它实际用的是 (b)），但读者会误以为在说圆柱一点函数。 + +### 2.2 被排除的观测量真的是动量 0 吗？——**成立（一阶）**，但有一个 note 自己没补的缺口 + +- `Θ_w = I⁰_{4,w}(p_c) − I⁰_{8,w}(1−p_c)`、`Δ_w(p) = I⁰_{4,w}(p) − I⁰_{8,w}(1−p)`。 + 按原文 `compass-snapshot-64159d56/sector-odd-spin4-anisotropy-20260914.md` §1.2， + 它是「两个 magnetic/topological **激发能**之差」，且 §2 明确说这是 **per-row** 量。 + per-row 能量 = 转移矩阵最大本征值的对数/行；对**平移不变**的转移矩阵，最大本征值落在**动量 0** 扇区 + （Perron–Frobenius / 平移不变）⇒ `Θ_w` 沿周长方向**是动量 0 的**。 + ⇒ 裁定：**`Θ_w`/`Δ_w` 是动量 0 量。成立。** +- ⚠️ **缺口（`#802` 自己点名的那条）**：`#802` 原文写 + 「**均值零列场在平移不变背景下一阶往往为零；需要先推导使用一阶还是二阶，不以此误判 spin 选择。**」 + 「矩阵元为 0」是**一阶**陈述。一个自旋 ±4 的无关算符，在**二阶**（经 `φ₄ φ_{−4}` 的自旋 0 组合） + 一般**可以**移动动量 0 能级，且幂次不同。note **从未证明** `Θ_w` 里的 `w^{−17/4}` 项是**一阶**效应。 + ⇒ 「spin4 的矩阵元 = 0 ⇒ 它不能给出 `w^{−17/4}`」这一步**依赖未证明的一阶假设**。**判定：条件性成立。** +- note 自己在 §3.6/§4.1 把这点标成**未决**（「`Θ_w` 到底是动量 0 的谱量，还是含缝/方向分辨的量」） + —— 这是诚实的；但它同时把 D1 当成「已交付的判据」，两者张力。 + +### 2.3 区分性 —— 8-arm 权重**成立**，但**不唯一** + +- `(21/8,21/8)`：Kac `h_{4,2}=h_{2,7}=(8²−1)/24=63/24=21/8` ✓，`x=2h=21/4` ✓，`spin=0` ✓。**成立。** + （`h(k)=(k²−1)/24`，`x(k)=(k²−1)/12`；8-arm 对应 `k=8` ✓。） +- 热族 level-4 手征后代：(h,h̄) = (5/8+4, 5/8) = (37/8,5/8)，`spin=+4`，`x=21/4` ✓。**成立。** +- **但动量规则是必要不充分**：它**只能杀 `spin≠0`，永远无法选出标量**。 + 在 `x=21/4` 这一层，能通过动量规则的候选（全是标量）：8-arm（外部）、 + 以及 note 自己承认「**本轮最大的空白**」的 `k=2..7` 标量 arm（`x=1/4,2/3,5/4,2,35/12,4`，其中 `k<8` 的都在 `21/4` **以下**， + 若幅度不零就会**压过** `w^{−17/4}`）。此外 identity 家族被**宇称（dual-even）**消去（不是被动量规则），`(ε,ψ̂)` log 对在 `x=5/4`。 + ⇒ **「排除 spin±4」不等于「唯一挑出 8-arm」**。note 的 §3.6/§1.5 也承认不能宣称谁胜出 —— **一致**； + 但 §0 第 4 条的标题「这是**可行的排除**」+ §3.4 D1「预测像不同」措辞偏强。 +- 另一处不对称：动量规则对 spin±4 给的是**可证的 0**，对 8-arm 只给**「不被排除」** + （任何 `Δ≠0` 的主场在平面/圆柱上 `⟨0|φ|0⟩` 本来就是 0）。8-arm 的**非零**性正是 `#768` 挂着的「非零性」缺口。 + ⇒ 「预测像不同」实际上是 **「一个可证为 0，另一个未被约束」**，只有在 8-arm 幅度确实非零时才成为区分。 + +### 2.4 ⭐ 它自己承认的附加结论（`(2,2)` spin-0 类）—— **本轮最重要发现** + +**先独立复算算术**：`(ii)`（= Verma，保留零范态）下 `q = [1,0,1,1,2,2,4]`，故在 `x=21/4` 处 +有 `(a,b)=(2,2)`、`spin=0`、维数 `q₂·q₂ = 1`、**动量 0**、C4 允许 —— **这个类确实存在**。 +（`(i)` 下 `q₂ = 0` ⇒ `(2,2)` 维数 0 ⇒ 不存在。两套计数我都独立复算，与 note §1.3 的表逐格一致。） + +**但是 note 由此得出「`(ii)/(iii)` 下热模**可以**贡献动量 0 观测量」，这一步不成立：** + +1. **在 `(ii)` 下，这个「多出来的 level-2 非导数类」就是零范态 `χ` 的方向本身。** + 理由（精确）：`V_2/L_{−1}V_1` 维数 1，只能由 `[L_{−2}|h⟩]` 张成； + 而 `χ = −3L_{−2}+2L_{−1}²` 给出 `[χ] = −3[L_{−2}] ≠ 0` ⇒ **整块 level-2 商类就是零范方向**。 + 零范场被所有正模湮灭 ⇒ 它的**关联函数恒为 0** ⇒ `(2,2)` 类在 `(ii)` 下**贡献 0**。 + 这与 note **自己 §1.2 的裁定**（「`(i)` 与 `(ii)` 的关联函数完全相同」）**完全一致，也与它的附加结论矛盾**。 +2. **在 `(iii)` 下**，note 自己说「代数不定，是物理模的性质」——`(2,2)` 类是否是真算子 + （有关联函数的 log 伙伴）**不由 Virasoro 代数决定** ⇒ **未建立**。 +3. ⇒ **裁定**：note 的措辞「只有保留零范态时……**才能进入**动量 0 观测量」**把 (ii) 与 (iii) 混为一谈**： + - `(ii)`（真的只是「零范态**保留**在态空间里」）⇒ **关联为 0，进不来**； + - `(iii)`（零范态成为 **log Jordan cell 的 bottom**）⇒ **可能**进来，但**未证明**。 + +**由此重新裁定 `#0` 第 4 条的强度**： +- 这条排除在 **(i) 与 (ii) 下都成立**（`(i)` 类被商掉；`(ii)` 类零范 ⇒ 关联恒 0）； +- **只有在 (iii) 且该类确为真算子时才失效** —— 而 note 自己主张物理模是 (iii)。 +- ⇒ **该排除是条件性的，note 没有把它标成条件，反而用它自己的 (iii) 结论去支持一个 (ii)-计数的论证。** + team-lead 担心的「(iii) 掏空这条排除」**未必成立**，但 **note 的论证方式确实不成立**， + 且 **§3.4 D3「(ii)/(iii) 下 `x=21/4` 处有 2 个独立的动量 0 幅度」对 (ii) 是错的**（(ii) 下仍是 1 个）。 + +--- + +## 3. V3：核「没有宣称什么」是否真的守住 + +| note 的「不能宣称」 | 我核的结果 | +|---|---| +| 不能宣称 level-4 类的差矩阵元**非零** | §1.5 明文守住 ✓。**但轻度泄漏**：§3.3 预测表里「主导 `Δ_w` = `A_S w^{−17/4}` / `A_4 w^{−17/4}`」**隐含两个幅度都非零**，§3.4 D3 还谈「2 个独立幅度」。这是「候选预测表」的读法可辩护，但字面上与 §1.5 的免责冲突。 | +| 不能把 `∂⁴ε` 当插入（其类为 0） | §1.4 ✓。**并注意**：这条是**平凡真**——`L_{−1}⁴|h⟩` 按定义就是 `L_{−1}(L_{−1}³|h⟩)` ∈ `L_{−1}V_3`，在 `V_4/L_{−1}V_3` 里类恒为 0，不需要任何计算。（theory768 的 `g2b_result.json: d4eps_in_derivative=true` 我未逐字重跑，但它平凡成立。） | +| 不能宣称 8-arm / spin4 谁胜出 | §3.6 ✓。**但**§0 第 4 条「构成对 spin4 的**可行排除**」与 §3.4 D1 的措辞**倾向 spin4 失败**；§3.6 又说不宣称。张力（与 §2.4 的条件性直接相关）。 | +| 不能宣称 `w^{−17/4}` 已被解释 | §2.5/§3.6 ✓（并且 §2.5 还列了 Jacobsen 4.0001(2) vs MZ16 `−3.42` 的冲突）✓ | + +**额外**：note 第 3 条 G1 用「identity 家族 dual-even ⇒ 在 `Δ_w` 里恒等消去」，依赖「`Δ_w` 是 dual-odd」。 +这条是**项目既有的对称性假设**，note 标为「本项目已确立」。我**未独立重算**，按「引用，未独立重算」记。 + +--- + +## 4. 发现的问题（最小复现） + +### 4.1 `issues-found.md` 的内容（详见该文件） + +1. **theory768 的 JSON flag `rank_equals_p_shift2: false` 与它自己的表和结论矛盾**（脚本 bug）： + `scripts/g2_virasoro.py` 里 `pnm2 = len(B[n-2])`，随后 `if rank[n] != pnm2`—— + 它把 `rank G_n` 拿去和 **`p(n−2)`** 比，而不是和 **`p(n)−p(n−2)`** 比。 + 于是 n=0,1,3,4,5,6 全部判为 DIFF ⇒ flag=false。 + 我用真确公式独立重算：**全 n≤6 成立**（我的 `rank_eq_p_minus_p_shift2_all=true`， + 同时复现了它的 `theory768_style_flag_p_shift2_only=false`）。 + ⇒ **只看 JSON 的读者会误判该声称失败**。 +2. **note §0#4 的附加结论与它自己 §1.2 矛盾**（§2.4）；`(ii)` 下 `(2,2)` 类是零范方向 ⇒ 关联恒 0。 +3. **note 把 (ii) 与 (iii) 混用**（同上）。 +4. **note 的「一点函数 `∝δ_{h,h̄}`」措辞**：无限圆柱上主场一点函数恒为 0；成立的是矩阵元/torus 迹。 +5. **note 用 (ii) 的类计数去支持 (iii) 的结论** —— 这是 §2.4 的机制。 +6. 我自己的实现曾有一个 complement 选基 bug（`cur == rank[n−1]` 应为 `cur == len(Bp)`）， + 会导致 `0,0,1,2`；**四路互检**才发现并修正（A/B/C/D 全给 `e₄=2`）。 + 记录在此以示：**单路「独立」实现也可能一致地错**。 + +--- + +## 5. 不能宣称什么（本次核验的边界） + +- 我**没有**独立重算 `#802` 的 `(T_g,N_g)`（`sector802` 已交付，规格 §4 禁止重复）。 +- 我**没有**验证 `Θ_w` 在实际扇区引擎里的数值定义（"每行能量差" 的精确闭合振幅/长宽比）； + §2.2 的「动量 0」结论基于**转移矩阵平移不变 ⇒ 基态动量 0** 这一标准结构，**非**对真实数据的实测。 +- 我**没有**判定 `(iii)`（c=0 log 模块）下载荷 `(2,2)` 类是否为真算子 —— 那需要 log 模块的具体构造， + Virasoro 代数**不足以**决定（note 自己也这么说）。所以 §2.4 的裁定是「**未建立**」，不是「错」。 +- 我**没有**判定 8-arm 的幅度非零（`#768` 的「非零性」缺口仍然开着）。 +- 我**没有**重跑 theory768 的 `g2b_derivatives.py` / `g2c_bilevel.py` / `g1g3_channels_predictions.py` + 的全部输出；我只独立复算了其中的 (i)/(ii) 类表与 `(a,b)` 权重算术（一致）。 +- 文献（He 2411.18696、VJS 1206.2312、Mathieu–Ridout 0708.0802、Javerzat et al. 2005.11830、 + Jacobsen 1507.03027、Mertens–Ziff 1603.07289）：**全部为「引用，未独立重算」**。 + +--- + +## 6. 一句话结论 + +**V1 四项全部成立**（`χ` 范数确为 0；`0,0,1,1` 成立**但它只是 `rank G_n=p(n)−p(n−2)` 与 `L_{−1}` 单射的算术推论**， +不是独立新事实）。**V2 的「动量 0 排除 spin±4」作为一阶矩阵元陈述成立、且 `Θ_w` 确为动量 0 量**， +但它 **(a)** 只在一阶成立（`#802` 要求的二阶问题未处理）、**(b)** 只杀自旋不选标量（不唯一）、 +**(c) 它自己承认的 `(2,2)` 附加结论不成立**——`(ii)` 下那个「多出来的类」正是**零范态方向、关联恒为 0**， +与 note §1.2 自相矛盾；`(iii)` 下则**未建立**。因此 **「可行排除」是条件性的**： +在 (i)/(ii) 下成立、在 (iii) 且该类为真算子时才失效——而 note 自己主张物理模是 (iii)。 +最载重的一条（`0,0,1,1` 与 `χ` 范数）**经得起独立复核**；最乐观的一条（排除 spin4 即区分两个候选）**不成立**。 diff --git a/docs/manuscripts/geometric-balance/void-pressure-equals-component-intensity-20260914.md b/docs/manuscripts/geometric-balance/void-pressure-equals-component-intensity-20260914.md new file mode 100644 index 000000000..42d4b468c --- /dev/null +++ b/docs/manuscripts/geometric-balance/void-pressure-equals-component-intensity-20260914.md @@ -0,0 +1,263 @@ +# In the fixed-subcritical dilute regime, void free energy equals winding-component intensity to first order + +2026-09-14. Author-level rare-event pressure argument joining the component-Poisson and fixed-width charge-free-energy descriptions. + +The important distinction is: + +- at finite width, the no-winding strip free energy `I^0_w` and the complete winding-component intensity `nu_w` are not identical; +- for fixed subcritical `p` and `w->infinity`, winding components become an exponentially dilute locally dependent gas, and the pressure/activity distinction disappears at first order. + +The result claimed here is + +\[ +\boxed{ +\frac{I^0_{G,w}(p)}{\nu^G_w(p)}\longrightarrow1, +\qquad w\to\infty, +\quad 0p_c`. + +## 1. Complete-component activity + +On the infinite cylinder `C_w x Z`, every occupied component is vertically finite almost surely. Count each horizontally essential component exactly once by the bottom-row/tie-break anchor from `poisson-birth-windows.md`. Its row intensity is + +\[ +\nu_w= +E[\#\{\text{complete winding-component anchors in row }0\}]. \tag{1.1} +\] + +The existing fixed-subcritical argument gives + +\[ +-\frac1w\log\nu_w\to\kappa_G(p)>0. \tag{1.2} +\] + +## 2. Linear-height localization is enough for pressure + +The birth-window proof used `H=w^2` to make localization errors superexponentially small. For the present first-order pressure statement choose instead + +\[ +H=Cw, \tag{2.1} +\] + +where `C` is a sufficiently large fixed constant depending on the declared subcritical `p`. + +Uniform cylinder height tails give + +\[ +0\le\nu_w-\nu_{w,H} +\le C_1w e^{-c_1H}. \tag{2.2} +\] + +Choose `C` so that + +\[ +c_1C>\kappa_G(p)+2\eta \tag{2.3} +\] + +for some `eta>0`. Then by (1.2), + +\[ +\boxed{\nu_w-\nu_{w,H}=o(\nu_w).} \tag{2.4} +\] + +Thus it is enough to compute the void pressure of the finite-window anchor field. + +## 3. Dependency graph and k-fold witness bound + +Let `I_i`, `i=(j,x)`, be the localized anchor indicators with height cutoff `H`. Each is measurable in `O(H)` rows and one circumference, so the dependency graph has maximum degree + +\[ +D_w=O(wH)=O(w^2). \tag{3.1} +\] + +Disjoint windows are independent. + +Let `B_w` be the first-span upper bound for the event that a window union of `O(H)` rows contains a horizontal winding witness. At fixed subcritical `p`, + +\[ +B_w\le\operatorname{poly}(w)e^{-\kappa_G(p)w}. \tag{3.2} +\] + +The nonmonotone anchors themselves are NOT fed into BK. Instead, if `k` distinct anchors occur, they belong to `k` distinct complete components. Choose one occupied horizontal-winding witness inside each component. These occupied witness site sets are disjoint. Hence, for every distinct tuple whose relevant windows lie in one connected dependency cluster, + +\[ +\{I_{i_1}=\cdots=I_{i_k}=1\} +\subset E_1\square\cdots\square E_k, \tag{3.3} +\] + +where the `E_a` are increasing enclosing winding events. Repeated SITE BK gives + +\[ +\boxed{ +E\prod_{a=1}^k I_{i_a} +\le B_w^k.} \tag{3.4} +\] + +This is stronger than the pair bound used for Chen--Stein. + +## 4. A local-dependence pressure lemma + +Consider a stationary family of zero-one variables on `m` rows, with finitely many types per row and a dependency graph of degree at most `D`. Suppose joint moments of every distinct connected `k`-set obey + +\[ +E\prod_{i\in S}I_i\le B^{|S|}. \tag{4.1} +\] + +Let + +\[ +Z_m=E\prod_{i\in\Lambda_m}(1-I_i) +=P(\text{no marked event in the length-}m\text{ window}). \tag{4.2} +\] + +Because joint moments factor across disconnected dependency-graph components, the logarithm of (4.2) has the standard connected-cluster expansion. The first-order contribution is + +\[ +-\sum_i EI_i. \tag{4.3} +\] + +Every higher term is supported on a connected index set. A fixed root belongs to at most + +\[ +(eD)^{k-1} \tag{4.4} +\] + +connected ordered/tree-coded `k`-clusters up to an inessential universal factor. The usual tree-graph/cumulant bound then gives, whenever `cDB<1`, + +\[ +\left|- rac1m\log Z_m +-\frac1m\sum_iEI_i\right| +\le +C\,n_{type}\,D B^2 +\sum_{r\ge0}(cDB)^r. \tag{4.5} +\] + +Here `n_type` is the number of anchor types per row (`w` in the raw site-anchor convention). The constants are universal and irrelevant for the exponential comparison. + +Equation (4.5) is the standard small-activity connected-cluster fact behind dependency-graph/LLL polymer expansions: **log void probability contains only connected event clusters.** Scott--Sokal / Dobrushin cluster-expansion criteria provide an abstract framework; the special moment bound (4.1) makes the present estimate elementary by tree counting. + +For the winding anchor field, + +\[ +DB_w=\operatorname{poly}(w)e^{-\kappa w}\to0, \tag{4.6} +\] + +so the expansion is absolutely convergent for all large `w`. + +## 5. Apply the pressure lemma + +Per row the one-anchor term is exactly + +\[ +\sum_{x=0}^{w-1}EI_{(0,x)}=\nu_{w,H}. \tag{5.1} +\] + +The higher connected-cluster correction is bounded by + +\[ +R_w +\le\operatorname{poly}(w)e^{-2\kappa w}. \tag{5.2} +\] + +Since + +\[ +\nu_w=e^{-\kappa w+o(w)}, \tag{5.3} +\] + +we have + +\[ +\boxed{R_w=o(\nu_w).} \tag{5.4} +\] + +Therefore the localized anchor void pressure `J_{w,H}` satisfies + +\[ +\boxed{J_{w,H}=\nu_{w,H}[1+o(1)].} \tag{5.5} +\] + +Together with (2.4), + +\[ +J_{w,H}=\nu_w[1+o(1)]. \tag{5.6} +\] + +## 6. From anchor void pressure to the safe-strip free energy + +The strip free energy + +\[ +I^0_{G,w} +=-\lim_{m\to\infty}\frac1m +\log P(\text{no horizontal occupied homology in an }m\text{-row strip})\tag{6.1} +\] + +is insensitive to `O(H)` boundary rows. + +Insert empty guard rows at the two strip ends. On the guarded interior, every horizontal essential component is a **complete** cylinder component and therefore has one anchor; conversely every localized complete winding component creates strip horizontal homology. Removing/adding the `O(H)` margins changes `-log` probability by at most a boundary cost independent of `m`, which vanishes after division by `m`. + +The only discrepancy is a component whose vertical span exceeds `H`; its per-row activity is the tail in (2.2), already `o(nu_w)` by the choice of `C`. + +Hence + +\[ +I^0_{G,w} +=J_{w,H}+o(\nu_w). \tag{6.2} +\] + +Combining with (5.6) proves (1). + +## 7. Quantitative form + +The argument yields the schematic estimate + +\[ +\boxed{ +I^0_{G,w} +=\nu_w ++O(\operatorname{poly}(w)e^{-2\kappa w}) ++O(w e^{-cCw}).} \tag{7.1} +\] + +The polynomial and constants have not been optimized. Since the leading activity is only `e^{-kappa w+o(w)}`, both errors are relatively negligible after choosing `C` as in (2.3). + +This is a pressure/activity statement, not an OZ prefactor theorem: it does not determine the polynomial prefactor of `nu_w` itself. + +## 8. Consequences + +### 8.1 Poisson births and void Perron roots are the same dilute gas to first order + +The component-Poisson theorem uses `nu_w` as its natural clock. The fixed-width charge theory uses `I^0_w` as the exponential no-winding cost. Equation (1) shows that in the fixed-subcritical large-width regime + +\[ +\boxed{\text{activity }\nu_w +\quad=\quad +\text{pressure }I^0_w\,[1+o(1)].} \tag{8.1} +\] + +Their finite-width difference is precisely a higher connected-cluster correction of the winding-component gas. + +### 8.2 The distinction reappears near criticality + +At the charge/eigenvalue root, `p_w->p_c` and the scaled width `w/xi(p_w)` is `O(1)`. The anchor gas is no longer exponentially dilute in `w`. There is no reason for `I^0_w/nu_w->1` in that regime. + +Thus the rare-component Poisson language and the critical transfer/CFT language are not competing descriptions. They are two regimes of the same object, separated by loss of small activity. + +### 8.3 Cluster corrections have a concrete meaning + +The first correction to `I^0=nu` is generated by connected pairs/triples of nearby winding components. A future second-order calculation can therefore target a winding-component virial coefficient rather than an unexplained finite-width discrepancy. + +This may be a cleaner route to finite-width corrections than fitting `I^0/nu-1` directly. + +## 9. Literature boundary + +Connected-cluster expansions for dependency graphs / abstract polymer gases are classical; Scott--Sokal (2003) and cluster-expansion Lovasz-local-lemma refinements give general nonvanishing/convergence frameworks. The percolation-specific content here is the component-anchor localization and the `k`-fold disjoint winding-witness BK bound that drives the activity parameter exponentially small. + +No claim of novelty is made for the generic pressure/activity expansion. + +## 10. Claim boundary + +The proof uses the existing author-level cylinder localization, full-component anchor construction, fixed-subcritical mass rate, and site BK inputs from #739. The connected-cluster estimate is standard but should receive a line-by-line combinatorial audit if promoted into the final manuscript. The result is only for fixed subcritical `p`; it is not a near-critical uniform theorem. diff --git a/docs/manuscripts/geometric-balance/ward-kb-locking-20260914.md b/docs/manuscripts/geometric-balance/ward-kb-locking-20260914.md new file mode 100644 index 000000000..82a670b23 --- /dev/null +++ b/docs/manuscripts/geometric-balance/ward-kb-locking-20260914.md @@ -0,0 +1,254 @@ +# note-kb-ward — issue #774: complete-component quality–boundary Ward checks + +**Machine**: Huawei Cloud `DevEnvC_XPk2PZ` (fresh), workdir `/workspace/ward774/`. +**Rule honoured**: **no computation on the local Mac**; every number below was produced by a +script executed on the cloud machine (logs and JSON are in `/workspace/ward774/out/`, +downloaded to `ward774-out/`). +**Engine**: pinned `tagged_winding_span.py`, +sha256 `9621acf490dbc4b7dba28f2d2f0f4c2e0f1ee9e42d3987d1f7815bd1c9d5c9a3`, +git blob `52f3611990ce2b1331d9e5296e0262f5e402e0d7` (printed by every run). +Only the shipped `build/lump/numeric_system/solve` and `activity_transfer` are used — **no new +transfer engine, no Monte Carlo, no GPU**. + +--- + +## 1. Convention (authoritative, BRIEF §5) + +* black = occupied, NN connectivity, probability `p`; white = empty, matching, `q = 1-p`. +* Component = **complete horizontal-winding (essential) NN cluster**, counted once (tagged). +* `K` = occupied sites of the component. +* `B` = **number of DISTINCT external sites NN-adjacent to the component** (the engine's + `v`-fugacity exponent). **Not** the number of contact edges — a site adjacent to the cluster + through two different contacts is counted once. +* `q = 1-p`, `S = K/p - B/q` (the **#774** sign convention; the giant-white package uses + `K/q - B/p` because there `q` is the white occupancy — see §8). +* `p_ref = 0.5927460507921` is used **only as a near-critical diagnostic reference**, never as a + new `p_c`. + +Component weight: for a fixed cluster `C`, the probability that `C` is exactly a complete +NN cluster of the configuration is `p^K q^B` (all `C` occupied, every site of its external +vertex boundary empty, everything else free). Hence `nu_w(p) = Σ_C p^{K} q^{B}`. + +## 2. `B` is the geometric distinct-boundary count — independently verified + +The `activity_transfer` labels each transition with `(k, nb)` and gives the source an +offset `nb`. I verified that `K = Σ k`, `B = source_nb + Σ nb` is the **geometric** pair by +brute-force enumeration, independent of the engine: + +* enumerate every connected NN site set `C` on the `w`-cylinder with minimum row 0, no empty + intermediate row, that winds horizontally (lift/spanning-tree winding test); +* `K = |C|`, `B = |{sites ∉ C, NN-adjacent to C}|`; +* compare the full `(K,B)` histogram with the histogram obtained from the engine's own + accepting paths. + +Result (`scripts/diag_marks.py`, `out/diag_marks.json`): + +| width | max rows | # components | brute = engine | +|---|---|---|---| +| 2 | 7 | 969 | **identical** | +| 3 | 5 | 5207 | **identical** | + +So the `nb` label is *not* an edge perimeter and *not* a mis-count; it is the distinct +external boundary. (Nitpick worth recording: for the NN case the *source* offset happens to +equal the occupancy of the component's first row, and the accept step adds the acceptance-row +occupancy, so the decomposition of `B` between source and transitions is not itself the +geometric one — only the total is. This is a label-semantics subtlety, not an error.) + +## 3. Algorithm (exact, two fugacities) + +The transfer matrix is a function of two fugacities `(u,v)`; with marks `(1,k,nb)` per +transition and the source carrying its own `v`-offset: + +``` +R(u,v), b(u,v), alpha(v) -> nu = alpha^T (I-R)^{-1} b, a rational function of (p,q) +E[K] = (u d/du) log nu |_{u=p, v=q} E[B] = (v d/dv) log nu |_{u=p, v=q} +d/dp log nu_w(p) = (u d_u - v d_v) log nu -> E[K]/p - E[B]/q = E[S] (I1) +``` + +All first and second derivatives (including the mixed one) are obtained by **implicit +differentiation of the resolvent**, i.e. by solving extra linear systems with the same matrix +`A = I - R(p,q)`; **no finite differences are used as a certificate**. `gauss_solve_many()` +does one Gauss–Jordan elimination with several right-hand sides; all arithmetic is +`fractions.Fraction`, so every certified check is an exact rational identity. + +**Cost.** `activity_transfer(w, matching=False)`: 12/68/340/1672/8244 states → +3/5/13/24/61 lumps for w=2..6 (~15 s at w=6); w=7: 40588 states → 126 lumps in 163 s, 268 MB; +w=8: see `out/w8.log` (state cap raised to 4·10⁶; the shipped builder hard-limits width to 6, +so for w=7,8 a **raise-only relaxed copy** of the *same* file is used — one line changed, +verified by diff, recorded in every output row as `engine`). + +## 4. Code validation (and one bug I made and fixed) + +My moment extractor is adapted from the #772 deliverable +`giant_white_volume.py::joint_activity` (unmodified upstream file). Before using it I +reproduced the **shipped** white-case numbers at `q=3/4` for w=2,3,4 +(`nu, mean_span, mean_occupation, mean_boundary, score_mean, score_variance, +residual_covariance_scaled, prospective_clt_variance`) — **all exactly equal** to +`giant-white-volume.json`, as is the shipped assertion `score_mean == -d_p log nu_black(1/4)`. +This is a validation of my code, *not* a re-derivation of #772's numbers as a deliverable. + +While doing this I found and fixed a bug of my own: my first version fed the *raw* `b`-vector +instead of the solved `y0` into the right-hand side of the first-moment solves, which gave +`E[L], E[K]` about 8× too small and made (I1)/(I2) fail. The validation above caught it. +**All numbers in this note are from the fixed code.** (Honesty note: the early, wrong run is +kept only in my logs; it is not reported as a result.) + +## 5. Main table — `p = p_ref` (black NN), widths 2..8 + +Float64 diagnostics at `p_ref` (`out/tA_float_ref.json` for w=2..7, +`out/tA_float_ref_w8.json` for w=8); exact rationals at `p_ref` for w≤6 and at `p=1/2` for w=7 +(`out/tA_exact_ref.json`, `out/tA_exact_half*.json`). `q/p = 0.687063`. +**w=8 completed** in 1820.8 s (≈30 min, 321 lumps, ≈2 GB peak) with the raise-only relaxed +activity builder — that is the cost of one extra width here; w=7 (40588 states → 126 lumps) +took 163 s. w=8 is float64; its exact-rational run was still going when I finalised, so the +exact certificates below stop at w=7. + +| w | `nu_w` | `E K` | `E B` | `E B/E K` | `mean_ratio` | `score_mean` | `score_var` | `shot_scale` | `ward2_ratio` | `Corr(K,B)` | cond | angle° vs `(p,q)` | `sm_eig/trace` | +|---|---|---|---|---|---|---|---|---|---|---|---|---|---| +| 2 | 0.149338 | 7.007 | 5.082 | 0.7253 | 1.055595 | -0.657163 | 30.018 | 50.580 | 0.59347 | 0.80419 | 32.8 | 19.91 | 0.02958 | +| 3 | 0.096361 | 15.077 | 10.497 | 0.6962 | 1.013346 | -0.339454 | 67.872 | 106.199 | 0.63910 | 0.90403 | 36.2 | 11.32 | 0.02692 | +| 4 | 0.072335 | 25.763 | 17.785 | 0.6903 | 1.004751 | -0.206497 | 121.193 | 180.555 | 0.67122 | 0.94013 | 49.1 | 7.77 | 0.01994 | +| 5 | 0.057782 | 39.210 | 26.993 | 0.6884 | 1.001982 | -0.131130 | 191.015 | 274.350 | 0.69624 | 0.95909 | 66.9 | 5.86 | 0.01474 | +| 6 | 0.048134 | 55.286 | 38.022 | 0.6877 | 1.000975 | -0.090975 | 276.893 | 386.600 | 0.71623 | 0.97010 | 87.9 | 4.69 | 0.01125 | +| 7 | 0.041251 | 73.953 | 50.837 | 0.6874 | 1.000537 | -0.066997 | 378.665 | 516.998 | 0.73243 | 0.97709 | 111.8 | 3.90 | 0.00886 | +| 8 | 0.036092 | 95.169 | 65.408 | 0.6873 | 1.000321 | -0.051516 | 496.174 | 665.236 | 0.74586 | 0.98183 | 138.5 | 3.34 | 0.00717 | + +All rows: (I1)(I2)(I3)(I4) pass — exactly for w≤6 at `p_ref` and for w=7 at `p=1/2`; float64 +at `p_ref` for w=7,8 (max |I1| diff 8.3e-14, max |I2| diff 3.5e-11). `nu_activity == nu_tagged` +for every row (float64 agreement ≥ 10 significant digits). + +**Extras (reported, not fitted as theorems).** `w^{55/48}(E B/E K - q/p)` = +0.1230, 0.0470, 0.0233, 0.0125, 0.0076, 0.0050, 0.0035 (w=2..8) — this **decreases**, i.e. the +deviation decays **faster** than `w^{-55/48}`. A 7-point log–log fit of +`E B/E K - q/p` gives an effective power ≈ `w^{-3.73}` (per-step exponent estimate rises +0.29→0.38 for `1-ward2_ratio`, see §7). `Var(S)/E K` = 4.28, 4.50, 4.70, 4.87, 5.01, 5.12, 5.21. + +## 6. Pre-declared λ points + +`c = 1/4` was fixed **before** running; `p = p_ref ± c·w^{-3/4}` (`w^{-3/4}` taken as its +12-digit decimal, `p_ref` as the rational `5927460507921/10^13`). No `c` was scanned +afterwards. Results (`out/tA_float_lam.json`): + +| w | `p_-` | mean_ratio(-) | ward2(-) | Corr(-) | `p_+` | mean_ratio(+) | ward2(+) | Corr(+) | +|---|---|---|---|---|---|---|---|---| +| 2 | 0.44410 | 0.78997 | 0.49806 | 0.76952 | 0.74140 | 1.24348 | 0.68743 | 0.86007 | +| 3 | 0.48307 | 0.83306 | 0.58655 | 0.87902 | 0.70242 | 1.12921 | 0.75404 | 0.93690 | +| 4 | 0.50436 | 0.86561 | 0.61808 | 0.92229 | 0.68113 | 1.08604 | 0.79266 | 0.96277 | +| 5 | 0.51798 | 0.88940 | 0.64300 | 0.94525 | 0.66751 | 1.06364 | 0.81719 | 0.97551 | +| 6 | 0.52753 | 0.90696 | 0.66351 | 0.95887 | 0.65796 | 1.05013 | 0.83445 | 0.98261 | +| 7 | 0.53465 | 0.92011 | 0.68065 | 0.96774 | 0.65084 | 1.04115 | 0.84732 | 0.98699 | + +Both sides converge to 1 in the same way, so the mean/boundary Ward relation is not a +one-sided coincidence at `p_ref`. + +## 7. The four certified checks + +Let `I3` = the 3×3 covariance `Cov(L,K,B)`; PSD is certified **exactly** by the sign of every +principal minor (not by a float eigenvalue). + +| check | statement | result | +|---|---|---| +| (I1) | `d/dp log nu == E[K]/p - E[B]/q` | w=2..6 **exact equality** (rational); w=7,8 float64, abs diff ≤ 1e-11 | +| (I2) | `d²/dp² log nu == Var(S) - (E[K]/p² + E[B]/q²)` | same | +| (I3) | `Cov(L,K,B)` PSD (all principal minors ≥ 0) | w=2..6 **yes**; smallest 3×3 minor > 0; 2×2 `Cov(K,B)` det > 0 | +| (I4) | `nu_activity == nu_tagged` (same p, same w) | **yes**, w=2..7 | + +Exact-run evidence (`out/tA_exact_ref.json`, `tA_exact_half*.json`; all rows +`exact=True`, `p = 5927460507921/10^13`): + +| w | I1 | I2 | I3 (2×2) | I3 (3×3) | I4 | seconds | +|---|---|---|---|---|---|---| +| 2 | exact | exact | PSD | PSD | equal | 0.0 | +| 3 | exact | exact | PSD | PSD | equal | 0.0 | +| 4 | exact | exact | PSD | PSD | equal | 0.4 | +| 5 | exact | exact | PSD | PSD | equal | 4.1 | +| 6 | exact | exact | PSD | PSD | equal | 140 | +| 7 | exact | exact | PSD | PSD | equal | 198 | + +(w=7 was certified exactly at `p=1/2` rather than `p=p_ref`: with `p_ref` as a rational with +denominator 10¹³ the 126-lump Gauss–Jordan elimination did not finish in a reasonable budget, +so I ran the *same* exact test at `p=1/2` (`--point given --p 1/2`, n_lumps 126, 198 s) and +left the `p=p_ref` exact w=7 job running/killed. The identities are calculus statements, so a +certification at a different `p` certifies the same code path. At `p_ref`, w=7 is float64 +(see the main table).) + +At `p_ref` the exact identities hold with **zero** difference (not "small"): the tagged +side uses analytic `d/dp, d²/dp²` of the resolvent (`tagged_density_derivatives`), and the +activity side uses the first/second moments of `S` — the two are different code paths built +from different representations of the same `nu(p)`. + +## 8. The four verdicts (#774) + +`MEAN_WARD` — **compatible**. +`E[B]/E[K]` at `p_ref`: 0.7253, 0.6962, 0.6903, 0.6884, 0.6877, 0.6874, 0.6873 for w=2..8 vs +`q/p = 0.687062`. `mean_ratio - 1` = 5.56e-2, 1.33e-2, 4.75e-3, 1.98e-3, 9.75e-4, 5.37e-4, 3.21e-4 +— monotone, and the deviation *shrinks faster than any of the pre-declared powers* +(`w^{55/48}·dev` is itself decreasing). Both pre-declared λ-sides converge to 1. This is a +**7-point (per-side 5-point) compatibility**, not a proof: the effective exponent 3.73 is an +empirical fit over w≤8, and I cannot exclude a limit such as `q/p + O(w^{-3})` with a tiny +residual bias. + +Log–log trend fits over w=2..8 (`out/kb-locking.json` → `trend_fits`): +`mean_ratio-1 ~ w^{-3.73}`, `1-ward2_ratio ~ w^{-0.34}`, `lambda_min/trace ~ w^{-1.07}`, +`angle(vs (p,q)) ~ w^{-1.29}`, `cond ~ w^{+1.09}`, `1-Corr(K,B) ~ w^{-1.71}`, +`E K/w^{91/48} ~ w^{-0.015}`, `E B/w^{91/48} ~ w^{-0.050}`. +(These are least-squares slopes of `log|quantity|` vs `log w` over the 7 widths — descriptive, +not a claim that a single power law holds.) + +`SECOND_WARD` — **compatible, but slowly and least well determined**. +`ward2_ratio = Var(S)/(E[K]/p² + E[B]/q²)` = 0.5935, 0.6391, 0.6712, 0.6962, 0.7162, +0.7324, 0.7459 (w=2..8). `ward2_ratio - 1` is monotone increasing toward 0 and its per-step log-slope is +still *rising* (0.29 → 0.38), i.e. the data have not flattened onto a plateau. A pure power +fit over w=2..8 gives `1 - ward2_ratio ≈ 0.58 w^{-0.34}` (fitted slope −0.34). Verdict: the trend is toward 1; +with only 7 widths I cannot rule out a small positive limit (`~1 - c w^{-0.4}` and +`→ 0.97` differ by less than the fit scatter over this range). So: compatible, **not** +established. + +`COVARIANCE_LOCKING` — **toward rank-one, along `(p,q)`**. +For `Cov(K,B)`: the principal eigenvector's angle with the critical ray `(p,q)` falls +19.91°, 11.32°, 7.77°, 5.86°, 4.69°, 3.90°, 3.34° (w=2..8; the angle with `(q,p)` also +falls but slowly, 29.2°→24.4°); `Corr(K,B)` rises 0.804 → 0.982; the condition number rises +32.8 → 138.5 and `λ_min/trace` falls 0.0296 → 0.00717 (`~ w^{-1.07}`). The **mean** vector `(E K, E B)` aligns with `(p,q)` to +within 0.03° at w=7 (`E K/E B` → `p/q`), so both the mean and the leading covariance +direction point at the same critical ray while the matrix loses rank in the other direction. +The λ-points show the same (Corr 0.77 → 0.98 on both sides). + +`STRONG_RANDOM_RAY` — **plausible, with an unexpected bonus**. +`E[K]/w^{91/48}` = 1.883, 1.878, 1.860, 1.855, 1.851, 1.848, 1.847 and +`E[B]/w^{91/48}` = 1.366, 1.308, 1.284, 1.277, 1.273, 1.271, 1.269 (w=2..8). Both are flat to +~1% over seven widths (`~ w^{-0.015}` and `~ w^{-0.050}` respectively), and their ratio equals `p/q` in the limit, i.e. **one constant +`A ≈ 3.16` gives both `E K ≈ A p w^{91/48}` and `E B ≈ A q w^{91/48}`**. So the strong random +ray `w^{-91/48}(K,B) ⇒ A(p,q)` is *supported* at `p_ref`. +Two honest caveats: (i) at `p_ref` the black infinite-cluster density is `θ(p_c)=0`, and the +finiteness is carried by `nu_w`; I additionally observed `nu_w·w = 0.2987, 0.2891, 0.2893, +0.2889, 0.2888, 0.2888, 0.2887` (w=2..8) — i.e. `nu_w ≈ 0.2888/w` at `p_ref` to 0.35%, which is what +makes `EK` scale as `w^{91/48}` while `(nu/w)EK = θ_w → 0`. (ii) These are 7 widths; the +visual flatness of `w^{-91/48}` over `w≤7` cannot by itself prove the exponent — the exponent +is exactly the ambiguity that a wider range would settle. + +## 9. What must NOT be claimed + +* Not a theorem about planar critical percolation. Everything above is a **finite-width + (w ≤ 8) exact/numerical family** on a cylinder; the "convergence" statements are + **compatibilities over 7 widths**, not limits. +* The `91/48` ray is **not** fitted from w≤8: I only checked that the pre-existing exponent + describes the data; I did not fit an exponent and then declare it. +* `B` here is the **distinct external boundary**; none of these numbers may be reused as + "edge perimeter" results (they would differ). +* `p_ref` is a diagnostic reference only; no claim about `p_c` is made. +* Black-cluster quantities here must not be extrapolated to white clusters (and vice versa): + the black `Corr(L,K)²` decreases while the white one increases (BRIEF §5). +* No Monte Carlo, no GPU, no new transfer engine was used, and no #275 source contract or + #760 gamma-vs-kappa result is touched. + +## 10. Reproduce + +``` +scripts/engine_relaxed.py # 1-line raise-only copy of the pinned engine (w<=8 for the activity builder) +scripts/taskA.py # all Task-A numbers; --widths --mode float|exact --point ref|lam --p ... +scripts/diag_marks.py # the independent brute-force B-semantics check +scripts/test_jm.py # validation of the moment extractor against the shipped #772 numbers +out/tA_float_ref.json out/tA_float_ref_w8.json out/tA_float_lam.json +out/tA_exact_ref.json out/tA_exact_half*.json out/diag_marks.json out/kb-locking.json +``` +Run as `cd /workspace/ward774 && PYTHONPATH=/workspace/mo/compat PY39COMPAT_MARKER=1 python3 scripts/taskA.py ...` diff --git a/docs/manuscripts/geometric-balance/white-component-mean-reward-20260914.md b/docs/manuscripts/geometric-balance/white-component-mean-reward-20260914.md new file mode 100644 index 000000000..6a6940632 --- /dev/null +++ b/docs/manuscripts/geometric-balance/white-component-mean-reward-20260914.md @@ -0,0 +1,279 @@ +# Mean volume and boundary rewards of the giant complementary white component + +2026-09-14. This note upgrades part of `supercritical-white-slab-bulk-20260914.md` from conjecture to an author-level theorem. The **samplewise slab LLN** remains open, but the component-Palm means are fixed by a local essential-membership limit plus exact mass transport. + +## 1. Setup + +Fix black NN site density + +\[ +0p_c(G8). \tag{1.1} +\] + +Work on the cylinder + +\[ +C_w\times\mathbb Z. \tag{1.2} +\] + +Black uses NN connectivity and white uses matching connectivity from the same Bernoulli labels. + +Let + +\[ +\nu_w=\nu_w^4(p)=\nu_w^8(q) \tag{1.3} +\] + +be the common row intensity of complete essential components. For a white essential component under component Palm, write + +\[ +L_w=\text{vertical span}, +\quad +N_w=\text{number of white sites}, +\quad +B_w=\text{number of distinct black external boundary sites}. \tag{1.4} +\] + +The previous alternating-barrier theorem gives + +\[ +\nu_wE_WL_w\to1. \tag{1.5} +\] + +## 2. Probability that a fixed cylinder site belongs to a white essential component + +Let + +\[ +\rho_w(q) +=P_q^{C_w\times\mathbb Z} +(0\text{ belongs to a white matching essential component}). \tag{2.1} +\] + +Let + +\[ +\theta_8(q) +=P_q^{\mathbb Z^2}(0\leftrightarrow\infty\text{ in }G8). \tag{2.2} +\] + +We claim + +\[ +\boxed{\rho_w(q)\to\theta_8(q).} \tag{2.3} +\] + +### 2.1 Upper bound + +If the cylinder component of the origin is horizontally essential, lift it to the plane. The lift contains a path from the origin to a nonzero horizontal translate of itself, hence it exits every Euclidean ball of radius `infinity`, the decreasing one-arm events converge to membership in the infinite cluster. Hence + +\[ +\limsup_{w\to\infty}\rho_w(q)\le\theta_8(q). \tag{2.5} +\] + +### 2.2 Lower bound by a high-probability white frame + +Because black NN density `p` is subcritical, black crossings of a rectangle in its long direction have probability at most `C e^{-cw}` uniformly over any fixed finite collection of aspect ratios. + +For the square-site matching pair, the standard finite-rectangle matching dichotomy says that failure of a white matching crossing in one direction forces a black NN crossing in the transverse direction. Consequently all white matching rectangle crossings in a fixed finite gluing scheme of scale `w` occur with probability + +\[ +1-O(e^{-cw}). \tag{2.6} +\] + +Choose constants + +\[ +0infinity`, + +\[ +\liminf\rho_w(q)\ge\theta_8(q). \tag{2.10} +\] + +Together with (2.5), this proves (2.3). + +The only planar input beyond subcritical black sharpness is the finite matching crossing dichotomy already inherent in the square-site 4/8 convention; no supercritical white correlation length is imported. + +## 3. Exact Campbell identity for white component volume + +Anchor every white essential component at its lowest row as in the preceding notes. For each component `C` with anchor row `a(C)`, send one unit of mass from `a(C)` to every white site of `C`, recording the target row. + +Stationarity in the vertical coordinate gives the discrete Campbell/mass-transport identity + +\[ +\boxed{ +\nu_w E_WN_w +=E[\text{number of row-0 white sites belonging to essential components}].}\tag{3.1} +\] + +Horizontal translation symmetry makes the right side + +\[ +w\rho_w(q). \tag{3.2} +\] + +Hence exactly at every finite width, + +\[ +\boxed{ +\frac{\nu_w}{w}E_WN_w=\rho_w(q).} \tag{3.3} +\] + +Using (2.3), + +\[ +\boxed{ +\frac{\nu_w}{w}E_WN_w\to\theta_8(q).} \tag{3.4} +\] + +Combine with (1.5): + +\[ +\boxed{ +\frac{E_WN_w}{wE_WL_w}\to\theta_8(q).} \tag{3.5} +\] + +This is a ratio-of-component-Palm-means theorem. It does not yet assert `N_w/(wL_w)->theta` samplewise. + +## 4. Boundary mean from the exact dual surface-excess identity + +The exact complementary component score on this branch gives + +\[ +\boxed{ +qE_WB_w-pE_WN_w +=pq\,\partial_p\log\nu_w^4(p).} \tag{4.1} +\] + +The black component-Palm activity estimate gives at every fixed subcritical `p` + +\[ +|\partial_p\log\nu_w^4(p)|=O_p(w). \tag{4.2} +\] + +Meanwhile `nu_w` is exponentially small in `w`. Multiply (4.1) by `nu_w/w`: + +\[ +q\frac{\nu_wE_WB_w}{w} +-p\frac{\nu_wE_WN_w}{w} +=O_p(\nu_w). \tag{4.3} +\] + +Using (3.4), + +\[ +\boxed{ +\frac{\nu_w}{w}E_WB_w +\to\frac{p}{q}\theta_8(q).} \tag{4.4} +\] + +Equivalently, + +\[ +\boxed{ +\frac{E_WB_w}{E_WN_w}\to\frac{p}{1-p}.} \tag{4.5} +\] + +This recovers the infinite-cluster one-site-flip density + +\[ +\beta_8(q)=\frac{p}{q}\theta_8(q) \tag{4.6} +\] + +at the level of giant-cylinder-component Palm means without separately proving a boundary mixing theorem. + +## 5. Mean reward vector + +Equations (1.5), (3.4), and (4.4) may be collected as + +\[ +\boxed{ +\left( +\nu_wE_WL_w, +\frac{\nu_w}{w}E_WN_w, +\frac{\nu_w}{w}E_WB_w +\right) +\longrightarrow +\left( +1, +\theta_8(q), +\frac{p}{q}\theta_8(q) +\right).} \tag{5.1} +\] + +Thus the three leading Palm means contain no unknown morphology amplitudes. + +Since `E L` is order `1/nu`, both additive rewards are order `w/nu`, as expected for an exponentially long supercritical slab. + +## 6. Separation from the stronger samplewise bulk law + +`supercritical-white-slab-bulk-20260914.md` conjectures + +\[ +\frac{N_w}{wL_w}\to\theta_8(q), +\qquad +\frac{B_w}{wL_w}\to\frac{p}{q}\theta_8(q) \tag{6.1} +\] + +in probability under component Palm. + +The present theorem is strictly weaker but already nontrivial: + +\[ +\frac{EN_w}{wEL_w}\to\theta_8(q), +\qquad +\frac{EB_w}{EN_w}\to p/q. \tag{6.2} +\] + +Proving (6.1) still requires controlling internal fluctuations/conditioning inside a random exponentially long slab. It should not be inferred from the mean identities alone. + +## 7. Archive-facing checks + +A component-Palm simulation of the huge white complementary component can therefore be checked against three levels: + +1. already proved: `nu E L ->1`; +2. proved here: `nu E N / w -> theta_8(1-p)`; +3. proved here without knowing `theta`: `E B / E N -> p/(1-p)`. + +Only after these mean checks pass should one test the stronger samplewise LLN or Gaussian bulk fluctuations. + +## 8. Claim boundary + +The Campbell identities are exact. The local essential-membership limit uses an author-level supercritical white frame built from the standard square-site matching crossing dichotomy plus subcritical black exponential crossing decay. The full samplewise slab LLN/CLT remains conjectural. \ No newline at end of file diff --git a/docs/research-bridges-post-compass-20260914.md b/docs/research-bridges-post-compass-20260914.md new file mode 100644 index 000000000..b09611a3a --- /dev/null +++ b/docs/research-bridges-post-compass-20260914.md @@ -0,0 +1,348 @@ +# 研究桥接推进：从校准文件到两个可判定机制 + +2026-09-14。本文接在 `docs/research-compass-beyond-exactness-20260914.md` 之后，目的不是再增加一层任务管理，而是把本轮真正改变判断的推理压成两个 bridge： + +1. Matching root 的 leading non-common correction 如何被拆成 microscopic regularization mismatch、spin character、thermal tangent/normal response 与最终 CFT module； +2. fixed-subcritical common-label process 如何从已有 component Poisson proof 进入 space × intensity-clock marked Poisson cloud，而不是继续求抽象 splitting kernel 的更多后果。 + +现有精确结果不重复证明。以下若无额外注明，新的 continuum/module 解释均为 conjecture/programme；AGG/BK 部分是对 #739 已写证明的重新组合，需独立审计后才能升级为 author-proof。 + +## A. Root correction：先分 regularization coupling，再谈 field 名称 + +固定宽度 safe transfer 的两条对象 + +```text +I4,w^0(p), +I8,w^0(1-p) +``` + +在 critical limit 流向同一个 magnetic/topological continuum sector。已有 `x_m=5/48`、level-one momentum 与 oblique magnetic-metric controls 支持这一点。因此最自然的 RG 组织不是先假定“两个 continuum state 的 matrix element 不同”，而是把它们看作同一 continuum state 的两个 microscopic regularizations。 + +写 + +```text +E_a(ell,p) + = ell^-1 Phi(X, + u_0^(a) ell^y0, + u_4^(a) ell^y4, ...), +X=t(p) ell^yt, +yt=3/4, +``` + +其中 `a=4,8`。matching/complement map 允许把 lattice couplings 分成 common/even 与 difference/odd： + +```text +u_j^+ = (u_j^(4)+u_j^(8))/2, +u_j^- = (u_j^(4)-u_j^(8))/2. +``` + +lower-dimensional common corrections 可以在每条 gap 里很大，却从 `Delta=I4-I8` 消掉。matching root 只对 `u^-` 敏感。 + +### A1. 当前最强 finite evidence 已经先确定 spin character + +oblique safe transfer 在同一个 square-site NN / complementary-matching microscopic model 上只改变 period direction，得到 + +```text +Delta(pc) ~ cos(4 theta) ell^-17/4, +p_root-pc ~ -cos(4 theta) ell^-4, +``` + +且 `(1,0),(1,1),(2,1),(3,1),(3,2)` 在 physical normalization 后 collapse 到共同 amplitude；near-node `(5,2)` 同样被显著压低。 + +因此当前最保守的优先排序应是： + +```text +leading observed irrep: spin 4 >> spin 0, +module identity: unresolved. +``` + +scalar `x=21/4` 仍可作为 subleading/hidden contribution，但不应再与 spin4 作为 leading observed amplitude 的等权默认解释。真正需要测的是同阶 spin-0 projection，而不是再拟合一个 exponent 4。 + +一个干净的 equal-circumference pair 是 + +```text +axis: u=(1,0), n=10, ell=10, +oblique: u=(3,4), n=2, ell=10, +H4(3,4)=-527/625. +``` + +同一 `ell` 上写 scaled critical mismatch + +```text +Y(theta)=B0+B4 H4(theta)+higher. +``` + +两点直接解 `B0,B4`，避免把不同 circumference 的 ordinary finite-size remainder误认成 angular harmonic。 + +### A2. RG-improvement interpretation + +若 leading difference coupling 是 thermal-family spin4， + +```text +y4 = 2-(xt+4) = -13/4, +xt=5/4. +``` + +则 critical mismatch 的 per-length exponent自动是 + +```text +y4-1 = -17/4, +``` + +而 root shift 是 relevant thermal tuning 对 irrelevant mismatch 的补偿： + +```text +p_w^*-pc ~ ell^(y4-yt)=ell^-4. +``` + +这把 matching root 解释成有限尺寸的 improvement condition：用一个很小的 thermal displacement 抵消两个 regularization 之间第一个 dual-odd irrelevant coupling。它也直接解释 square/kagome 的 4 vs 6：若更高旋转对称性强制 `u_4^-=0`，下一允许 coupling 才决定 pseudo-critical shift。 + +### A3. 更强猜想：TANGENT_SPIN4 + +仅有 root shift 仍不能说明 spin4 是否改变 universal charge curve 的形状。设 physical dimensionless charge function + +```text +Q_{ell,theta}(X) + = F(X)+u4(ell) H4(theta) G4(X)+..., +u4~ell^-13/4. +``` + +一般分解 + +```text +G4(X) + = [G4(0)/F'(0)] F'(X) + G4_perp(X), +G4_perp(0)=0. +``` + +第一项只平移 root；把每个方向自己的 root 移回0后，只剩 `G4_perp`。 + +强猜想为 + +```text +G4(X)=c4 F'(X), +``` + +即 leading spin4 对这个 observable 完全沿 thermal tangent。则 + +```text +Q_{ell,theta}(X) + = F(X+c4 u4 H4)+o(u4), +``` + +所以会同时出现清楚的 orientation-dependent root shift 与近乎为零的 fixed-b/intrinsic normal response。 + +最直接的检验不是更多 root，而是 root-centered oblique curves。进一步用各自 root slope 归一 thermal metric： + +```text +s_theta=dQ/dp|p*, +Psi_theta(y)=Q_theta(p*+y/s_theta). +``` + +若 `Psi_axis-Psi_oblique` 的 cos4 component 比 raw critical/root signal 明显降一个阶，则支持 tangent mechanism；若仍保持 `ell^-13/4`，则 spin4 有真正 normal shape component，后续 module/matrix-element identification才有高价值 target。 + +这也是 #802 中 `(T,N)` 的一个直接 actual-site realization： + +```text +T = root translation, +N = root-centered, slope-normalized shape response. +``` + +### A4. c=0 module identity：generic-Q 解简并比 Q=1 贴标签更干净 + +generic-Q Potts 表示论给一个关键事实：diagonal `(h_{r,1},h_{r,1})` Kac fields 的 null descendants 在 generic Q 上是真正 quotiented；energy `Phi_{2,1}` 因此有明确的 Kac module。到 `Q=1,c=0`，energy `Phi_{2,1}` 与 2-hull `Phi_{0,2}` 在 `(5/8,5/8)` 碰撞并形成 logarithmic pair。 + +于是它们的 spin4 descendants也在 `x=21/4` 碰撞： + +```text +Q4 Phi_{2,1}, +Q4 Phi_{0,2}. +``` + +所以 `Q=1 + x=21/4 + cos4theta` 仍不能决定 module。若 generic-Q 两支 amplitudes 在 `Q->1` regular，则得到 ordinary `ell^-17/4`; 若系数以 `±1/(Q-1)` 相消，极限会生成 + +```text +ell^-17/4 [A+B log ell]. +``` + +因此 module/Jordan 判别应优先做 generic-Q branch tracking 或 Q-velocity，而不是自由拟合 `A+B log L`。这条路线与 TANGENT_SPIN4 正交：前者问 field/module，后者问该 field 对目标 observable 的 correction function 是 tangent 还是 normal。 + +## B. Common-label bridge：原 Poisson proof 可能已经包含 no-merger 与 marked process + +#780 定义 early parameter `p_-` 与 final parameter `p_+`。对每个 `p_+` final essential component C，令 `n_C` 是它包含的 distinct `p_-` essential lineages 数，并定义 merger factorial density + +```text +M_w(p_-,p_+) + = E sum_{C: final anchor row 0} binom(n_C,2). +``` + +原任务把 `M_w/nu_- ->0` 当作新的 blocker。重新读取 #739 的 `poisson-birth-windows.md` 后，这一步似乎可以由旧 AGG/BK/localization 估计直接推出。 + +### B1. 短 final component：merger pair 是旧 `b2` pair 的子集 + +取 `H=w^2`。若 final C 的 span `L(C)<=H` 并含两个 early components，则它们的 unique anchors 行距 `<=H`；early components 本身也是 C 的子集，所以 span也 `<=H`，由 #739 的 localized anchor indicators 精确计数。 + +#739 §6 已证明，对任意两个重叠 anchor windows 中的不同 full winding components，各自存在 disjoint occupied winding witnesses，site-BK 给 + +```text +E[I_i I_j] <= U_w(p_-)^2, +``` + +其中 + +```text +U_w(p)=poly(w,H) exp[-(w-1) kappa(p)]. +``` + +每行的 close-pair 数至多 `O(w^2 H)` 个，所以 + +```text +M_w^short <= poly(w) exp[-2 kappa(p_-) w]. +``` + +这就是原 component-Poisson proof 的 `b2/m`，只是按 lineage-merger 语义重新读取。 + +### B2. 长 final component：旧 localization tail 是 superexponential + +对 `L(C)>H`，粗界 + +```text +n_C <= |C| <= w L(C) +``` + +给 + +```text +M_w^long + <= (w^2/2) E sum_{C:anchor0} L(C)^2 1{L(C)>H}. +``` + +#739 §4 的 strip-crossing proof 对任意 `r>=w` 给 unnormalised span-tail intensity + +```text +E[# {C:anchor0,L(C)>r}] <= C_I w exp(-c_I r) +``` + +uniformly on compact subcritical intervals。用 tail-sum identity 得 + +```text +M_w^long <= poly(w,H) exp(-c_I H) + = exp[-Theta(w^2)]. +``` + +因此 + +```text +limsup (1/w) log M_w <= -2 kappa(p_-), +M_w/nu_w(p_-) ->0, +``` + +因为已有 `-log nu_w/w -> kappa(p_-)`。这甚至不要求 `p_+-p_-=O(1/w)`；任意固定 compact-subcritical pair 都有相同尺度分离。 + +若审计无缺口，no-macroscopic-merger 不再需要新的大计算。 + +### B3. no-merger 以后，mark process 直接继承同一个 AGG dependency graph + +在 bounded intensity-clock window 上固定 top parameter `p_+`。对每个 localized final component C，用同一份 U-label filtration定义 first essential birth mark `tau(C)`。rare merger可先任意 tie-break，其总质量由 `M_w` 控制。 + +由于 final component 不碰 guard rows，所有较低 p 的 ancestors 也是同一 local window 内的子图；`tau(C)` 是 local label function。把 parameter window 分成有限 bins `A_r`，定义 typed indicators + +```text +I_{j,x,r} + =1{final p_+ anchor at (j,x), tau(C) in A_r}. +``` + +不相交 windows exact independent；重叠 windows 的 pair bound仍由两个 p_+ disjoint winding witnesses控制。因此 AGG process theorem 原样给有限 mark bins 的 independent Poisson approximation。 + +### B4. mark intensity 不需要 p-analyticity + +对任意中间 p，每个 p-essential component 唯一属于一个 p_+ final component，故单位长度精确有 + +```text +nu_w(p)=E sum_C n_C(p). +``` + +而 birth mark `<=p` 的 final-component intensity 是 + +```text +mu_w((-,p])=E sum_C 1{n_C(p)>=1}. +``` + +逐 C 有 + +```text +0 <= n_C-1{n_C>=1} <= binom(n_C,2), +``` + +所以 + +```text +0 <= nu_w(p)-mu_w((-,p]) <= M_w(p,p_+). +``` + +若参数仅按 intensity ratio 定义 + +```text +nu_w(p_w(x))/nu0 -> Lambda(x), +``` + +则 `M_w/nu0->0` 立即给 + +```text +mu_w((x1,x2])/nu0 + -> Lambda(x2)-Lambda(x1). +``` + +这已经识别 marked Poisson 的 mean measure，不需要先证明 `nu(p)` analytic，也不必先写 `p_w=p0+x/(vw)`。 + +birth-anchor 与 final-anchor 的位置差至多 O(H) on localized event，而 + +```text +H nu0 = w^2 exp[-kappa(p0)w+o(w)] ->0, +``` + +因此在 rescaled longitudinal coordinate 上 anchor motion 自动消失。 + +若 finite-mark partition -> PRM 的标准升级及 torus/cylinder coupling无隐藏问题，目标 birth cloud应为 + +```text +sum_C delta_(nu0 y_C, Lambda_w(tau(C))) + => PPP(dy dLambda). +``` + +此后 #785 的 exact clock kernel/Laguerre hierarchy才成为已建立 model map 的 consequence，而不再是 bridge 本身。 + +## C. 近期待验证顺序 + +按信息增量排序，而不是按文件数： + +1. **#780 proof audit first**：审计 B1/B2 的四个 pathwise/counting 点；若通过，写成独立 lemma，并立即尝试 B3/B4 的 typed-AGG marked process。此路线理论收益远大于再做 common-label static tables。 +2. **#802 equal-ell / centered-curve test**：axis `(1,0),n=10` 与 `(3,4),n=2` 共用 `ell=10`; 同时输出 critical mismatch、thermal derivative、root 与 5--7 个 root-centered curve points。先判 `spin0 vs spin4`，再判 `tangent vs normal`。 +3. **#768 改用 RG factorization 语言**：区分 microscopic odd coupling `u_4^-` 与 universal magnetic matrix element；不要把两者都叫“sector matrix-element difference”。 +4. **#586 generic-Q reserve**：只有 actual-site centered-curve 说明 spin4 normal component确实需要 field identity时，再投入 generic-Q energy/2-hull branch tracking。若 tangent-only 已解释当前 observable，则 module研究仍有数学价值，但不是 root 机制的前置。 + +## D. 当前大胆但可失败的统一猜想 + +一个较强的工作图景是： + +```text +finite matching topology + -> fixes which global channels can differ; +microscopic regularization mismatch + -> supplies a small dual-odd spin4 coupling; +thermal RG response + -> converts it into the L^-4 pseudo-critical root shift; +rare complete-component AGG/BK structure + -> supplies a Poisson space×intensity birth cloud; +record/splitting kernels + -> are consequences of that cloud, not independent mechanisms. +``` + +在这个图景中，项目近期真正未知的低维对象很少：root 侧是 `spin irrep + tangent/normal + module` 三层；common-label 侧是 `local marked birth map` 一层。若后续验证继续支持这种压缩，应主动减少派生模型任务，而把资源集中到能击穿这两个 bridge 的反例上。 + +## Claim boundary + +- oblique root/free-energy values、existing AGG/BK/localization inequalities 是现有 branch 的 deterministic/author-proof inputs，本文不重新认证。 +- `regularization mismatch` 与 `TANGENT_SPIN4` 是新的 RG interpretation/conjecture。 +- energy–2-hull generic-Q collision 是文献支持的 representation-theory input；其对 Matching-One safe-sector correction 的具体投影仍未识别。 +- `M_w/nu_->0` 与 marked-PPP route 是从现有 #739 proof 组合出的 candidate author-proof；在逐行审计写入 branch 前不得升级成已证 theorem。 diff --git a/docs/research-compass-beyond-exactness-20260914.md b/docs/research-compass-beyond-exactness-20260914.md new file mode 100644 index 000000000..8df888618 --- /dev/null +++ b/docs/research-compass-beyond-exactness-20260914.md @@ -0,0 +1,181 @@ +# 从精确结果回到真正的问题：研究方向校准 + +2026-09-14。回应所有者关于再次陷入“精确陷阱”的担忧。此次读取的研究分支快照为 #771 `e9b473d7d5d64e2cd30d9c85c3277a6fab75f850`；同时读取 main README、研究入口、路线图、#650 最新分配评论、#275 及本分支的主张账本与扇区修正说明。 + +这是一份研究判断，不是新的定理、验收规则或发表计划。没有重证所有旧论证，没有把账本上的标签当作正确性证书。它不撤回原始数据，不关闭问题，也不要求中断团队已经运行的工作。 + +## 1. 判断：真正的问题不是太精确，而是可解性开始替代问题的重要性 + +我们已经有多次真正改变理解的发现。但近期也出现了一个重复模式：找到一个可处理的极限对象，随后不断求它的矩、联合变换、条件分布、时间反演、谱系与生成元。每一项都可能正确，也可能令人愉快；然而，它们越来越多地回答“这个已选定模型还有什么性质”，而不是“为什么原始点渗流会选择这个模型，它在哪些尺度失效，哪个微观差异决定原始响应”。 + +我此前反复用每轮结果数、精确检查数和新文件数组织汇报，强化了这个倾向。这些数量说明执行范围，不能衡量认识的增量。将一次近似的后果计算得更精确，也不能减小尚未控制的模型近似误差。 + +这里需要区分两种完全正当的研究动机： + +- 因为一个数学对象本身美丽而研究它；这无需为项目主线作贡献的证明。 +- 因为它能解释 Matching One 而研究它；这需要指出连接在哪里。 + +问题不在第一种动机，而在把它包装成第二种。自由探索可以继续，但不应让每个新推论自动成为全项目下一项任务。 + +## 2. 将许多结果压缩成四项认识 + +### 2.1 拓扑约束减少了真正独立的全局变量 + +有限 matching 对偶、rank 出生反射、rank-one 同调方向的持久性，以及 `W4-W8=r4-1`，提供的是同一组全局约束的不同侧面。它们告诉我们哪些联合分布不可能，而不只是给某个参数点一个漂亮的数。 + +特别地，同参数黑白绕行簇数的广延部分只沿 `(1,1)` 方向波动；差值是有界拓扑量。这个事实与内部体积/边界具有两个涨落方向并不矛盾：对象不同，信息尺度也不同。参见 [R1]。 + +### 2.2 根的位置与整条出生分布，不是同一个问题 + +现有几何论证区分了根一致性与全出生分布集中所需的周期条件；指数长条的两中心理论，又将局部连接代价和全系统的绕行机会数联系起来。这是从特殊几何向一般结构的推进。 + +这些全尺寸结论应沿各自的实际论证和输入使用，不能由本文的方向判断再升级一次。对于已经有完整作者推导的几何问题，后续应回应具体数学缺口，而不是默认再扩大几何普查。参见 [R2]。 + +### 2.3 许多不同的分布，是同一个随机时钟的不同读法 + +稀有屏障、白色间隔、内部体积与评分形成了一条重要连接。指数长度、位置取样下的 Gamma 长度、Gaussian–exponential 混合的 Laplace 评分、随机信息反演中的 Student 分布，以及切分/合并谱系，不能被当作许多相互独立的微观机制。 + +最有解释力的共同结构是： + + 随机区间长度 / 暴露量 + + 区间内的条件噪声 + + 取样与历史的读取方式。 + +从 Bernoulli 簇到这个结构的映射，是研究连接；模型已经确定以后的一批分布公式，是模型的后果。后果可以帮助验证接口或发现新的结构，但它们不反向提供多份独立证据来支持映射本身。参见 [R3]。 + +### 2.4 状态维数、噪声与记忆，都相对于观察问题 + +完整微观状态、强合并、标量递推阶数、有限精度有效秩、单时刻分布、整条参数过程，是不同对象。查询孔洞的第二内部方向、单时刻相同而记忆不同的过程，以及旧的 Jordan 可见性反例，都指向这一认识。 + +这已经是一项足够强的概念收获。不需要无限增强反例，才能承认“相同指数、低秩或单时刻分布不能识别机制”。original-U 的实际候选仍须给出原始源与正规化上的预测，不能被新的容易区分的观察量替代。参见 [R4]。 + +## 3. 最重要的断点：典型对象的精确理论，仍可丢掉决定根的信息 + +在内部参数上，定义 + + chi = P0 + P2, + vartheta = log(P2/P0). + +精确地， + + M = chi tanh(vartheta/2). + +所以根由 `vartheta=0` 决定，而 `chi` 决定这两个扇区一共出现得多频繁。在长条的 rank-one 平台里，`chi` 可以极小。此时大部分配置、绝大部分簇统计都可以已被一个极好的极限模型描述，而决定根的比值仍然没有被近似到需要的精度。参见 [R1,R5]。 + +一个简单的代数说明：若 `P0=epsilon a(p)`、`P2=epsilon b(p)`、`P1=1-epsilon(a+b)`，其中 a,b 正且有界，那么 epsilon→0 时所有模型都趋向 rank-one；但根的位置由 `a=b` 决定，可以不同。这不是新的 site 反例，只说明弱极限丢弃了什么信息。 + +因此，以下箭头不能省略： + + 完整簇的典型极限 + --需额外的扇区相对误差/自由能比较--> + P2/P0 或两个无绕行扇区的自由能差。 + +普通的绝对 o(1) 误差，不足以确定两个远小于误差的概率谁大。知道完整簇密度，也不等于知道无绕行概率的全部指数率；特定区域中的渐近等价需要自己的证明。 + +这解释了当前的“精确陷阱”：我们可能在把一个已经丢掉目标信息的对象解得越来越彻底。不是这些解没有数学价值，而是不能期待原问题的答案从丢掉的信息中自行恢复。 + +## 4. 不把不同极限区间拼成一套已经成立的统一理论 + +| 区间 | 主要机制 | 不能自动运送过去的结论 | +|---|---|---| +| 固定小宽度，p→0 | 最小局部图案、离散屏障与孔洞 | 不自动给固定 p、大宽度的记忆和缝合 | +| 固定严格亚临界 p，w→∞ | 稀有长连接、有限相关长度、巨大对偶间隔 | 不自动给 p→pc 时的一致误差 | +| log(m)/w→d>0 | 指数机会数与连接代价竞争 | 不等于固定长宽比的二维临界极限 | +| p随w趋向pc，m/w有限或增长 | 临界多尺度连接、拓扑扇区耦合 | 不能直接套独立 Poisson 屏障或旧固定-p余项 | + +边界上的连接往往比区域内部再求一个公式更重要。 + +D’Alimonte–Manolescu 的近临界 OZ 工作确实提供有价值的工具，但本轮核对的原文定义是键随机簇模型；它不能直接成为 square-site 完整周期簇的精确缝合定理 [L3]。这一点是模型区别，不是发表门槛。 + +同理，Poisson 记录及 Dirichlet 切分/合并已有成熟文献 [L2,L4]。真正需要研究的是我们如何进入这些对象，哪些参数、条件化和拓扑信息能够带过去，而不是是否还能给它们加一个新的可求解标记。 + +## 5. 下一阶段只给两个问题持续的主要注意力 + +### 主问题 A：Matching One 的第一项非公共修正，为什么能够留下？ + +使用已有扇区自由能差 + + Delta_w(p)=I^0_(4,w)(p)-I^0_(8,w)(1-p). + +根由 Delta_w=0 决定。在满足局部非退化性和余项控制时，根偏移由 `-Delta_w(pc)/Delta'_w(pc)` 决定。这不是新发现；它把当前未知量定位清楚了：我们已经知道应该比较什么，但尚未完整解释哪些较低阶修正必定在差值中消失。 + +研究的第一目标不是给 21/4 配一个更漂亮的场名，而是识别： + + 哪个修正在两个扇区中真的相同？ + 哪个修正具有非零差矩阵元？ + 这种相同来自恒等式、对称性、特定格子，还是仅是现有尺寸上的近似？ + +现有 spin-four 热族假说比仅猜一个标量维数更有结构，但仍是候选。旋转允许性不证明它最低、不证明振幅非零，也不排除更低的标量/对数通道；总导数、零向量与接触项必须按实际插入处理。#771 的最新说明已经承认这一点 [R5]。Jacobsen 的扇区本征值框架提供直接起点，而不是另建一套猜指数方法 [L1]。 + +可实际推进的两条路，不要求同时开新队列： + +1. 在已有真实 site 转移和源上，分离平均扇区修正、差值及热导数。让两个具体机制给出不同的角度/对称性/参数扰动响应，而不是只看同一条根曲线。改变几何时须保留或显式控制模参数、热漂移和已有算术混杂；不能把这些变化事后解释为 spin。 +2. 从两个扇区的矩阵元与允许插入出发，排除至少一个真实的低阶竞争项。一个明确的非消去机制或反例，也比继续匹配更多幂次有价值。 + +复用 #768/#585 和已有 charge 分析；没有新的前向区分时，不增加角度/宽度生产。这个问题与 original-U 可能最终相接，但在源映射完成前不是对 #275 的替代解答。 + +### 桥接问题 B：稀疏图案的切分描述，能否到达固定亚临界参数的热窗口？ + +保留 #780，但将注意力集中到实际映射，而不是继续求切分模型的更多两时刻公式。 + +固定 p0 这个结果将改变我们关于原模型的哪一个判断，或带来哪一种值得独立研究的新结构？ + +能将两个现象压缩成一个原因、证明一条跨区间映射、找到一个击穿猜想的反例，都可能是大步。反之，一个已经确定的随机模型又多一个闭式矩，通常是局部完善；除非那个矩揭示了此前没有意识到的结构。 + +合理的近期开端是：把“第一项 sector-odd 修正”中的两个真实竞争机制写清，再确定一个不会同时被两者解释的现有响应；并由另一条有界分析处理热窗口中的谱系并合。不是重新开二十个可能方向。 + +若阶段结束时只是公式更多，而“什么是共同修正、什么打破共同性、弱极限丢掉了什么”仍原封不动，应当换问题，而不是继续加精度。 + +## 8. 来源和读取范围 + +内部来源按本轮快照读取，本文只综合其结构，不重复宣称完成它们的全部证明： + +- [R1] `manuscripts/geometric-balance/round2-claim-ledger-20260914.md`，尤其 A/B/H/I；`neutral-gas-limit-collapse-20260914.md` 的内容亦已在对话中读取。 +- [R2] `docs/ROADMAP.md`、`docs/RESEARCH-FRONTIER.md`，以及 #739/#771 的范围说明。入口仍含较早分配，故不将其自动视为新任务。 +- [R3] `monotone-cut-mark-filtration-20260914.md`、`tagged-cut-records-and-future-clock-20260914.md`、`circular-coalescent-and-topological-clock-20260914.md`、`shared-label-gaussian-sheet-and-lineage-memory-20260914.md`；正文与本轮上下文已有的实际Git读回互相对照。 +- [R4] #275 当前正文、`exploration-noise-and-random-information-20260914.md`，以及查询孔洞交付。高阶矩/联合过程所需输入仍各自保留。 +- [R5] `charge-sector-scaling-hierarchy-20260914.md`、`sector-odd-spin4-anisotropy-20260914.md`。后者明确将 spin-four 热族的非零差矩阵元列为猜想，而不是本文替它完成证明。 +- 当前协调 #650 的两条路线评论已读；本文改变的是默认分析注意力，不沿用“先发表一篇”的目标。 + +外部原始来源，本轮只做针对性核对，不进行新的一轮广泛综述： + +- [L1] J. L. Jacobsen, *Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras*, arXiv:1507.03027v1。HTML摘要、模型与扇区构造段已读；用于确定方法先例及对象，不宣布具体修正场已被识别。https://arxiv.org/html/1507.03027v1 +- [L2] J. Bertoin and C. Goldschmidt, *Dual random fragmentation and coagulation and an application to the genealogy of Yule processes*, arXiv:math/0408128。本轮读取arXiv摘要；具体Kingman对应的正文核对记录来自前轮，未冒充本轮重读全证明。https://arxiv.org/abs/math/0408128 +- [L3] L. D’Alimonte and I. Manolescu, *Near-critical Ornstein–Zernike theory for the planar random-cluster model*, arXiv:2510.13648v3。HTML模型定义与条件混合相关段已读；其配置空间为边标签。https://arxiv.org/html/2510.13648v3 +- [L4] A. Gnedin, *Corners and Records of the Poisson Process in Quadrant*, arXiv:0709.1285。本轮读取摘要/书目信息；不将标准记录过程的继续求解当成新的渗流机制。https://arxiv.org/abs/0709.1285 + +结论：不要因为一个对象能被精确计算，就让它不断决定我们下一步问什么。保留数学兴趣，但把主要推理力量放在真正跨过模型、尺度与观测之间的连接上。 \ No newline at end of file diff --git a/docs/research-status/independent-audit-batch-20260914.md b/docs/research-status/independent-audit-batch-20260914.md new file mode 100644 index 000000000..fb68af964 --- /dev/null +++ b/docs/research-status/independent-audit-batch-20260914.md @@ -0,0 +1,126 @@ +# 独立核验批次：2026-09-14 外包核验产出（第三方审计，非自有推导） + +**提交者身份**：这是**操作者侧的独立核验批次**，不是项目所有者的推导。 +本批全部内容来自**独立子代理**在**孤立云机**上跑的验证/复算/审计，目标是 +「把合作者交付的包真的跑起来 + 对承重恒等式做对抗性审计」。 + +> ⚠️ **口径声明（重要）** +> 1. 本批是**第三方复现与审计**，**不声称**任何新的物理结论。 +> 2. 所有数字都由子代理在云机脚本中产出；本机 Mac 未做任何计算。 +> 3. 严格区分 **精确恒等式 / 有限枚举事实 / 有限宽度拟合 / 条件定理 / 假设 / 猜想**。 +> 4. **不与 STATUS 科学判决冲突、不改 `#275` 原始 U 合同、不改任何原件。** +> 5. 本分支**未被合并**；本批**未新增 issue、未删除数据、未操作服务器**。 +> 6. 部分交付的**笔记由子代理在被中断前写出**；被中断者的**原始输出**由操作者原样归档， +> 并在文件名中标注 `raw` / `telemetry`。 + +--- + +## 一、本批包含什么（按与当前 P0 的关联度排序） + +### A. `#802`（P0）—— 匹配根的切向—法向联合响应 + +| 文件 | 内容 | 强度 | +|---|---|---| +| `docs/manuscripts/geometric-balance/root-tangential-normal-response-20260914.md` | 控制表逐位复现 + `Δ_w`/`Δ'_w` + `(T,N)` 联合响应 + 四项裁定 V1–V4 | **控制表复现：有限枚举事实**；指数：**有限宽度拟合** | +| `results/research-dispatch/sector-root-response-20260914.json` | 同上，机器可读 | —— | +| `docs/manuscripts/geometric-balance/closure-amplitude-and-delta-exponent-20260914.md` | 闭合幅度 `A_r` 与 `Δ_w` 的**独立性**（幅度进不去 `Δ_w`）+ 外推指数 | **精确恒等式 + 有限宽度外推** | + +**要点（可复核）** +- 控制表 `w·I4`/`w·I8`（w=4..8）逐位复现，最大偏差 **4.87e-11** + （控制表只印 10 位小数 ⇒ 实为复现到其全部打印精度）。 +- `Δ_w(p_c)` = `+3.275880e-03 … +1.483071e-04`（w=4..8，全非零）； + `Δ'_w(p_c)` = `+2.4659 … +2.0236`（全正）。导数用 **Feynman–Hellmann 解析式**。 +- **结构性结论**：一阶下 `N = 0 ⟺ g` 与 `∂_p` 平行 `⟺ T = −1`。 + 即「法向响应为零」与「纯热重新调定」在一阶是同一件事； + 盲点的定量内容是倍数比 `φ ≈ 1.9e-4`（内禀读出比切向分量大约 5×10³ 倍）。 +- **`Δ_w` 只由 Perron 根构成 ⇒ 闭合幅度无法进入它**（`identical_Delta_after_amplitude_removal = true`； + 有限 `m` 迹估计 `D(m) − Δ_w → 0` 如 `e^{−mΔ}/m`）。 +- **独立验证**：Alexander duality **222720 次检查 0 违例**；反射律 `P8_p(k)=P4_p(2−k)` + **12 例最大差 0.000e+00**；FH 实现 vs 精确代数恒等式 **4.13e-13**。 +- **不能宣称**：`Δ_w`/`Δ'_w`/根偏移**同源，是一项数据生产不是三份独立证据**； + `w≤8` 的拟合**不是** exponent measurement；未识别 8-arm / spin±4。 + +### B. `#768`（P0）—— 首个非公共扇区修正的通道与预测 + +| 文件 | 内容 | +|---|---| +| `docs/manuscripts/geometric-balance/first-noncommon-correction-channels-20260914.md` | G1 四件事分开 / G2 等级计数 / G3 预测像 | +| `results/research-dispatch/theory-768-channels-20260914.json` | 通道清单与 `(T,N)` 预测表 | +| `results/research-dispatch/theory-768-level-dimensions-20260914.json` | 模去 `L_{−1}` 的等级维数 | +| `results/research-dispatch/theory-768-virasoro-20260914.json` | 等级计数的精确有理数输出 | + +**要点（可复核）** +- **`0,0,1,1` 成立**，但**只对「把 level-2 奇异向量商掉的不可约商」这一个模**： + `p(n)=1,1,2,3,5,7,11`；`rank G_n = p(n)−p(n−2)`；奇异向量只在 level 2（维数 1）； + `χ = −3L_{−2}+2L_{−1}²`，**范数恰为 0**。 +- **null 的去向**：(i) 与 (ii) **代数上都能站住，但关联函数完全相同**（零范且被正模湮灭）； + **只有 (iii) 改变物理**，而 `c=0` 渗流的物理模**就是 (iii)**（能量算符是零范态且是对数多重态 bottom field）。 +- **G1：只证明消去两类**——identity 家族（dual-even，在差里恒等消去）与所有 **C4 禁止**通道 + （spin `1,2,3 mod 4`）。**`k=2..7` 的标量 arm 通道（`x=1/4,2/3,5/4,2,35/12,4`）未被证明消去**。 +- **G3 的关键（本轮可操作的排除）**：在平移不变圆柱/环面上，权重 `(h,h̄)` 的一点函数 `∝ δ_{h,h̄}`。 + 因此**热族 level-4 手征后代（spin ±4, `x=21/4`）在动量 0 观测量中矩阵元为 0**； + 而 scalar 8-arm（`(21/8,21/8)`，spin 0）不零 ⇒ **预测像不同**，构成可行的排除。 +- **不能宣称**：不能宣称 level-4 类的**差矩阵元非零**；不能宣称 8-arm 或 spin4 谁胜出； + 不能宣称 `w^{−17/4}` 已被解释。**`∂⁴ε` 显式验证其类为 0，不得当作插入。** + +### C. `#775`（P1，当前最小产物）—— 精确 rank 表 L=3..6 + +| 文件 | 内容 | +|---|---| +| `results/geometric-consistency/rank-sector-C-L5-20260914.json` | **L=5 全枚举**（和 = 2²⁵ 精确） | +| `results/geometric-consistency/rank-sector-C-L6-20260914.json` | **L=6 全枚举**（和 = 2³⁶ 精确） | +| `results/geometric-consistency/ml-exponent-20260914.json` | `M_L(p_c)` 与 `2−x` 拟合 | +| `results/geometric-consistency/ml-definition-check-20260914.json` | **定义核验**（L=3 与 MZ16 多项式逐项相同） | + +**要点** +- `C[L,j,k]` 对 **L=3,4,5,6** 精确整数；L=5 和 = `2²⁵`、L=6 和 = `2³⁶`（无漏配置）； + 并查集 vs 通用覆盖 BFS **0/25 万**差异；L=6 交叉 **0/5 万**差异。 +- **L=4 的 `rank_totals` `[36559, 19932, 9045]` 与独立仲裁的 `W4−W8` 分布逐位吻合。** +- **定义已澄清**：L=3 时 `P2−P0` 与 **Mertens–Ziff 2016** 的多项式**逐项相同** + `[−1,0,0,6,0,0,0,−18,18,−4]`；交叉绕行配置数 91 = `C2_total`。 +- `M_L(p_c)`（L=3..6）= `0.02496990, 0.01033324, 0.00445693, 0.00236193`； + 拟合 `rms_log`：`−3.25`(Jacobsen) `0.0428`、`−3.42`(MZ16) `0.0396`、 + `−4.0` `0.0226`、free `−4.205` `0.0193`(dof=1)。 + ⇒ **两个候选都不被支持，数据偏好更陡；`L≤6` 判不了 `21/8`。** +- **不能宣称**：`L≤6` 只是**有限宽度兼容性**，不是指数测量；L=7/8 未产出 + （`#650` 已取消该阶梯）。 + +### D. `21/8` 的文献定标（**更正一条仓库内判词**） + +| 文件 | 内容 | +|---|---| +| `docs/manuscripts/geometric-balance/literature-calibration-frontier-20260914.md` | 长程检索（10 轮 / 30 次查询 / 4 次全文核对） | +| `docs/manuscripts/geometric-balance/correction-21-8-20260914.md` | 操作者侧更正说明 | + +**要点**：文献里**有两份互相冲突的测量**，不是一致的否定。 +- **支持整数 4**：**Jacobsen 2015**, *J. Phys. A* **48** 454003, arXiv:`1507.03027` §7.1 式 (36)—— + **方形格点 site 渗流（同一模型）**、周期 TL 转移矩阵特征值、**n ≤ 21**、**非 MC**， + `Δ₁ = 4.000 1(2)`；原文「**It appears inevitable to admit that `Δ₁ = 4` exactly**」；式 (40) `Δ_k = 2(k+1)`。 + 该文同时给出 `p_c = 0.59274605079210(2)`（即本项目惯用的 `p_ref`）。 +- **冲突**：**Mertens & Ziff 2016**, *PRE* **94** 062152, arXiv:`1603.07289`，`2−x = −3.42`、斜率 `−4.07`， + **作者自认「larger systems are needed」**。 +- ⇒ 正确表述是「**生死未判定**」，**不是**「已被文献否掉」。 +- 另：`√(cE)·Z ~ Laplace` 是**命名恒等式**（Gauss–Laplace transmutation，Ding–Blitzstein arXiv:`1510.08765`）。 +- `#775` 那张按**同调秩**分辨的整数表**无人发表过**（已发表的 `Z_0/Z_1/Z_2` 是**绕行方向型**）。 + +### E. P1/P2 已完成或负面结果的归档 + +| 文件 | 内容 | +|---|---| +| `docs/manuscripts/geometric-balance/gamma-w-parameter-dependence-crosscheck-20260914.md` | **独立确认**「`γ_w−κ = A e^{−cw}` 里的 `c` 不普适」；并澄清「受限转移阵」在该参数族下**是恒等** | +| `docs/manuscripts/geometric-balance/cavity-field-98-negative-20260914.md` | `#779`：`z_c(p)` **强 w-依赖**；w=8 时 `z_c(1/8)=1.17878`（距 `9/8` 差 4.8%）；`Z_p(z)` **未**达 `1/(9−8z)` ⇒ **负面** | +| `docs/manuscripts/geometric-balance/query-hole-definition-difference-20260914.md` | `#777`：`S_D(0)` 与八点多项式**不一致但属定义差异**（真查询孔洞多出 4 个十点矩形） | +| `docs/manuscripts/geometric-balance/ward-kb-locking-20260914.md` | `#774`：四项 Ward verdict 全 compatible（w=2..8）；`residual_covariance_scaled` **命名不清**（不是 (6.6) 的 `Σ`） | +| `docs/manuscripts/geometric-balance/audit-780-monotone-fragmentation-20260914.md` | `#780` 包审计：9 条声称无错误结论；`216`/`49` 是**参数网格大小而非独立证据** | +| `docs/manuscripts/geometric-balance/reproduce-genealogy-noise-20260914.md` | 谱系/噪声包 layer1 逐字复现通过 | + +--- + +## 二、本批**没有**完成的事（明确列出） + +1. **`#801` 的七机作业映射**：本批**未**回填到 issue（操作者的红线原为禁止仓库写操作）。 + **操作者侧的映射表已单独落盘**，待授权后回填。 +2. **`#780` 的共同标签并合缺口**：一项子代理任务**被中断，零产出**；未计入本批。 +3. **时钟可实现性 / 有限误差**（`#780` 的下一份产物）：**未做**。 +4. **`p_0 → p_c` 的有效范围**：**未做**（属待研究的有效范围边界）。 +5. 本批**不含**任何 L5–8 尺寸阶梯（`#650` 已取消）。 diff --git a/docs/research-status/recent-update-research-focus-20260914.md b/docs/research-status/recent-update-research-focus-20260914.md new file mode 100644 index 000000000..dc287f3c9 --- /dev/null +++ b/docs/research-status/recent-update-research-focus-20260914.md @@ -0,0 +1,159 @@ +# 近期更新后的研究选择：机制接口、三点源与有界计算 + +2026-09-14。回应所有者“阅读近期分析，思考后续任务”。这是研究综合与任务建议，不是全仓正确性认证或新的自动生产队列。 + +## 阅读范围 + +先比较 #771 的 `19b5f3b9201756b57c5045592323fc313b8cd1c9` 到 `1c93410cafdbdc0e6b64a558a9b54c761c354716`：40 个新增提交、40 个变化路径。重点读源正规化复核、数值底噪、斜向独立实现/成本墙、两阶段改进和 V14/W22 碰撞。工作期间又读取 `60d70ab1b985809e82661fac7f1abff6bb0840c1` 新增的三份 source grading / reverse audit / action spectrum 说明。近期其他 PR 主要读摘要，不宣称所有分支逐文件核验。 + +结论：现在最值得主动投入的仍是实际 rank 投影的 Ward 接口，以及一个真正独立的微观源响应。post-H4 残差值得保留，但大整数圆不应成为其他工作都必须等候的前置。 + +## 1. 应当吸收，不再重复采购的更新 + +`audit-802-source-normalization-20260914.md` 已确认： + +- 水平黑/白 pair 满足 `H_W=N-2K+H_B`，不是两个独立非热源。 +- 真实 safe 压力为 `I=log Z_row-log lambda_raw`。原先的平均压力源导数和法向值需要补正规化，零源的扇区差/热导数/根不因此作废。 +- `N=0` 表示在所选观察中沿热切向，给 `T=-a`，不强制 `T=-1`。 + +故直接复用修复后的表，不再为“两个独立 pair 源”各跑一套宽度。旧原始数据和勘误历史保留。 + +#801 评论 `5662731595` 已收到操作者作业回填，含已完成、停止、继续和预留项目。不能继续说“尚未收到七机清单”。但它是操作者快照，不是本轮实时主机遥测；只需更新发生变化的作业，不再复制空表。 + +## 2. 数值精度、状态成本和可辨识性是三件不同的事 + +`delta-p-intrinsic-floor-and-n1105-gate-20260914.md` 将旧 Ω 差异主要定位到 pc_ref 口径，并报告紧设置下小几何实现离散度约 `4.68e-16`。这是有用校准，不是未构建的大几何上的统一认证误差。 + +随后 `oblique-twist-independent-check-and-cost-wall-20260914.md` 报告 N325 的 `(1,18)` 构造在截断前已超过 250 万状态，仍未闭合；其他试探也越过成本界。N1105 的巨大状态数是外推，不当作实测，但现有增长已经不支持直接扩大同一 BFS 表示。C++ 能改常数，不能自动减少必须存储/访问的状态集合。 + +建议 #807 暂缓当前表示下的 N1105 生产；#808 的 N377 也不因信号较大就自动获准大算。新算法若实质减少状态或给可控截断，先在已完成对象上量出收益。这里不声称任何算法都不可能。 + +同圆角向投影只分开某一周长的有效谐波系数。即使零数值误差，`q_m(ell)=a_m+b_m/ell^2+...` 中的渐近 a_m 仍与径向修正混合。多角度不能替代径向余项控制。 + +safe-root 是无穷长圆柱根。这里常用的 N=|u|² 是整数圆编号/平方周长；只有实际闭合第二周期后，才对应 N 站点的方形环面。不要混用两种观察。 + +## 3. N377 的预测基线也需要误差，而不只是求根器 + +新 `two-stage-angular-radial-improvement-20260914.md` 记录 + +``` +p_perp4(5)=0.5927362453692130 +p_perp4(10)=0.5927459750664773 +pc_hat=(128*p_perp4(10)-p_perp4(5))/127 + =0.5927460516782668... +C7_hat=-0.7661178948... +``` + +预先提出指数七、并与独立参考作比较，是值得保留的相容性线索。但两尺度消去不提供未知余项的界，也没有纯化角向标量。 + +本轮按已打印输入重新计算： + +``` +pc_hat-pc_ref = +8.8616678268e-10 +C7_hat*377^(-7/2) = -7.3637812859e-10 +绝对值之比 = 1.2034126874... +pc_hat+C7_hat*377^(-7/2)-pc_ref = +1.4978865409e-10 +``` + +这不是新 N377 根，也不把 pc_ref 当严格真值。它说明当前外推基线的偏移已大于所计划的下一修正，不能把基线当无误差靶点，否则连残差符号都可能被改变。 + +axis/(3,4) 的 H4-null 组合对 H8 的增益是 `429/625`，对 H12 约 `0.52885504`。N377 把 H8 增益压至 `0.0036146395...`，但 H12 约 `-0.1378082`、H16 约 `-0.9810512`。它是选择性抑制，不是所有高谐波消失。p_perp4 不应提前重命名为已纯化的 p_H0。 + +## 4. 接住最新三点源：一阶响应不需要另建长记忆自动机 + +`bernoulli-chaos-source-grading-20260914.md` 的微观分级应成为 #802 下一步的起点。固定基准 p0，令 `a_v=n_v-p0`。互补并同时换基准 q0=1-p0 时 `a_hat=-a`。三个不同站点的乘积是确切 exchange-odd；在基准独立测度下，与常数、一次和二次 chaos 正交。这不是 continuum spin，也不证明 RG 保持此分级。 + +选择一个明确的 D4 三点源，w≥4： + +``` +H3 = sum_(x,y) [a_(x,y)a_(x+1,y)a_(x+2,y) + +a_(x,y)a_(x,y+1)a_(x,y+2)]. +``` + +p0 固定，改变 p 时不要重新中心化源。用白掩码实现同一物理源时必须保留互补负号。 + +对现有带行掩码 M 的 safe 权重 `W(i,M,j;p)`，定义 T=Σ_M W，Tr=lambda*r，l^T r=1： + +``` +M_x(i,j)=sum_M W(i,M,j;p)*(n_x(M)-p0) +M_h(i,j)=sum_M W(i,M,j;p) + *sum_x (n_x-p0)(n_(x+1)-p0)(n_(x+2)-p0) +``` + +这里 n 是黑色指示；matching 相也使用同一物理指示。平稳 Doob 路径的三步展开给 + +``` +mu3_G = (l^T M_h r)/lambda + + sum_x (l^T M_x^3 r)/lambda^3. +``` + +第一项计水平三点，第二项计三个连续新行的同列三点。它是 Markov 奖励的一阶恒等式，实际意义是：**不用为有限 g 的三行相互作用先建立扩大的自动机，就能得到 g=0 响应。** 仅对均匀权重有效的强压缩不能自动用于位置标记 M_x；需保留兼容表示。 + +纵向三点耦合不同行，所以有限 g 的总正规化不是逐行 Z_row 的简单乘积。设 f 为无 safe 限制的完整压力，则 + +``` +I_G,g = f_g - mu3_G +f_g|g=0 = 2w(p-p0)^3. +``` + +只有 p=p0 时 f_g 为零，在每个宽度自己的 root 处仍需按实际 p 计算。二阶需要 Green–Kubo/接触项，不能用三步公式包办。 + +#802 的下一小产物可明确为：已有一两个可负担宽度上的 H3 两相平均、正规化 (T,N)，对照修正后的 H2。基准乘积测度下正交，不意味着在 rank 条件/Doob 相中没有热投影；T≠0 完全可能。源真正独立仍不保证两个 CFT 候选可区分，需 #768 给出相应预测，不能从源数目推断机制已识别。 + +本轮仅以有限 Bernoulli 源代数和两态 Markov 奖励核对上述接口，没有生产实际 safe 相的 H3 数据。 + +## 5. Ward 主线：先检验实际投影，再扩大模块叙事 + +已有条件候选为 + +``` +p*_Lambda-pc = A Re[u^-4 E4(v/u)] + o(|u|^-4). +``` + +真正缺少的是实际 rank 投影的泛函、共同热正规化、null 后裔是否解耦，以及应力张量运输是否有接缝项。对固定准初级约定，可以把目标压成 + +``` +A_r = F_r(U4+Ubar4) + -(pi^4/12) Re[u^-4 E4(tau)] F_r(epsilon). +``` + +#768/#585 先确定一个真实扇区/charge 差上的 A_r，或给出具体异常项。普通 torus trace 的 Ward/Zhu 递推不自动认证这个非局部投影；不需要先替整个 c=0 理论定下模块身份。 + +#802 同时保留两压力：`D=I4-I8`、`A=(I4+I8)/2`。在固定 D 比较 A，才是所选两压力观察的横截响应。单条 D 曲线移根/归一斜率不等于消去任意热重参数化。 + +先找已有档案中未用于选择机制的有限模参数。若有三个可比几何，设 `F_i=Re[u_i^-4 E4(v_i/u_i)]`，用 + +``` +C=(F2-F3)p1+(F3-F1)p2+(F1-F2)p3 +``` + +同时消去 pc、A；保留共同数值误差和几何余项。没有这类档案时，先确定 finite-torus rank 闭合的真实计算接口，不能把普通 safe 迹自动当 rank0 概率。此前 L5/L6 方形/圆柱比值已用于提出支持，不再称为新的前瞻留出。 + +## 6. Post-H4 与 action spectrum 应怎样保留 + +`v14-w22-log-collision-20260914.md` 中的权重碰撞和 `d_Q Delta x=-15/(2pi sqrt(3))` 是明确代数。generic-Q diamond/Jordan 结构有外部理论支持，但实际 residual 的标量成分、source lift 和耦合仍未识别。纯幂不排除对数伙伴，维数相等也不强迫指定读出的 log 振幅非零。 + +#586/#61 可以继续做最小实际 Q-lift/parity 接口，而不是马上开 Q×角度×宽度大网格。更直接的理论分叉是：后 H4 剩余来自新线性插入，还是已有扰动的二阶混合？需要连通二次插入、热反演和接触项；两个一点函数的乘积不等于二阶响应。一阶 spin4 零点也不自动消掉 spin4 与 spin−4 配成的标量二阶项。 + +最新 action-spectrum 说明正确地不再令并合、质量局部性和慢谱分裂共享一个未经证明的精确代价。BK 给并合的上概率界，不给其指数恰好等于一份 kappa。保留这个区分即可，不再立刻生成三套指数拟合。近临界常数会退化，不能只写 w*kappa→infinity 就替代联合一致性。 + +#780 已有无并合与两个时钟构造。下一步只关闭一个具体有限误差/适用范围缺口；不重复更多 Poisson 核、Gamma 矩或时钟存在性工作。 + +## 7. 建议的下一轮 + +- #768/#585：一个真实 rank 投影 Ward/null/接缝关系，理论优先。 +- #802/#61：一个归一化正确、与 H2 真独立的 H3 小响应；两压力一起读。 +- #807/#808/#589：暂缓现有表示的大圆生产；重新给径向/高谐波/基线误差和算法成本。C++ 移植本身不等于状态复杂度突破。 +- #780：具体概率输入和有限窗口误差的收束，不再占用默认静态生产。 +- #586：声明 Q-lift 后的模块/耦合问题，长期保留。 + +不新开重复 issue，不修改原始 U 合同或 STATUS，不删除数据、不合并 PR、不操作主机。自主数学兴趣继续；共享算力不必为了保持满载而扩大。 + +## 来源与执行范围 + +内部主要来源均在 `docs/manuscripts/geometric-balance/`： +`audit-802-source-normalization-20260914.md`；`delta-p-intrinsic-floor-and-n1105-gate-20260914.md`；`oblique-twist-independent-check-and-cost-wall-20260914.md`；`two-stage-angular-radial-improvement-20260914.md`；`v14-w22-log-collision-20260914.md`；`thermal-null-ward-root-mechanism-20260914.md`；`two-observable-response-and-angular-alias-20260914.md`；`intrinsic-birth-clock-construction-20260914.md`。增量三稿为 `bernoulli-chaos-source-grading-20260914.md`、`reverse-audit-irrep-source-action-spectrum-20260914.md`、`topological-action-spectrum-beyond-witness-count-20260914.md`。 + +定向外部阅读：Gaberdiel–Lang arXiv:0810.0106v2 §2；Grans-Samuelsson et al. arXiv:2007.11539v3 摘要/出版说明；Yifei He arXiv:2411.18696v2 §§3.2.1、3.4、6 中高能量 Kac/log 结构的条件。它们不自动证明 actual-site 投影。 + +本轮只执行短算术重读：已打印的 Richardson/N377 数字、精确谐波泄漏、有限整数圆最小值、碰撞代数、源正交与三步奖励身份。没有重跑转移、L5/L6 枚举、Monte Carlo 或全仓 CI。上述数值重组不是新的 pc/大几何根或渐近证书。 diff --git a/docs/root-response-and-bridge-targets-20260914.md b/docs/root-response-and-bridge-targets-20260914.md new file mode 100644 index 000000000..88c54009c --- /dev/null +++ b/docs/root-response-and-bridge-targets-20260914.md @@ -0,0 +1,142 @@ +# 从内禀曲线的盲点到两项可区分的研究任务 + +2026-09-14。服务 #768/#802 与 #780 的短分析，不再生成一套尺寸普查。本文的三个部分分别是精确微分/概率反例、明确假设下的 Virasoro 商空间计算、以及严格计数上界和待证概率估计。均未识别实际 square-site 连续场。 + +## 1. 消去热坐标，可能同时消去真正移动根的修正 + +在 P0,P1,P2>0 的局部参数区间定义 + + b(p,g) = (1/2) log(P0/P2), + e(p,g) = log[P1/(2 sqrt(P0 P2))]. + +设 b_p != 0。沿匹配根 b(p*(g),g)=0 求导，严格有 + + T_g := dp*/dg = -b_g/b_p. (1) + +把热坐标 p 消去、在固定 b 上看内禀形状，则 + + N_g := (d/dg)e(p(b,g),g) + = e_g - (e_p/b_p)b_g. (2) + +这是隐函数定理/链式法则，不是新发明的几何原理。#773 已有物理源商和 moving-root 公式；本轮的意义是指出旧任务如何过度解释这个商。 + +在 source 向量里加 a∂p，N 不变，而 T 改变。即 N 丢弃纯热切向，T 保存实际热坐标中的位移。两者都要保留，不能拿一个更“不变”的观察量替代原问题。 + +若原始权重为 w(p,g;ω)、H=∂g log w，则 + + b_g = [E(H|r=0)-E(H|r=2)]/2, + e_g = E(H|r=1)-[E(H|r=0)+E(H|r=2)]/2. (3) + +全局正规化的导数在这些差里抵消。H 必须是实际概率变化的评分；高阶导数要另带 contact 项。内禀拓扑源 s(r−1) 只平移 b，不提供独立微观干预证据。 + +### 一个不会靠更高精度解决的反例 + +令 y>0，a 任意，在临界邻域取 + + b_L(p) = -L^y (p-pc-a L^-4), + c_L(p) = C0(b_L(p)), + +其中 C0 为任意正的偶函数。由 + + P0=e^b/[2(c+cosh b)], + P1=2c/[2(c+cosh b)], + P2=e^-b/[2(c+cosh b)] + +得到一族正规化、严格正的概率。每个 a 的内禀曲线都是同一个 C0，odd 部分严格为零；匹配根却为 pc+aL^-4。它是局部三概率族的逻辑反例，不是声称已经构造了对应的 Bernoulli site 模型。所需的 matching 对偶伙伴可直接按 P_hat,r(p)=P_(2-r)(1-p) 定义，仍不获得微观实现。 + +因此“单一 odd 修正导致根按 L^-4 移动”并不推出“内禀 O_L(b) 必有非零 L^-13/4 首项”。后一结论需要该修正在法向投影不为零。这是 #775 的数据解释必须补上的假设，不是删掉已有数据的理由。 + +更一般，令 x=a_t L^y(p-pc)，u=L^-ω， + + b=B0(x)+uB1(x)+o(u), e=E0(x)+uE1(x)+o(u). + +若 B0(0)=0、B0'(0)!=0，且展开局部一致，则 + + p*-pc = -u B1(0)/(a_t L^y B0'(0)) + o(u L^-y), + N(x) = u [E1(x)-E0'(x)B1(x)/B0'(x)] + o(u). + +当 (B1,E1)=a(x)(B0',E0')，领先法向响应完全消失，根仍可移动。反过来，B1=0、E1!=0 可改变形状而不改变首阶根。非零根响应与非零形状响应必须由候选分别预测。 + +### 直接的下一步 + +#802 使用同一个物理源同时报告 (T,N)，优先区分“热方向重新调定”与“真正形状改变”。对已有 raw rank-by-occupancy 表先做便宜重读，不增加默认尺寸。候选可以有允许的 nuisance 幅度，但不能两个预测像都任意拟合后再宣称分离。没有非零信号锚点的锥都含零，设计必须对齐相同信号约定。 + +## 2. spin-four 机制有一个比大扫描更小的表示论岔路 + +[R1] 报告 square/kagome 的不同根修正序列，并将旋转对称性作为可能原因；相同主临界扇区消去并不独自确定第一个剩余项。本节是条件性的模块筛选，而不是为 spin4 贴一个已证标签。 + +取 Virasoro 代数 + + [L_m,L_n]=(m-n)L_(m+n)+(c/12)(m^3-m)δ_(m+n,0), + +c=0、h=5/8 时 + + χ = (L_-2-(2/3)L_-1²)|h> + +是 level2 奇异向量：L1χ=[3-(2/3)(4h+2)]L_-1|h>=0，L2χ=[4h-(2/3)6h]|h>=0，更高正模也消失。 + +**假设**实际相关热模块确实把 χ 及其后裔商掉。仅在这个代数商里，再模去 L_-1 导数，得到： + +|层级n|Verma维数|商掉χ后维数|再商掉总导数的维数| +|---:|---:|---:|---:| +|0|1|1|1| +|1|1|1|0| +|2|2|1|0| +|3|3|2|1| +|4|5|3|1| + +脚本用负模 PBW 交换关系和有理消元重建这张表。这里没有假定字符系数的相减自动等于导数秩，而是显式算出两个子空间之和。 + +在这个商中，level2 的 L_-2 等价于 (2/3)L_-1²，不留下独立的非导数项。level3 有一维，但其 spin3 不被 C4 允许；level4 留有一维，并允许反射偶的 spin±4 组合。若反手征维数仍为5/8，其总维数为 5/4+4=21/4。 + +这为候选提供一条更具体的路径：**先检查 null 如何实现，再算该 level4 非导数类在两个扇区中的差矩阵元。** 不能将它简化为 ∂⁴ε；后者本身是总导数。tensor-product 商中的 (2,2) 也不能当作一个自动独立的低阶标量后裔。 + +但有四个尚未跨越的条件： + +1. h=5/8 并不决定实际模块。相同最高权可以属于不把 null 设零的 Verma/不可分解表示；[R2] 明确区分退化场与相同维数但不满足 null 方程的场。 +2. 上述计数不排除其他 primary/module 的较低维 dual-odd 项；6-arm、identity-family、logarithmic伙伴等仍须分别定位。 +3. 一个非导数类不保证其周期扇区差矩阵元非零；contact、seam及边界/规范归一化仍可改变映射。 +4. square/kagome 的不同指数不是这个具体 thermal module 选择的证明。self-matching 控制会杀掉所有odd项，也不能单独挑出spin4。 + +所以本轮没有证明 square-site 的 L^-4 机制。完成的是一个有具体输入和可失败位置的检索/分析问题；它比再把同一根算多几位更有信息。 + +## 3. 固定参数热窗口的桥接，用一个实际并合量来判断 + +在同一批 Uniform 标签上取 p_-0，并且一个估计根处的函数误差/残差≤ε，则平均值定理给根误差≤ε/d0。重要的是误差对两扇区之差和导数的传输，不只是两个大本征值分别有几位数字。 + +若 P0、P2 都很小，绝对 o(1) 的分布逼近并不控制 log(P2/P0)。反之，整张原始系数表能生成很多诊断，但不制造很多份独立信息。这两条是本轮七机调整的数学理由。 + +## 5. 执行范围与来源 + +本轮脚本只计算2条奇异向量等式、levels0..4的PBW商/导数秩、27个正概率族控制及48条切向混合恒等式。它不是新的物理普查，不提供field identification或大宽度外推。运行结果在 `results/research-dispatch/root-response-null-control.json`。 + +[R1] J. L. Jacobsen, *Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras*, arXiv:1507.03027v1，§7–8。已读取HTML中的square/kagome修正与leading-sector cancellation段落；其旋转解释不是实际插入定理。https://arxiv.org/html/1507.03027v1 + +[R2] S. Ribault, *Conformal field theory on the plane*, arXiv:1406.4290v4，§2.1.2、§2.3.1。已读取奇异向量/退化场和同权非退化场区别。这里只引用表示论范围，不当作渗流热模块的身份证明。https://arxiv.org/html/1406.4290v4 + +内部输入：#773 的 source quotient、#768 关于内禀 odd 曲线的评论及后续pair-even修正；#771 的 charge-sector 说明；#780 的共同标签过程任务。本轮修正的是过强的可识别性推断，不撤回其正确代数或既有原始数据。 diff --git a/notes/topological-pivotal-pair-atlas-20260914.md b/notes/topological-pivotal-pair-atlas-20260914.md new file mode 100644 index 000000000..60e9ab261 --- /dev/null +++ b/notes/topological-pivotal-pair-atlas-20260914.md @@ -0,0 +1,371 @@ +# Topological pivotal-pair atlas — exact raw joint table for issue #769 + +**Author:** rev769 (cloud VM `DevEnvC_551oUR`, `/workspace/rev769/`) +**Date:** 2026-09-14 +**Scope:** issue #769 Phase A (exact L=5 main product) + Phase B (D4 displacement-orbit +structure). Phase C is **not** completed as a classifier; only raw material is saved. +**Honesty flags** are in §8. Everything below is *computed by me* unless marked +"引用 (quoted, not independently recomputed)". The T1 control values are **not** +引用 — they are independently reproduced. + +--- + +## §T0 What is derivable and what must be recomputed (with reasons) + +This list is itself part of the delivery (task spec §0). + +### T0-A. Derivable **without** any new enumeration + +| # | quantity | why it is derivable | +|---|---|---| +| A1 | `M(p) = E_p[X]`, and **all** its `p`-derivatives `M′(p), M″(p), …` | `X = r_black − 1` depends on `ω` only through the rank `r_black`. Hence the *entire* `p`-dependence of `M` is a functional of the rank marginal `C[L,j,k] = #{r=j, |ω|=k}` alone. Computed **exactly** (rational) from the reused #775 tables `C_L3..5.json`. | +| A2 | the two Russo/translation constraints `N·Σ_{d≠0}J_d = M″(p)` and `N·E[Δ₀X] = M′(p)` | these are identities of the object definition, with the left sides from the pair atlas and the right sides from A1. Used as the **independent** side of the T2 check. | +| A3 | D4 displacement orbits: representative, multiplicity, torus distance | pure lattice combinatorics, independent of any rank computation. | +| A4 | `Σ_orbit mult = N − 1` | arithmetic check on A3 (2/5/5 orbits at L=3/4/5). | +| A5 | `Δ_vX ∈ {0,1,2}` and hence `Δ₀Δ_dX ∈ {−2,…,+2}` | a priori: `r_black` is monotone under adding black sites, so a single-site difference cannot be negative and cannot exceed 2. This is why **no** value outside `{−2..2}` occurs (§T4) and why the sign convention `Δ_vX = X(v=1) − X(v=0)` is *forced*, not chosen. | + +### T0-B. **Not** derivable → must be produced by extending the same enumeration + +| # | quantity | why the marginal `C[L,j,k]` cannot give it | +|---|---|---| +| B1 | histogram of `Δ₀X` by occupancy of the other `N−1` sites (the counts `{0:161,1:90,2:5}` etc.) | `C[L,j,k]` is a **marginal** histogram; `Δ₀X = X(0=1) − X(0=0)` is a **differential** observable of the pair (site 0, rest). The only functional of it fixed by the marginal is its **mean** (via Russo, = A2). The *distribution* is not fixed. | +| B2 | for every displacement `d`: histogram of `Δ₀Δ_dX` over `{−2..2}` by occupancy of the remaining `N−2` sites | two-site **mixed** discrete difference — no functional of the rank marginal determines it. The sum rule A2 fixes only `Σ_d J_d`, i.e. one number out of the whole orbit table, and only for the *signed* moment (not for `A_d = E|Δ₀Δ_dX|`). | +| B3 | `J_d, A_d, P⁺_d, P⁻_d` | linear functionals of the B2 histograms. | + +### T0-C. Reuse, not re-production + +The expensive component — the rank / winding kernel — is **copied verbatim** from +`rank775`'s delivery (issue #775): + +* path **A** (`pivotal.c`) reuses `rank775-work/ranksec_brute.c::rank_of` + (union-find with `Z²` offsets, forward edges only); +* path **B** (`pivotal_bis.c`) reuses `rank775-work/rank_bis.c::rank_uf` + (generic-adjacency union-find) and `rank_bis.c::rank_cov` + (universal-cover `Z²` BFS, both edge orientations). + +Only the **dimension** was added (site-0 state × displacement orbit) plus the +occupancy-count histogram accumulation. No second rank production was written. + +### T0-D. Explicitly not done (constraints of #769) + +no reconstruction of `A_d` from the one-dimensional `M(p)` polynomial (forbidden); +no Monte Carlo; no L≥6; no STATUS / #275 contract change; no arm-count classifier +(Phase C only saved raw material, §T5); no assertion of an exponent anywhere. + +--- + +## §T1 Control-value reproduction — **all 12 numbers reproduced** + +Frozen reference value `p_c = 0.59274605079210` (as given by #769). All mine are +**exact rational** evaluations (`Fraction`, `p_c` treated as the exact decimal +`59274605079210/10^14`); the table prints 20 significant digits. + +### T1 control table + +| L | quantity | #769 quoted | mine (20 digits) | delta | +|---|---|---|---|---| +| 3 | counts | `{0:161, 1:90, 2:5}` | `{0: 161, 1: 90, 2: 5}` | **0 (exact)** | +| 3 | Ed0 | 0.4458689104616698 | 0.44586891046166977571 | -5.551e-17 | +| 3 | jump2 | 0.1039113209 | 0.10391132091753346805 | +1.753e-11 | +| 3 | A2 | 23.04926674699 | 23.049266746993427819 | +3.428e-12 | +| 3 | Mpp | 2.336146088861 | 2.3361460888611493896 | +1.492e-13 | +| 3 | cancel | 9.86636 | 9.8663636049532082272 | +3.605e-06 | +| 4 | counts | `{0:24639, 1:7840, 2:289}` | `{0: 24639, 1: 7840, 2: 289}` | **0 (exact)** | +| 4 | Ed0 | 0.31159887509994 | 0.31159887509993999399 | +0.000e+00 | +| 4 | jump2 | 0.07434091137 | 0.074340911371403713232 | +1.404e-12 | +| 4 | A2 | 45.48519754007 | 45.485197540074412686 | +4.412e-12 | +| 4 | Mpp | 2.933170180195 | 2.9331701801949563475 | -4.352e-14 | +| 4 | cancel | 15.50718 | 15.507179858568993614 | -1.414e-07 | + +Legend: `Ed0 = E[Δ₀X]`; `jump2` = rank-jump-2 share of `M′`; `A2 = A2_abs`; +`Mpp = M″`; `cancel = A2_abs/|M″|`. + +Every deviation is **at or below the precision at which #769 quoted the value** +(e.g. the `cancel` deviations are ≤ 3.6e-6 while the quoted values have 6 digits). +The `counts` are equal **exactly**, as integers. + +### Convention judgement (T1 asked me to decide this myself) + +* the counts sum to `2^{L²−1}` (256 / 32768): they are counts over the configurations + of the **other `L²−1` sites**, i.e. the distribution of `Δ₀X`; +* `Δ₀X = X(v=1) − X(v=0)` **does not depend** on the state of site 0 itself, so there + is no "site 0 fixed black vs white" ambiguity for the counts. The state of site 0 + enters only *inside* each term of the difference, which is exactly how I evaluate it; +* the **direction** is forced and not a free choice: `r_black` is monotone under adding + black sites, hence `Δ₀X ≥ 0` (my enumeration produced **zero** negative values, and + the L=3 list is `{0:161, 1:90, 2:5}` with no 3-value); +* the decisive evidence for this reading is the simultaneous exact match of + **3 integer counts + 5 derived quantities at two sizes** with two independent code + paths. Nothing was substituted. + +--- + +## §T2 The two required identities — **both hold exactly** + +| L | `N·E[Δ₀X]` (atlas) | `M′(p)` (exact from reused `C[L,j,k]`) | equal? | `N·Σ_{d≠0}J_d` (atlas) | `M″(p)` (exact from reused `C[L,j,k]`) | equal? | +|---|---|---|---|---|---|---| +| 3 | 4.0128201941550279814 | 4.0128201941550279814 | **yes** | 2.3361460888611493896 | 2.3361460888611493896 | **yes** | +| 4 | 4.9855820015990399038 | 4.9855820015990399038 | **yes** | 2.9331701801949563475 | 2.9331701801949563475 | **yes** | +| 5 | 5.8827716932504600604 | 5.8827716932504600604 | **yes** | 3.4161276168223449860 | 3.4161276168223449860 | **yes** | + +The comparison is performed as an **exact `Fraction` equality** (`Mpp_enum − ddEX == 0` +and `Mprime_enum − dEX == 0`), not as a float tolerance. The right column is obtained +by differentiating `M(p) = Σ_k G[k] p^k q^{N−k}` symbolically, with +`G[k] = Σ_j (j−1) C[L,j,k]` from the **reused #775 rank tables** — i.e. it uses only +the rank marginal and **none** of my pair data. Both identities therefore hold to +exact arithmetic, at all three sizes. + +`M″` at L=3 equals the #769 control value `2.336146088861` (see §T1): the identities +and the controls are satisfied simultaneously by the same numbers. + +> Note for readers: these identities are consequences of Russo's formula plus +> translation invariance, so their holding is a **consistency/validation** result +> (it certifies that the pair atlas is the correct joint object), not an independent +> piece of physics evidence. The non-trivial content is that the *independently +> enumerated* pair table reproduces the second derivative of the rank-marginal +> polynomial exactly. + +--- + +## §T3 Phase A — exact L=5 (main product) + +### T3 identity / L=5 table + +| L | counts (j=0,1,2) | sum | E[Δ₀X] | jump2 share | J_sum | M′ | M″ | A2_abs | cancel_ratio | +|---|---|---|---|---|---|---|---|---|---| +| 3 | 161,90,5 | 256 (=2^8) | 0.44586891046166977571 | 0.10391132091753346805 | 0.25957178765123882107 | 4.0128201941550279814 | 2.3361460888611493896 | 23.049266746993427819 | 9.8663636049532082272 | +| 4 | 24639,7840,289 | 32768 (=2^15) | 0.31159887509993999399 | 0.074340911371403713232 | 0.18332313626218477172 | 4.9855820015990399038 | 2.9331701801949563475 | 45.485197540074412686 | 15.507179858568993614 | +| 5 | 13800255,2906128,70833 | 16777216 (=2^24) | 0.23531086773001840242 | 0.052085344508862943781 | 0.13664510467289379944 | 5.8827716932504600604 | 3.4161276168223449860 | 70.908531625515400319 | 20.756991418099877127 | + +**L=5 facts** + +* full enumeration of `2^25 = 33 554 432` configurations; the base enumeration is + `2^24` configurations of the sites `≠ 0` (33.5 M rank evaluations, 0.88 s wall, + 16 threads). Per-`d` pair totals are each exactly `2^23 = 8 388 608`, and the + single-site total is exactly `2^24` — the enumeration is complete and exact; +* `Δ₀Δ_dX` support at L=5 is the **full** `{−2,−1,0,1,2}` (at L=3 only `{−1,0,1}`; + L=4 already reaches the full range); +* L=6+ was **not** attempted (not an acceptance condition for #769). + +### T3 cross-validation — **three code paths, byte-identical at L=3, 4 and 5** + +| path | rank kernel | algorithm | +|---|---|---| +| A | union-find + `Z²` offsets (`ranksec_brute.c::rank_of`) | build `j0[γ]` once (array), then table-lookup pairs | +| B-uf2 | generic-adjacency union-find (`rank_bis.c::rank_uf`) | recompute **all four** paired-flip ranks `X(0=a,d=b)` directly | +| B-cov | universal-cover `Z²` BFS (`rank_bis.c::rank_cov`) | same direct four-rank recomputation, structurally different kernel | + +Result: the **single-site histogram and all per-`d` pair histograms are identical as +integers** for A vs B-uf2 and A vs B-cov, at **L=3, L=4 and L=5** (not only L=3,4). +So the L=5 numbers satisfy the #769 requirement "两个实现路径至少在 L=3,4 完全一致后才信 L=5" +with margin. + +--- + +## §T4 Phase B — D4 displacement-orbit structure + +`p_c = 0.59274605079210`. Orbit-level sums; `A_d`, `J_d`, `P±_d` are **per pair** +(i.e. divided by the orbit multiplicity). `hist identical = yes` means every member of +the orbit produced an identical exact integer histogram (internal symmetry check, all +**yes**). + +**L3** (N=9, P_pivotal=0.42270349674060299395, P_piv²=0.17867824615673296593) + +| rep d | mult | torus dist | J_d·mult | A_d·mult | J_d | A_d | P⁺_d | P⁻_d | A_d/P_piv² | hist identical | +|---|---|---|---|---|---|---|---|---|---|---| +| (0,1) | 4 | 1.0000 | 0.967840 | 1.652645 | 0.241960 | 0.413161 | 0.327561 | 0.085601 | 2.312320 | yes | +| (1,1) | 4 | 1.4142 | -0.708268 | 0.908384 | -0.177067 | 0.227096 | 0.025015 | 0.202082 | 1.270977 | yes | + +**L4** (N=16, P_pivotal=0.30001660292132312676, P_piv²=0.090009962028450872520) + +| rep d | mult | torus dist | J_d·mult | A_d·mult | J_d | A_d | P⁺_d | P⁻_d | A_d/P_piv² | hist identical | +|---|---|---|---|---|---|---|---|---|---|---| +| (0,1) | 4 | 1.0000 | 0.528192 | 1.130011 | 0.132048 | 0.282503 | 0.206411 | 0.075227 | 3.138573 | yes | +| (0,2) | 2 | 2.0000 | 0.191503 | 0.432418 | 0.095751 | 0.216209 | 0.155789 | 0.060229 | 2.402057 | yes | +| (1,1) | 4 | 1.4142 | -0.452616 | 0.837505 | -0.113154 | 0.209376 | 0.047877 | 0.160971 | 2.326146 | yes | +| (1,2) | 4 | 2.2361 | -0.043599 | 0.350775 | -0.010900 | 0.087694 | 0.038397 | 0.049297 | 0.974267 | yes | +| (2,2) | 1 | 2.8284 | -0.040157 | 0.092115 | -0.040157 | 0.092115 | 0.025979 | 0.061965 | 1.023385 | yes | + +**L5** (N=25, P_pivotal=0.22918274392381965818, P_piv²=0.052524730112451096451) + +| rep d | mult | torus dist | J_d·mult | A_d·mult | J_d | A_d | P⁺_d | P⁻_d | A_d/P_piv² | hist identical | +|---|---|---|---|---|---|---|---|---|---|---| +| (0,1) | 4 | 1.0000 | 0.384887 | 0.842637 | 0.096222 | 0.210659 | 0.153181 | 0.057219 | 4.010669 | yes | +| (0,2) | 4 | 2.0000 | 0.166644 | 0.514443 | 0.041661 | 0.128611 | 0.084998 | 0.043475 | 2.448575 | yes | +| (1,1) | 4 | 1.4142 | -0.343235 | 0.697267 | -0.085809 | 0.174317 | 0.044125 | 0.129912 | 3.318754 | yes | +| (1,2) | 8 | 2.2361 | -0.016390 | 0.561085 | -0.002049 | 0.070136 | 0.033962 | 0.036061 | 1.335287 | yes | +| (2,2) | 4 | 2.8284 | -0.055261 | 0.220909 | -0.013815 | 0.055227 | 0.020688 | 0.033977 | 1.051453 | yes | + +The complete exact integer histograms (all `d`, all `Δ∈{−2..2}`, all occupancy bins +`k = 0..N−2`) are in `results/topological-pivotal-pair/orbit-table-20260914.json`; +they are `p`-independent, so any other reference `p` can be used without re-enumerating. + +### T4 readings + +1. **Signed interaction changes sign with distance and does not decay monotonically.** + At L=5 the per-pair signs run `+ (d=1), − (√2), + (2), − (√5), − (2√2)`; the + diagonal pair is strongly *negative* and the NN pair strongly *positive*. The + signed magnitudes at `√5` and `2√2` are ~2 orders of magnitude below the NN one. + ⇒ "fast sign change": `compatible`, but not a clean one-sign decay. +2. **Absolute interaction is concentrated at short distance.** + per-pair `A_d` at L=5: `0.2107 (1) > 0.1743 (√2) > 0.1286 (2) > 0.0701 (√5) > 0.0552 (2√2)`. + At L=4 the ordering is *not* monotone in distance (`0.2162` at dist 2 vs `0.2094` at + dist √2) — a clear finite-size/torus artefact and a warning against reading this as + a radial profile. +3. **Near/far comparison against "two independent pivotals".** + `A_d / P_piv²` is **≈ 1.0 at the largest available separation** (L=3: 1.27 at √2, + which is the L=3 diameter; L=4: 1.02 at 2√2; L=5: 1.05 at 2√2), i.e. the far field is + at the scale of the product of two single-point pivotal probabilities, while the + NN pair is **2.31 / 3.14 / 4.01 ×** that product at L=3 / 4 / 5. So there *is* a + short-distance excess over the independent-pivot scale, and its size grows with L. +4. **Against the named fusion scales — `unresolved`.** + `A_d` is a bounded lattice diagnostic (`≤ 2`), and the only scale I can compare it to + honestly is the single-point pivotal product in 3. With 2/5/5 orbits at three sizes + and a torus diameter of ≤ 2√2, I **cannot** distinguish "single-point pivotal + probability squared" from "6-arm fusion" from "8-arm fusion" magnitudes: + ⇒ write `unresolved`. The near/far **pattern** is `compatible`; the **fusion + identification is unresolved** (it would need the arm witness classifier, §T5). + +### Maximal / representative pair comparisons (as requested) + +* nearest-neighbour `(0,1)` vs diagonal `(1,1)`: opposite sign of `J`, both with large + `A`; this sign opposition is the main source of the cancellation in §D1. +* intermediate `(0,2)` (dist 2) and `(1,2)` (dist √5): small signed `J`, moderate `A` + (the L=5 `(1,2)` orbit has multiplicity 8, the largest). +* maximally separated `(2,2)` (dist `2√2 = 2.828`, the L=5 diameter): smallest `A`, + negative `J`, and `A_d/P_piv² ≈ 1.05`. +* the `L=3` torus has only **two** orbits (the diameter is √2), so L=3 cannot separate + "nearest" from "farthest" at all — a hard finite-size limitation. + +--- + +## §T5 Phase C (bonus) — **not** completed as a classifier; raw material saved + +The issue explicitly allows, when an exact arm classifier is too hard, saving +representative configurations + lifted homology. That is what was done +(`witness_L3/4/5.json`; 90 witnesses at L=5, 5 per (orbit, `Δ`-value) with +`Δ ≠ 0`). Each witness records, for the **four** paired-flip states +`(s₀,s_d) ∈ {0,1}²`: the rank `r_black`, the number of black components, the explicit +list of black winding vectors in `Z²`, plus the black-site lists of the two extreme +states. Example worth reading (L=5, orbit `(0,1)`, `Δ=+1`): rank is `0` in all three +states except `(s₀=1,s_d=1)`, where a single `+x` winding appears — i.e. the two sites +must be occupied *together* to close a winding cycle. + +**No arm count is claimed anywhere.** Local degree counts would be exactly the +"guessing from local degree" that #769 forbids. + +--- + +## §D1–D3 The three required verdicts + +### D1 — Is the smallness of `M″` caused by increasingly severe ± pivotal-pair cancellation? + +**Verdict: `compatible` (on the three computed sizes).** + +| L | A2_abs | abs(M″) | cancel_ratio | A2_abs/N² | rank-jump-2 share | +|---|---|---|---|---|---|---| +| 3 | 23.049266746993427819 | 2.3361460888611493896 | 9.8663636049532082272 | 0.284559 | 0.10391132091753346805 | +| 4 | 45.485197540074412686 | 2.9331701801949563475 | 15.507179858568993614 | 0.177677 | 0.074340911371403713232 | +| 5 | 70.908531625515400319 | 3.4161276168223449860 | 20.756991418099877127 | 0.113454 | 0.052085344508862943781 | + +`cancel_ratio` increases monotonically `9.87 → 15.51 → 20.76`: the signed total `M″` +grows only `2.34 → 2.93 → 3.42` while the absolute pair activity grows `23.05 → 45.49 +→ 70.91`. So the *surviving fraction* `|M″|/A2_abs` falls `0.1014 → 0.0645 → 0.0482`. +Independently, the share of `M′` carried by rank-jump-2 events falls +`0.1039 → 0.0743 → 0.0521` (as #769's Round-5 note reported for L=3,4 — 引用 for the +trend direction, but my values are independently reproduced). +What I **cannot** say: that `cancel_ratio` follows a power law; three finite sizes +cannot certify an exponent. + +### D2 — Does the absolute pair interaction show a clear near "fusion" region and a far "two-pivotal" region? + +**Verdict: near/far scale separation `compatible`; fusion identification `unresolved`.** + +Supporting facts are in §T4: per-pair `A_d` decreases with torus distance at L=5 +(0.211 → 0.055), and `A_d/P_piv²` goes from `4.01` (NN) to `1.05` (maximally +separated). The far field is therefore at the order of magnitude of the product of two +single-point pivotal probabilities — the natural "two (quasi-)independent pivotal" +scale — while the NN field sits a factor 2.3–4.0 above it, with the factor *growing* +with L. + +Why `unresolved` for the fusion reading: +* three sizes, and only five orbits (two at L=3) — no exponent may be written; +* the L=4 ordering is non-monotone in distance, showing the "profile" is still + dominated by finite-size/torus effects; +* no arm classifier was run, so nothing here identifies a 6-arm or 8-arm object; +* `A_d` is order-1-bounded, so "compatibility" between the far-field level and any of + {pivotal², 6-arm fusion, 8-arm fusion} magnitudes cannot be turned into an argument, + in either direction. → `unresolved` for all three named scales. + +### D3 — Is there a fact that directly opposes #768? + +**Verdict: `unresolved` / nothing found.** + +No configuration class in this atlas can be certified as "matching-odd and not removed +by signed balance": that classification needs the Phase-C arm witness, which was +deliberately not attempted (§T5). The nearest adjacent observation is that the **total +signed** pair interaction `Σ_{d≠0}J_d` is positive at all three sizes +(0.2596 / 0.1833 / 0.1366) and *decreasing*, while the absolute activity grows — so any +candidate "first non-common correction" that reads `M″` (or `Σ_d J_d`) directly as an +unsuppressed amplitude is, at these sizes, sitting inside a progressively stronger +cancellation. This is a **constraint**, not a counterexample, and I do not claim it +refutes or confirms any arm candidate in #768. + +--- + +## §6 Files + +Cloud (`/workspace/rev769/out/`, all downloaded to `rev769-out/`): + +* `controls-20260914.json` — T1/T2/T3 exact-rational controls + identity checks +* `orbit-table-20260914.json` — T4 D4 orbit atlas (exact integer histograms + moments) +* `pivotal_L{3,4,5}_pathA.json`, `pivotal_L{3,4,5}_pathB_uf2.json`, + `pivotal_L{3,4,5}_pathB_cov.json` — raw per-`d` integer histograms, three paths +* `witness_L{3,4,5}.json` — Phase C raw material +* `note-tables.md` — machine-rendered markdown tables used in this note + +Scripts (`rev769-work/`, also on the cloud in `/workspace/rev769/scripts/`): + +* `pivotal.c` (path A), `pivotal_bis.c` (path B), `witness.c` +* `analyze_pivotal.py` (exact-rational post-processing), `compare_paths.py`, + `make_note_tables.py` + +Reproduction (on the cloud VM; `PYTHONPATH` not needed for these): + +```sh +gcc -O2 -march=native -pthread -o pivotal pivotal.c +gcc -O2 -march=native -pthread -o pivotal_bis pivotal_bis.c +gcc -O2 -march=native -o witness witness.c +for L in 3 4 5; do + ./pivotal $L --threads 16 --out out/pivotal_L${L}_pathA.json + ./pivotal_bis $L --kernel uf2 --threads 16 --out out/pivotal_L${L}_pathB_uf2.json + ./pivotal_bis $L --kernel cov --threads 16 --out out/pivotal_L${L}_pathB_cov.json +done +python3 compare_paths.py # -> ALL_IDENTICAL +python3 analyze_pivotal.py # -> controls-20260914.json, orbit-table-20260914.json +``` + +All computation was performed on `DevEnvC_551oUR`; nothing was computed on the Mac. + +--- + +## §8 What I cannot claim + +1. **No exponent is certified.** L=3,4,5 are three finite sizes. Every statement about + the L-dependence of `cancel_ratio`, `A2_abs`, `A_d` is `compatible / incompatible / + unresolved` only — never "measured exponent". +2. **The absolute influence is not a CFT operator.** `A_d = E|Δ₀Δ_dX|` and `A2_abs` + are pure lattice diagnostics; #769 explicitly forbids calling them CFT operators, + and I attach no field-theoretic identity to them. +3. **The identities of §T2 are consistency checks**, implied by Russo + translation + invariance; they are not independent evidence about #768. +4. **`M″` is not "small" in absolute terms** — it grows with L; the small quantity is + the *ratio* `|M″|/A2_abs`. +5. **The near/far structure is not an asymptotic two-region structure.** The L=5 torus + diameter is `2√2`; the "far" region is 2–3 lattice spacings. +6. **Phase C is not a classifier.** No arm count, 6-arm, 8-arm or four-cluster label is + claimed anywhere in this delivery. +7. **L=6+ was not computed** (not an acceptance condition). No MC was added; no STATUS, + `#275` contract, or original data was modified. +8. Numbers quoted from #769/#775 are marked 引用; the T1 control values are **not** + 引用 — they were independently reproduced here. diff --git a/results/geometric-consistency/coalescent-bulk-clock.json b/results/geometric-consistency/coalescent-bulk-clock.json new file mode 100644 index 000000000..94f91c30d --- /dev/null +++ b/results/geometric-consistency/coalescent-bulk-clock.json @@ -0,0 +1 @@ +{"conditioning":"Terminal cells uniformly relabelled; cyclic order forgotten. 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--git a/results/geometric-consistency/oblique-magnetic-metric-controls-20260914.json b/results/geometric-consistency/oblique-magnetic-metric-controls-20260914.json new file mode 100644 index 000000000..493971205 --- /dev/null +++ b/results/geometric-consistency/oblique-magnetic-metric-controls-20260914.json @@ -0,0 +1,17 @@ +{ + "schema": "oblique-magnetic-metric-v1", + "target_2pi_5_over_48": 0.6544984694978736, + "definition": "scaled gap = n*(a^2+b^2)*I0 because one Bezout t-step has physical normal height 1/|u|", + "claim_boundary": [ + "deterministic safe-transfer values evaluated at the nearby charge-coexistence root", + "magnetic 5/48 interpretation additionally uses percolation universality / the pDTL sector dictionary" + ], + "records": [ + {"direction": [1,1], "n": 4, "I0": 0.08294509721199142, "scaled_gap": 0.6635607776959314, "ratio_to_target": 1.0138461869972137}, + {"direction": [1,1], "n": 5, "I0": 0.06605187490118845, "scaled_gap": 0.6605187490118846, "ratio_to_target": 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charge-coexistence roots", + "reference_pc": 0.59274605079, + "prediction": "p_root-pc ~ -A*cos(4theta)/ell^4", + "interpretation_boundary": [ + "roots are deterministic safe-transfer outputs", + "reference_pc is used only to form the displayed scaling combinations", + "the spin-four angular law and common amplitude are scaling conjectures", + "small n and directions close to a cos(4theta) node can have relatively large higher-order contamination" + ], + "records": [ + {"direction": [1,0], "n": 8, "charge_root_p": 0.5926727605746268, "physical_circumference": 8.0, "cos_4theta": 1.0, "root_minus_pc": -0.00007329021537316738, "shift_times_ell4": -0.3001967221684936, "A_estimate": 0.3001967221684936}, + {"direction": [1,0], "n": 9, "charge_root_p": 0.5927006240698091, "physical_circumference": 9.0, "cos_4theta": 1.0, "root_minus_pc": -0.000045426720190921976, "shift_times_ell4": -0.2980447111726391, "A_estimate": 0.2980447111726391}, + {"direction": [1,1], "n": 4, "charge_root_p": 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diff --git a/results/geometric-consistency/rank-sector-C-L5-20260914.json b/results/geometric-consistency/rank-sector-C-L5-20260914.json new file mode 100644 index 000000000..e3b03a758 --- /dev/null +++ b/results/geometric-consistency/rank-sector-C-L5-20260914.json @@ -0,0 +1,11 @@ +{ + "L": 5, + "N": 25, + "threads": 8, + "mask_lo": 0, + "mask_hi": 33554432, + "rank0": [1, 25, 300, 2300, 12650, 53120, 176900, 478700, 1068575, 1982350, 3054880, 3869650, 3931075, 3067350, 1723100, 639850, 141575, 15900, 550, 0, 0, 0, 0, 0, 0, 0], + "rank1": [0, 0, 0, 0, 0, 10, 200, 2000, 13000, 60600, 213280, 581350, 1229100, 1970750, 2301450, 1861060, 994350, 340550, 72700, 9000, 520, 0, 0, 0, 0, 0], + "rank2": [0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 600, 6400, 40125, 162200, 432850, 767850, 907050, 725125, 407450, 168100, 52610, 12650, 2300, 300, 25, 1], + "wall_seconds": 9.855 +} diff --git a/results/geometric-consistency/rank-sector-C-L6-20260914.json b/results/geometric-consistency/rank-sector-C-L6-20260914.json new file mode 100644 index 000000000..4146a7bd5 --- /dev/null +++ b/results/geometric-consistency/rank-sector-C-L6-20260914.json @@ -0,0 +1,11 @@ +{ + "L": 6, + "N": 36, + "threads": 8, + "mask_lo": 0, + "mask_hi": 68719476736, + "rank0": [1, 36, 630, 7140, 58905, 376992, 1947780, 8347320, 30254904, 94088944, 253787076, 598517172, 1241138298, 2270804400, 3669278472, 5226303348, 6525511038, 7070366736, 6544591544, 5064164712, 3186853893, 1579450836, 595366164, 164526840, 31987068, 4124196, 313290, 10656, 72, 0, 0, 0, 0, 0, 0, 0, 0], + "rank1": [0, 0, 0, 0, 0, 0, 12, 360, 5436, 54336, 399780, 2288088, 10538070, 39962304, 126774180, 339782112, 772416936, 1485756360, 2397393616, 3199280040, 3466778256, 2990089320, 2015358768, 1046966400, 415851744, 125773992, 28780308, 4889208, 589680, 45576, 1704, 0, 0, 0, 0, 0, 0], + "rank2": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 1332, 22896, 244548, 1817100, 9944136, 41373504, 133150140, 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a residual / assess fit quality" + } + } +} \ No newline at end of file diff --git a/results/geometric-consistency/rank1-slope-harmonic-L3-20260914.json b/results/geometric-consistency/rank1-slope-harmonic-L3-20260914.json new file mode 100644 index 000000000..4f22d45cc --- /dev/null +++ b/results/geometric-consistency/rank1-slope-harmonic-L3-20260914.json @@ -0,0 +1,39 @@ +{ + "L": 3, + "N": 9, + "p": "1/2", + "configurations_checked": 512, + "rank_one_configurations": 162, + "rank_one_cardinality_counts": { + "3": 6, + "4": 36, + "5": 72, + "6": 48 + }, + "slope_counts": { + "0,1": 78, + "1,-1": 3, + "1,0": 78, + "1,1": 3 + }, + "z4_unnormalized_real": "75/256", + "z4_unnormalized_real_float": 0.29296875, + "z4_unnormalized_imag": "0", + "z4_conditional_real": "25/27", + "z4_conditional_real_float": 0.9259259259259259, + "z4_conditional_imag": "0", + "integrated_rank_one_probability": "3/20", + "integrated_z4_real": "19/140", + "integrated_z4_imag": "0", + 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b/results/geometric-consistency/safe-transfer-momentum-spectrum-w5-w9-20260914.json @@ -0,0 +1,46 @@ +{ + "schema": "safe-transfer-momentum-spectrum-v1", + "claim_boundary": [ + "deterministic small-width transfer spectroscopy", + "the first excited eigenvalue is numerically twofold degenerate", + "translation phases identify lattice momenta +/-1", + "the magnetic level-one interpretation additionally uses the pDTL/CFT sector dictionary" + ], + "records": [ + { + "width": 5, + "charge_root_p": 0.5922358232050263, + "safe_states": 51, + "G4": {"lambda0": 0.8750628303941339, "lambda1": 0.22849736881152655, "scaled_gap": 6.713854973622314, "gap_over_2pi": 1.0685432062540978, "rotation_eigenvalues": [{"real": 0.3090169943749476, "imag": -0.9510565162951536}, {"real": 0.30901699437494756, "imag": 0.9510565162951536}], "max_phase_error": 2.01e-16}, + "G8_complement": {"lambda0": 0.8750628303941315, "lambda1": 0.265080282167181, "scaled_gap": 5.971314791322344, "gap_over_2pi": 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0.5927006240698073, + "w9_local_root": 0.5927006240691101, + "difference": -6.972200594645983e-13 + }, + "oblique_validation": { + "direction": [5, 2], + "n": 2, + "repository_root": 0.5927450615814472, + "local_root": 0.5927450615720459, + "difference": -9.401368572525826e-12 + } + }, + "pairs": [ + { + "ell": 5.0, + "orientation_1": {"u": [1,0], "n": 5, "H4": 1.0, "root": 0.5922358232050258}, + "orientation_2": {"u": [3,4], "n": 1, "H4": -0.8432, "root": 0.5931582013380546, "states_G4": 45, "states_G8": 147}, + "pc4perp": 0.5927362453692130, + "A4_hat": 0.3127638526162537, + "pc4perp_minus_diagnostic_reference": -9.805420786990204e-06, + "thermal_shape": { + "D_axis": 3.447892045996305, + "D_oblique": 3.4505542673624325, + "a2_axis": 0.021411069573031176, + "a2_oblique": 0.023067129472255116, + "a3_axis": 0.029039391466043542, + "a3_oblique": 0.029138601736312553 + } + }, + { + "ell": 8.06225774829855, + "ell_squared": 65, + "orientation_1": {"u": [1,8], "n": 1, "H4_exact": 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"H4_exact": "4633/7225", "H4": 0.6412456747404844, "root": 0.5927198385492305, "states_G4": 4997, "states_G8": 11865}, + "orientation_2": {"u": [6,7], "n": 1, "H4_exact": "-6887/7225", "H4": -0.9532179930795848, "root": 0.5927850166194443, "states_G4": 2767, "states_G8": 27659}, + "pc4perp": 0.5927460512227809, + "A4_hat": 0.2953416668, + "pc4perp_minus_diagnostic_reference": 4.327808111214137e-10, + "thermal_shape": { + "D_1": 3.39549610, + "D_2": 3.39551688, + "relative_D_difference": 6.12e-06, + "a2_1": 0.0141685, + "a2_2": 0.0143268, + "a3_1": 0.0239713, + "a3_2": 0.0239978, + "normalized_curve_difference_scale": "approximately 9e-6 to 3.8e-5 on |y|<=0.5" + } + }, + { + "ell": 10.0, + "orientation_1": {"u": [1,0], "n": 10, "H4": 1.0, "root": 0.5927163956300879, "states_G4": 8953, "states_G8": 8953}, + "orientation_2": {"u": [3,4], "n": 2, "H4": -0.8432, "root": 0.5927709164472407, "states_G4": 7259, "states_G8": 75541}, + "pc4perp": 0.5927459750664773, + "A4_hat": 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only about 4.3e-10 from the diagnostic reference after the fit; this is an exploratory finite-size result, not a rigorous enclosure.", + "Equal-ell physical thermal slopes are orientation-independent to 1e-4--1e-5, and root/slope-normalized curves differ by only O(1e-5--5e-5) on |y|<=0.5.", + "These finite controls strongly disfavor an orientation-independent exponent-four term as the dominant explanation of the observed root shift and support a predominantly thermal-tangent spin-four correction.", + "Same-ell angular projection acts as a finite-size improvement operator: it removes the observed H4 irrep before any residual exponent or module assignment is attempted." + ], + "claim_boundary": [ + "All new roots are finite deterministic transfer outputs from locally executed copies/reimplementations of committed algorithms, with published control roots reproduced to about 1e-11 or better.", + "The projected threshold estimates are not rigorous threshold enclosures; higher-irrep and scalar finite-size corrections remain.", + "No exponent is inferred from the projected residuals.", + "The angular-irrep hierarchy and TANGENT_SPIN4 interpretation remain scaling conjectures." + ] +} diff --git a/results/research-control-20260914/cross-node-post-h4-projector-20260914.json b/results/research-control-20260914/cross-node-post-h4-projector-20260914.json new file mode 100644 index 000000000..295b93c2a --- /dev/null +++ b/results/research-control-20260914/cross-node-post-h4-projector-20260914.json @@ -0,0 +1,105 @@ +{ + "schema": "cross-node-post-h4-projector-v1", + "date": "2026-09-14", + "purpose": "Replace infeasible same-ell multi-angle tomography by a bounded near-equal-ell pair that nearly nulls complementary angular harmonics.", + "algorithm_provenance": { + "semantics": "local execution of scripts/oblique_charge_transfer.py lifted-gain safe transfer on #771 branch", + "eigensolver": "scipy sparse eigs, tol=1e-12; brent root xtol/rtol about 1e-14", + "validation": { + "direction": [3, 2], + "n": 2, + "repository_root": 0.5928240591851011, + "local_root": 0.5928240591852450, + "difference": 1.44e-13 + }, + "warning": "The shipped oblique script used looser eig/root tolerances. Small post-H4 residuals must use a single p_c convention and tight tolerances." + }, + "directions": [ + { + "direction": [5, 2], + "n": 2, + "ell_squared": 116, + "ell": 10.770329614269007, + "H4_exact": "41/841", + "H4": 0.04875148632580262, + "H8_exact": "-703919/707281", + "H8": -0.9952465851620501, + "safe_states_G4": 22994, + "safe_states_G8": 131677, + "row_memory_G4": 5, + "row_memory_G8": 7, + "tight_root": 0.5927450615721411, + "repository_loose_root": 0.5927450615814472 + }, + { + "direction": [3, 2], + "n": 3, + "ell_squared": 117, + "ell": 10.816653826391967, + "H4_exact": "-119/169", + "H4": -0.7041420118343196, + "H8_exact": "-239/28561", + "H8": -0.008368054339833897, + "safe_states_G4": 19018, + "safe_states_G8": 206197, + "row_memory_G4": 3, + "row_memory_G8": 5, + "tight_root": 0.5927612086097266 + } + ], + "leading_H4_projector": { + "definition": "q4_i=H4_i/ell_i^4; choose weights w_i summing to one with sum w_i q4_i=0", + "weights": [0.934200379537736, 0.06579962046226387], + "A4_hat": 0.2932542820236196, + "pc_hat_perp_H4": 0.5927461240410858, + "diagnostic_pc": 0.5927460507921, + "residual_vs_diagnostic_pc": 7.324898587679485e-08, + "raw_angular_weights_after_projection": { + "H8": -0.9303103523915915, + "H4_squared": 0.034844823804204124 + }, + "interpretation": "This pair is close to an H4-node/H8-node cross filter: H8 survives with O(1) weight while H4^2 is suppressed to about 3.5 percent in the raw angular projector." + }, + "comparison_to_existing_equal_ell_projector": { + "axis_1_0_n10_vs_3_4_n2": { + "ell": 10.0, + "pc_hat_perp_H4": 0.5927459750664773, + "residual_vs_same_diagnostic_pc": -7.572562266133787e-08, + "effective_H8_weight": 0.6864, + "effective_H4_squared_weight": 0.8432 + }, + "key_observation": "At nearly the same physical scale, the two H4-projected residuals have opposite signs while their effective H8 weights also have opposite signs. A single angle-independent residual cannot naturally explain this sign reversal without additional channels." + }, + "reference_free_model_checks": { + "pure_scalar_ell_minus_7_fit_on_axis_34_ell5_ell10": { + "fit_pc": 0.5927460516782668, + "fit_amplitude": -0.7661178948280093, + "predicted_cross_node_pc_hat": 0.5927460061955850, + "observed_cross_node_pc_hat": 0.5927461240410858, + "miss": -1.178455008e-07, + "note": "The H4 ell^-6 dressing leakage from the 0.43 percent ell mismatch is about 1.6e-10 for A2 about 0.66, negligible relative to this miss." + }, + "pure_H4_squared_ell_minus_6_fit_on_axis_34_ell5_ell10": { + "fit_pc": 0.5927461295061164, + "fit_amplitude": -0.18315896479616048, + "predicted_cross_node_pc_hat": 0.5927461255146714, + "observed_cross_node_pc_hat": 0.5927461240410858, + "miss": 1.47e-09, + "caveat": "This three-point geometry fit is excellent but its fitted asymptotic pc is about 7.9e-8 above the independent diagnostic pc. Therefore it cannot be promoted as the full asymptotic mechanism without an additional channel." + } + }, + "diagnostic_H8_amplitudes_assuming_root_power_27_over_4": { + "N65_same_ell_projector": -0.5005840192, + "N85_same_ell_projector": 0.0062896510, + "ell10_axis_34_projector": -0.6203911276, + "cross_node_projector": -0.7306793201, + "note": "N85 is a near-cancellation geometry. The other three signs/magnitudes are consistent with a negative H8-like component plus radial dressing, but no exponent is inferred." + }, + "claim_boundary": [ + "This is deterministic finite transfer evidence, not an asymptotic decomposition theorem.", + "The diagnostic pc is not used to locate any root; it is used only in the displayed residual/amplitude diagnostics.", + "The cross-node result overturns the earlier ranking of a pure scalar ell^-7 residual as the default post-H4 explanation; it does not uniquely identify the H8 continuum field.", + "A mixed H4^2 ell^-6 contribution can mimic the new projector very well over these scales but gives a shifted fitted pc, so a multi-channel explanation remains live.", + "Do not revive N1105 under the current row-memory solver; the low-memory cross-node route is the intended bounded replacement." + ] +} diff --git a/results/research-control-20260914/equal-ell-spin4-pilot.json b/results/research-control-20260914/equal-ell-spin4-pilot.json new file mode 100644 index 000000000..79e90c76f --- /dev/null +++ b/results/research-control-20260914/equal-ell-spin4-pilot.json @@ -0,0 +1,123 @@ +{ + "schema": "equal-ell-spin4-pilot-v2", + "date": "2026-09-14", + "purpose": "Bounded post-compass diagnostic for spin-4 numerator, scalar thermal denominator, reference-free critical cancellation, and tangent-vs-normal charge response.", + "provenance": { + "repository_algorithm": "scripts/oblique_charge_transfer.py at 6192f7df2e16627a9e5d2bef17761d7af5ea8eed", + "execution": "Local copy of the repository algorithm; no repository-wide CI and no external compute.", + "control": { + "direction": [5, 2], + "n": 2, + "published_root": 0.5927450615814472, + "local_root": 0.5927450615720459, + "difference": -9.401368572525826e-12 + } + }, + "new_oblique_pilot": { + "direction": [3, 4], + "n": 2, + "physical_circumference": 10.0, + "cos_4theta": -0.8432, + "safe_states_G4": 7259, + "safe_states_G8": 75541, + "row_memory_G4": 4, + "row_memory_G8": 7, + "charge_root_p": 0.5927709164472407, + "physical_thermal_slope_indicator": { + "definition": "|u| * d_p Theta_row(p_root) * ell^(1/4)", + "value": 3.3918787405216855 + }, + "root_slope_normalized_curve": { + "definition": "Q= n*|u|^2*Theta_row; y=Q_p(root)*(p-p_root)", + "points": [ + {"y": -0.5, "p": 0.566557134262247, "Q": -0.49951877568091263}, + {"y": -0.25, "p": 0.5796640253547438, "Q": -0.2495230386586551}, + {"y": 0.25, "p": 0.6058778075397375, "Q": 0.25121323253987055}, + {"y": 0.5, "p": 0.6189846986322344, "Q": 0.5063680240938175} + ], + "local_coefficients_Q_equals_y_plus_a2_y2_plus_a3_y3": { + "a2": 0.0134624997637573, + "a3": 0.02356480057684157 + } + } + }, + "near_equal_circumference_control": { + "axis": { + "direction": [1, 0], + "n": 9, + "ell": 9.0, + "root": 0.5927006240698073, + "physical_thermal_slope_indicator": 3.396596345288809, + "descriptive_local_a2": 0.0143802939, + "descriptive_local_a3": 0.0241007713 + }, + "oblique": { + "direction": [2, 1], + "n": 4, + "ell": 8.94427190999916, + "cos_4theta": -0.28, + "safe_states_G4": 2866, + "safe_states_G8": 17051, + "row_memory_G4": 2, + "row_memory_G8": 3, + "charge_root_p": 0.5927592096219316, + "physical_thermal_slope_indicator": 3.3970869975426115, + "local_a2": 0.014587913300682766, + "local_a3": 0.024155436009950897, + "normalized_curve_points": [ + {"y": -0.8, "Q": -0.8025853778409048, "Q_minus_y": -0.0025853778409047345}, + {"y": -0.4, "Q": -0.3991841749509856, "Q_minus_y": 0.0008158250490143959}, + {"y": 0.4, "Q": 0.4039074495488476, "Q_minus_y": 0.003907449548847586}, + {"y": 0.8, "Q": 0.822140277631288, "Q_minus_y": 0.022140277631288008} + ] + }, + "comparison": { + "relative_ell_mismatch": 0.006191010000093334, + "relative_thermal_slope_difference_oblique_over_axis_minus_1": 0.00014445155572095825, + "relative_a2_difference": 0.014437772956337653, + "relative_a3_difference": 0.002268144765027118, + "reference_free_pc_hat": 0.5927461435419498, + "reference_free_A4_hat": 0.2986532567271586, + "pc_hat_minus_diagnostic_reference_0_59274605079": 9.275194978730639e-08, + "normalized_curve_residual_difference_after_axis_interpolation": [ + {"y": -0.8, "oblique_minus_axis": 0.0004779664765172994}, + {"y": -0.4, "oblique_minus_axis": 0.00005319644967300775}, + {"y": 0.4, "oblique_minus_axis": -0.00004776900812846017}, + {"y": 0.8, "oblique_minus_axis": 0.00005580918564660574} + ] + } + }, + "axial_comparators": { + "w8": { + "physical_thermal_slope_indicator": 3.403146112835565, + "local_normalized_a2_from_direct_derivatives": 0.015644049288630238, + "local_normalized_a3_from_direct_derivatives": 0.024826840034183777 + }, + "w9": { + "physical_thermal_slope_indicator_from_root_centered_curve": 3.396596345288809, + "descriptive_local_normalized_a2": 0.0143802939, + "descriptive_local_normalized_a3": 0.0241007713 + } + }, + "reference_free_two_orientation_inversions": [ + {"pair": "axis9 + (2,1)n4", "pc_hat": 0.5927461435419498, "A4_hat": 0.2986532567271586}, + {"pair": "axis8 + (2,1)n4", "pc_hat": 0.592746072183, "A4_hat": 0.300284}, + {"pair": "axis8 + diagonal(1,1)n5", "pc_hat": 0.592745892926, "A4_hat": 0.299550}, + {"pair": "axis8 + (3,2)n2", "pc_hat": 0.592745970903, "A4_hat": 0.299870}, + {"pair": "axis9 + diagonal(1,1)n5", "pc_hat": 0.592746173374, "A4_hat": 0.298849}, + {"pair": "axis9 + new(3,4)n2", "pc_hat": 0.5927458798726768, "A4_hat": 0.29692332262678434} + ], + "proposed_exact_equal_ell_pair": { + "axis": {"direction": [1, 0], "n": 10, "ell": 10.0, "H4": 1.0}, + "oblique": {"direction": [3, 4], "n": 2, "ell": 10.0, "H4": -0.8432, "root_already_measured": 0.5927709164472407}, + "spin4_cancel_estimator": "pc_hat=(625*p_34+527*p_axis10)/1152", + "amplitude_estimator": "A4_hat=10000*(625/1152)*(p_34-p_axis10)" + }, + "interpretation_boundary": [ + "The new oblique values are deterministic outputs of a local copy of the committed transfer algorithm, with committed roots reproduced to about 1e-11 or better.", + "The reference-free pc inversions use only finite-width roots and the assumed leading cos(4 theta)/ell^4 form; their agreement is evidence for that form, not an independent theorem for pc.", + "The axis-w9 normalized coefficients use the existing descriptive root-centered curve fit rather than a dedicated derivative calculation; the axis-w8 values use direct derivative controls.", + "The near-equal-circumference axis9/(2,1)n4 comparison is the strongest current finite diagnostic for TANGENT_SPIN4: root and raw orientation effects differ strongly, while physical thermal slope and root/slope-normalized local shape nearly coincide.", + "The next decisive bounded calculation is the dedicated axial w=10 root/derivative, which completes the exact equal-circumference pair with the already measured (3,4)n2 control." + ] +} diff --git a/results/research-control-20260914/even-omega2-same-ell-angular-decomposition-20260914.json b/results/research-control-20260914/even-omega2-same-ell-angular-decomposition-20260914.json new file mode 100644 index 000000000..86f0e0374 --- /dev/null +++ b/results/research-control-20260914/even-omega2-same-ell-angular-decomposition-20260914.json @@ -0,0 +1,48 @@ +{ + "schema": "even-omega2-same-ell-angular-decomposition-v1", + "date": "2026-09-14", + "definition": "At each fixed physical ell, solve (ell*I0 - 2pi*5/48)*ell^2 = A0(ell)+A4(ell)*cos(4theta) from two orientations at their charge roots.", + "target_2pi_5_over_48": 0.6544984694978736, + "pairs": [ + { + "ell_squared": 25, + "ell": 5.0, + "orientations": [ + {"direction":[1,0],"n":5,"H4":1.0,"root":0.5922358232050267,"scaled_gap":0.6672979451886933,"states_G4":51,"states_G8":51}, + {"direction":[3,4],"n":1,"H4":-0.8432,"root":0.5931582013380804,"scaled_gap":0.6658695720920044,"states_G4":45,"states_G8":147} + ], + "A0": 0.3006133422394473, + "A4": 0.019373550031046183 + }, + { + "ell_squared": 65, + "ell": 8.06225774829855, + "orientations": [ + {"direction":[1,8],"n":1,"H4":0.8788165680473373,"root":0.5926839084966005,"scaled_gap":0.6597557372947834,"states_G4":1394,"states_G8":2153}, + {"direction":[4,7],"n":1,"H4":-0.48449704142011835,"root":0.5927803979494244,"scaled_gap":0.6593733135920737,"states_G4":1134,"states_G8":6216} + ], + "A0": 0.32569878726259877, + "A4": 0.01823317870775144 + }, + { + "ell_squared": 85, + "ell": 9.219544457292887, + "orientations": [ + {"direction":[2,9],"n":1,"H4":0.6412456747404844,"root":0.5927198385491780,"scaled_gap":0.6585618113002926,"states_G4":4997,"states_G8":11865}, + {"direction":[6,7],"n":1,"H4":-0.9532179930795848,"root":0.5927850166195059,"scaled_gap":0.6582319478504072,"states_G4":2767,"states_G8":27659} + ], + "A0": 0.33410784870195587, + "A4": 0.01758484298271412 + } + ], + "observations": [ + "The common omega~2 correction is predominantly angular-scalar, but a nonzero spin4 component is resolved at equal physical circumference and is not an artifact of mixing different ell values.", + "A4(ell) drifts smoothly from about 0.0194 to 0.0176 over ell=5 to 9.22; a simple asymptotic extrapolation is not claimed from three sizes.", + "This supplies the even-spin4 parent channel needed by a T4 x I4 mixed q=2/H8 mechanism, while A0 supplies the separate scalar parent channel for T4 x S0 H4 dressing." + ], + "claim_boundary": [ + "All values are deterministic safe-transfer finite-size controls evaluated at the orientation-specific charge root.", + "The extraction gives angular coefficients of the common magnetic-gap correction, not a direct normalized CFT coupling.", + "Do not form a universal mixed-amplitude ratio without deriving the lattice-to-CFT normalization and contact-term convention." + ] +} diff --git a/results/research-control-20260914/gaussian-h8-node-q2-control-20260914.json b/results/research-control-20260914/gaussian-h8-node-q2-control-20260914.json new file mode 100644 index 000000000..5251cc672 --- /dev/null +++ b/results/research-control-20260914/gaussian-h8-node-q2-control-20260914.json @@ -0,0 +1,123 @@ +{ + "schema": "gaussian-h8-node-q2-control-v2", + "date": "2026-09-14", + "freeze_provenance": { + "H8_node_control": "Issue #768 comment 5666850020 was posted before the (5,1) target roots were evaluated.", + "positive_H8_control": "Issue #768 comment 5666992538 was posted before the (7,1),n=1 parent root was evaluated." + }, + "model": "p(u,n)=pc+c4*H4*ell^-4+c46*H4*ell^-6+c8*H8*ell^-6+higher", + "training_lineage": { + "parent": [2,1], + "child": [3,1], + "gaussian_multiplier": "1+i up to square symmetries", + "H4_parent": -0.28, + "H4_child": 0.28, + "H8_common": -0.8432, + "n_values": [2,3], + "tight_roots": { + "parent_n2": 0.5929714652070754, + "child_n2": 0.5926956531470028, + "parent_n3": 0.5927885674009905, + "child_n3": 0.5927359840351146 + }, + "difference_fit": { + "definition": "D_n=p_child-p_parent=alpha*n^-4+beta*n^-6", + "alpha": -0.004136260375773327, + "beta": -0.0011069303415566429 + }, + "four_root_saturated_decomposition": { + "pc_fit": 0.5927460854444604, + "c4": -0.2954471696980948, + "c46": -0.11998694156938763, + "c8": -0.13631120371404618, + "note": "This is post-reveal training and is not itself new evidence." + } + }, + "frozen_H8_node_control": { + "parent": [3,2], + "child": [5,1], + "gaussian_relation": "(3+2i)(1+i)=1+5i", + "H4_parent_exact": "-119/169", + "H4_child_exact": "+119/169", + "H8_common_exact": "-239/28561", + "H8_common": -0.008368054339833897, + "resource_preflight": { + "child_n1_states_G4": 64, + "child_n1_states_G8": 99, + "child_n2_states_G4": 14278, + "child_n2_states_G8": 33757, + "n3_stopped_above_state_cap": 500000 + }, + "frozen_prediction": { + "alpha": -0.0015387334649554147, + "beta": -0.00004371729829274529, + "D1": -0.00158245076324816, + "D2": -0.00009685392434553756, + "predicted_H8_piece_of_beta": -4.542909034280429e-7, + "interpretation": "Because H8 is near zero, beta is a negative control for c8 and mainly measures c46 H4 dressing." + }, + "target_roots": { + "parent_n1": 0.5940218205591057, + "child_n1": 0.5924348977628815, + "parent_n2": 0.5928240591852278, + "child_n2": 0.5927268521423491 + }, + "observed_differences": { + "D1": -0.0015869227962241883, + "D2": -0.00009720704287874415 + }, + "observed_alpha_beta": { + "alpha": -0.0015447759826718126, + "beta": -0.00004214681355237578, + "relative_alpha_error_vs_frozen": 0.003927, + "relative_beta_error_vs_frozen": -0.03592 + } + }, + "two_lineage_difference_only_decomposition": { + "description": "Solve the two Gaussian-lineage beta coefficients for c46 and c8. pc cancels completely.", + "c46": -0.11561405460044301, + "c8": -0.13817818581653674, + "leading_c4_from_alpha_least_squares": -0.29558821336234653, + "comparison_to_training_saturated_values": { + "c46_training": -0.11998694156938763, + "c8_training": -0.13631120371404618 + }, + "interpretation": "The H8 q=2 coefficient remains stable when extracted using an H8-node negative control and no external pc." + }, + "frozen_positive_H8_control": { + "parent": [7,1], + "parent_n": 1, + "child": [3,4], + "child_n": 2, + "gaussian_relation": "(7+i)(1+i)=6+8i=2(3+4i)", + "H4_parent": 0.8432, + "H4_child": -0.8432, + "H8_common": 0.42197248, + "frozen_prediction": { + "alpha": 0.00012461999075356528, + "beta": 0.000001285523679808176, + "D_pred": 0.00012590551443337347, + "known_child_root": 0.5927709164472407, + "parent_root_pred": 0.5926450109328073 + }, + "target_parent_root": { + "root": 0.5926447378755775, + "safe_states_G4": 494, + "safe_states_G8": 765, + "row_memory_G4": 7, + "row_memory_G8": 8 + }, + "observed": { + "D": 0.00012617857166319357, + "D_minus_prediction": 2.7305722982010126e-7, + "relative_D_error": 0.002168747183544509 + }, + "interpretation": "This is an overdetermining held-out control with positive common H8. The frozen q=2 model predicts the child-parent difference to about 0.22 percent at n=1 despite strong preasymptotic conditions." + }, + "claim_boundary": [ + "n=1 is strongly preasymptotic; close agreement is a deterministic finite control, not an asymptotic theorem.", + "The decomposition establishes a robust H8 angular response with the q=2/root-ell^-6 scaling template within multiple Gaussian lineages. It does not by itself identify whether the H8 response is a nonlinear T4 x I4 response tensor or a distinct continuum block.", + "The simplest locked H0:H8=1:1 aligned-monomial model is not assumed. A general second-order spin4 x spin4 response has separate spin0 and spin8 tensor channels unless factorization is proved.", + "All primary Gaussian differences eliminate pc. External pc is not used in the alpha/beta, c46/c8, or positive-H8 held-out tests." + ] +} diff --git a/results/research-dispatch/audit-802-source-normalization-20260914.json b/results/research-dispatch/audit-802-source-normalization-20260914.json new file mode 100644 index 000000000..43eb13d2a --- /dev/null +++ b/results/research-dispatch/audit-802-source-normalization-20260914.json @@ -0,0 +1,1181 @@ +{ + "meta": { + "p_c": 0.5927460507921, + "team_json": "sector-root-response-20260914.json (commit 723f629)", + "analysis_note": "docs/research-bridges-post-compass-20260914.md / /tmp/concurrent-out/two-observable-response-and-angular-alias-20260914.md (\u00a77)", + "engine_weight_reading": "sector802_lib.build_matrices: R_ij = sum p^k (1-p)^(w-k) exp(gamma h) -> UNNORMALISED row weight (no /Z_row)", + "local_computation": "none; every number produced on DevEnvC_NePnUn" + }, + "C1a_identity_full_enumeration": { + "3x3": { + "n_configs": 512, + "N": 9, + 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Delta=-3.2758796085e-03 Dp=2.4659477621e+00 | A_open4=0.617686 A_open8=2.506993 | p_ch=0.591417170853 w4*off=0.340193 | 0.2s +w=5 Delta=-1.1766036623e-03 Dp=2.3063319729e+00 | A_open4=0.646017 A_open8=3.764601 | p_ch=0.592235823205 w4*off=0.318892 | 0.9s +w=6 Delta=-5.2257299199e-04 Dp=2.1894276107e+00 | A_open4=0.703049 A_open8=5.885781 | p_ch=0.592507356206 w4*off=0.309348 | 6.3s +w=7 Delta=-2.6522246900e-04 Dp=2.0980576358e+00 | A_open4=0.777206 A_open8=9.372740 | p_ch=0.592619633400 w4*off=0.303528 | 42.6s +w=8 Delta=-1.4830712223e-04 Dp=2.0235972867e+00 | A_open4=0.872497 A_open8=15.174843 | p_ch=0.592672760575 w4*off=0.300197 | 413.9s +/workspace/sectorA/scripts/s802b_c1_amplitude.py:135: RuntimeWarning: invalid value encountered in log + lw = np.log(np.array(w, float)); ly = np.log(np.array(y, float)) +-> /workspace/sectorA/out/closure-amplitude-raw.json (463.9s) +{ + "Delta_3param": { + "C": NaN, + "p": NaN, + "D": NaN, + "rms_log": NaN + }, + "Delta_pure": { + "C": NaN, + "p": 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'4.1232', '4.0826'] +w^17/4 D : ['-1.185995', '-1.099646', '-1.059961', '-1.035803', '-1.021632'] +w^4 D : ['-0.838625', '-0.735377', '-0.677255', '-0.636799', '-0.607466'] +w^4 off : ['0.340193', '0.318892', '0.309348', '0.303528', '0.300197'] +w^1/4 Dp : ['3.487377', '3.448771', '3.426639', '3.412651', '3.403271'] diff --git a/results/research-dispatch/delta-p-intrinsic-floor-20260914.json b/results/research-dispatch/delta-p-intrinsic-floor-20260914.json new file mode 100644 index 000000000..e8946910b --- /dev/null +++ b/results/research-dispatch/delta-p-intrinsic-floor-20260914.json @@ -0,0 +1,490 @@ +{ + "answer": { + "question": "N1105 should run now?", + "answer": "GO, conditional. With ONE common p_c and tight solver/root tolerances the per-orientation p_root is reproducible to F ~ 4.7e-16, so S/F ~ 5.8e3. With the shipped settings (two different p_c; ARPACK tol=1e-10; brentq xtol=3e-11) the scatter is F ~ 6.3e-12 > S, i.e. NO-GO.", + "S_over_F_tight": 5793.1588515823305, + "S_over_F_shipped_settings": 0.4295785079783902, + "verdict_tight": "GO", + "verdict_shipped": "NO-GO", + "fix_first": [ + "use a single common p_c for all four orientations", + "dense eig for n<=2500, else ARPACK tol<=1e-13", + "root-finder xtol<=1e-15 (NOT scipy brentq default 2e-12, NOT 3e-11)" + ], + "how_to_verify_fixed": [ + ">=2 independent paths on the SAME geometry must agree in p_root to <=1e-15", + "the axis (1,0) control: three independent automata must give identical safe-block spectra" + ] + }, + "p_c_common": 0.5927460507921, + "signal_S": { + "small_Cmix_0.0073": 2.71e-12, + "large_Cmix_0.0898": 3.33e-11 + }, + "step0_convention": { + "ell": 8.0, + "ell4": 4096.0, + "p_c_oblique_file": 0.59274605079, + "p_c_closure_file": 0.5927460507921, + "p_c_difference_closure_minus_oblique": 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a/results/research-dispatch/essential-lineage-merger-control.json b/results/research-dispatch/essential-lineage-merger-control.json new file mode 100644 index 000000000..9fe943bff --- /dev/null +++ b/results/research-dispatch/essential-lineage-merger-control.json @@ -0,0 +1,122 @@ +{ + "width": 3, + "height": 3, + "vertical_boundary": "free", + "physical_masks_crosschecked": 512, + "common_label_pairs": 19683, + "pairs_with_a_merger": 7, + "minimal_winding_witnesses": 3, + "explicit_merger": { + "early": 455, + "late": 463, + "ancestor_counts": [ + 2 + ], + "pairs": 1 + }, + "exact_probabilities": [ + { + "p_minus": { + "fraction": "1/5", + "decimal": 0.2 + }, + "p_plus": { + "fraction": "1/4", + "decimal": 0.25 + }, + "merger_probability": { + "fraction": "721/125000000", + "decimal": 5.768e-06 + }, + "factorial_pair_expectation": { + "fraction": "721/125000000", + "decimal": 5.768e-06 + }, + "lineage_loss_expectation": { + "fraction": "721/125000000", + "decimal": 5.768e-06 + }, + "one_witness_probability_at_p_plus": { + "fraction": "12097/262144", + "decimal": 0.046146392822265625 + }, + "two_disjoint_witnesses_probability_at_p_plus": { + "fraction": "95/131072", + "decimal": 0.00072479248046875 + }, + "BK_upper": { + "fraction": "146337409/68719476736", + "decimal": 0.0021294895705068484 + } + }, + { + "p_minus": { + "fraction": "1/4", + "decimal": 0.25 + }, + "p_plus": { + "fraction": "1/3", + "decimal": 0.3333333333333333 + }, + "merger_probability": { + "fraction": "217/7077888", + "decimal": 3.0658863208912034e-05 + }, + "factorial_pair_expectation": { + "fraction": "217/7077888", + "decimal": 3.0658863208912034e-05 + }, + "lineage_loss_expectation": { + "fraction": "217/7077888", + "decimal": 3.0658863208912034e-05 + }, + "one_witness_probability_at_p_plus": { + "fraction": "2107/19683", + "decimal": 0.10704669003708785 + }, + "two_disjoint_witnesses_probability_at_p_plus": { + "fraction": "79/19683", + "decimal": 0.0040136158105979775 + }, + "BK_upper": 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b/results/research-dispatch/intrinsic-clock-response-controls.json new file mode 100644 index 000000000..0cd01fa73 --- /dev/null +++ b/results/research-dispatch/intrinsic-clock-response-controls.json @@ -0,0 +1 @@ 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+{"check_summary":{"PBW_commutators":2,"module_dimension_sequences":2,"positive_law_cases":27,"response_mixing_equalities":48,"singular_equalities":2},"counterfamily":[{"L":4,"amplitude":"-1/5","intrinsic_odd_part":"0 exactly (c=2cosh b)","root":"767/1280"},{"L":8,"amplitude":"-1/5","intrinsic_odd_part":"0 exactly (c=2cosh b)","root":"12287/20480"},{"L":16,"amplitude":"-1/5","intrinsic_odd_part":"0 exactly (c=2cosh b)","root":"196607/327680"},{"L":4,"amplitude":"0","intrinsic_odd_part":"0 exactly (c=2cosh b)","root":"3/5"},{"L":8,"amplitude":"0","intrinsic_odd_part":"0 exactly (c=2cosh b)","root":"3/5"},{"L":16,"amplitude":"0","intrinsic_odd_part":"0 exactly (c=2cosh b)","root":"3/5"},{"L":4,"amplitude":"1/5","intrinsic_odd_part":"0 exactly (c=2cosh b)","root":"769/1280"},{"L":8,"amplitude":"1/5","intrinsic_odd_part":"0 exactly (c=2cosh b)","root":"12289/20480"},{"L":16,"amplitude":"1/5","intrinsic_odd_part":"0 exactly (c=2cosh 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Agreement ~1e-12 relative.", + "cw_bracket_width_max": 2.942091015256665e-14 + }, + "sampling": { + "value": 0.0, + "comment": "no Monte Carlo anywhere: the transfer matrices and the finite-torus brute force are exact enumerations, so the sampling error is identically zero." + }, + "finite_width": { + "comment": "SYSTEMATIC, not statistical: Delta_w and Delta'_w are only computed for w=4..8 and the local log-slopes drift with w.", + "Delta_local_slopes": [ + -4.588769756817091, + -4.451600006471014, + -4.3995634204722665, + -4.353161412053569 + ], + "Delta_slope_drift_per_step": [ + 0.13716975034607692, + 0.0520365859987475, + 0.04640200841869735 + ], + "Delta_prime_local_slopes": [ + -0.299886995944021, + -0.2853103507975162, + -0.27653568140796503, + -0.270612161438631 + ], + "root_shift_local_slopes": [ + -4.2888827608730695, + -4.166289655673498, + -4.123027739064302, + -4.082549250614939 + ], + "comment2": "5 widths => at most 4 local slopes; three-point/four-point agreement is 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\ No newline at end of file diff --git a/results/research-dispatch/theory-768-channels-20260914.json b/results/research-dispatch/theory-768-channels-20260914.json new file mode 100644 index 000000000..ef2416345 --- /dev/null +++ b/results/research-dispatch/theory-768-channels-20260914.json @@ -0,0 +1,436 @@ +{ + "marker": "PY39COMPAT_MARKER=1", + "G1": { + "kac_scalar_channels": [ + { + "k": 0, + "h": "-1/24", + "x": "-1/12", + "x_float": -0.08333333333333333, + "x_le_21_4": true, + "kac_labels": [ + [ + 2, + 3 + ], + [ + 4, + 6 + ], + [ + 6, + 9 + ] + ] + }, + { + "k": 1, + "h": "0", + "x": "0", + "x_float": 0.0, + "x_le_21_4": true, + "kac_labels": [ + [ + 1, + 1 + ], + [ + 3, + 4 + ], + [ + 5, + 7 + ], + [ + 7, + 10 + ] + ] + }, + { + "k": 2, + "h": "1/8", + "x": "1/4", + "x_float": 0.25, + "x_le_21_4": true, + "kac_labels": [ + [ + 2, + 2 + ], + [ + 4, + 5 + ], + [ + 6, + 8 + ] + ] + }, + { + "k": 3, + "h": "1/3", + "x": "2/3", + "x_float": 0.6666666666666666, + "x_le_21_4": true, + "kac_labels": [ + [ + 3, + 3 + ], + [ + 5, + 6 + ], + [ + 7, + 9 + ] + ] + }, + { + "k": 4, + "h": "5/8", + "x": "5/4", + "x_float": 1.25, + "x_le_21_4": true, + "kac_labels": [ + [ + 2, + 1 + ], + [ + 4, + 4 + ], + [ + 6, + 7 + ] + ] + }, + { + "k": 5, + "h": "1", + "x": "2", + "x_float": 2.0, + "x_le_21_4": true, + "kac_labels": [ + [ + 3, + 2 + ], + [ + 5, + 5 + ], + [ + 7, + 8 + ] + ] + }, + { + "k": 6, + "h": "35/24", + "x": "35/12", + "x_float": 2.9166666666666665, + "x_le_21_4": true, + "kac_labels": [ + [ + 4, + 3 + ], + [ + 6, + 6 + ] + ] + }, + { + "k": 7, + "h": "2", + "x": "4", + "x_float": 4.0, + "x_le_21_4": true, + "kac_labels": [ + [ + 3, + 1 + ], + [ + 5, + 4 + ], + [ + 7, + 7 + ] + ] + }, + { + "k": 8, + "h": "21/8", + "x": "21/4", + "x_float": 5.25, + "x_le_21_4": true, + "kac_labels": [ + [ + 4, + 2 + ], + [ + 6, + 5 + ] + ] + } + ], + "notes": [], + "c4_allowed_spin_s": { + "0": true, + "1": false, + "2": false, + "3": false, + "4": true, + "5": false, + "6": false, + "7": false, + "8": true + }, + "c4_rule": "square lattice C4: spin s allowed iff s ≡ 0 mod 4", + "thermal_chiral_tower": [ + { + "level": 0, + "x": "5/4", + "x_float": 1.25, + "spin": "±0", + "c4_allowed": true, + "irrelevant_vs_21_4": "below" + }, + { + "level": 1, + "x": "9/4", + "x_float": 2.25, + "spin": "±1", + "c4_allowed": false, + "irrelevant_vs_21_4": "below" + }, + { + "level": 2, + "x": "13/4", + "x_float": 3.25, + "spin": "±2", + "c4_allowed": false, + "irrelevant_vs_21_4": "below" + }, + { + "level": 3, + "x": "17/4", + "x_float": 4.25, + "spin": "±3", + "c4_allowed": false, + "irrelevant_vs_21_4": "below" + }, + { + "level": 4, + "x": "21/4", + "x_float": 5.25, + "spin": "±4", + "c4_allowed": true, + "irrelevant_vs_21_4": "equal" + }, + { + "level": 5, + "x": "25/4", + "x_float": 6.25, + "spin": "±5", + "c4_allowed": false, + "irrelevant_vs_21_4": "above" + }, + { + "level": 6, + "x": "29/4", + "x_float": 7.25, + "spin": "±6", + "c4_allowed": false, + "irrelevant_vs_21_4": "above" + } + ], + "identity_chiral_tower": [ + { + "level": 0, + "x": "0", + "spin": "±0", + "c4_allowed": true + }, + { + "level": 1, + "x": "1", + "spin": "±1", + "c4_allowed": false + }, + { + "level": 2, + "x": "2", + "spin": "±2", + "c4_allowed": false + }, + { + "level": 3, + "x": "3", + "spin": "±3", + "c4_allowed": false + }, + { + "level": 4, + "x": "4", + "spin": "±4", + "c4_allowed": true + }, + { + "level": 5, + "x": "5", + "spin": "±5", + "c4_allowed": false + } + ], + "scalar8arm_chiral_tower": [ + { + "level": 0, + "x": "21/4", + "spin": "±0", + "c4_allowed": true + }, + { + "level": 1, + "x": "25/4", + "spin": "±1", + "c4_allowed": false + }, + { + "level": 2, + "x": "29/4", + "spin": "±2", + "c4_allowed": false + }, + { + "level": 3, + "x": "33/4", + "spin": "±3", + "c4_allowed": false + }, + { + "level": 4, + "x": "37/4", + "spin": "±4", + "c4_allowed": true + } + ], + "scalars_below_21_4": [ + { + "k": 1, + "x": "0", + "x_float": 0.0 + }, + { + "k": 2, + "x": "1/4", + "x_float": 0.25 + }, + { + "k": 3, + "x": "2/3", + "x_float": 0.6666666666666666 + }, + { + "k": 4, + "x": "5/4", + "x_float": 1.25 + }, + { + "k": 5, + "x": "2", + "x_float": 2.0 + }, + { + "k": 6, + "x": "35/12", + "x_float": 2.9166666666666665 + }, + { + "k": 7, + "x": "4", + "x_float": 4.0 + } + ] + }, + "G3": { + "identity_T": "-b_g/b_p", + "identity_N": "-b_g*e_p/b_p + e_g", + "N_equals_ep_times_T_when_eg0": true, + "counterexample_T": "L**(-4)", + "counterexample_N": "0", + "counterexample_N_zero": true, + "counterexample_dpda_equals_Lm4": true, + "counterexample_root_shift": "p*-pc = a L^-4 (任意 a, 任意 C0)", + "note_counterexample": "b_L 与 C0 任意时 N==0 而 T=aL^-4，说明内禀曲线看不见纯热切向位移；非零 N 需要额外 even 响应", + "predictions": [ + { + "candidate": "scalar_8arm", + "operator": "(h,hbar) = (21/8, 21/8); 独立 primary; Kac label h_{4,2}=h_{2,7}", + "x": "21/4", + "spin": "0", + "in_thermal_module": false, + "theta_w_leading": "A_S * w^(-17/4)", + "theta_prime_leading": "B * w^(-1/4) (thermal primary x=5/4)", + "T_leading": "-(A_S/B) * w^(-4)", + "N_leading": "e_p * T = -(e_p A_S/B) w^(-4) [若源纯 dual-odd]", + "log_term": "无来自该算符的首阶 log（其模内 level<4 无手征伴随）", + "free_amplitudes": 1, + "companions_below_21_4": "无（其手征后代全部 >= 25/4）", + "image_in_TN": "1 维射线 N = e_p T" + }, + { + "candidate": "thermal_spin4", + "operator": "(h,hbar) = (5/8+4, 5/8) 及其反手征共轭; thermal 模 level-4 手征后代", + "x": "21/4", + "spin": "±4", + "in_thermal_module": true, + "theta_w_leading": "A_4 * w^(-17/4) * (1 + lambda * log w) [lambda != 0 条件性地由 (iii) log 模给出]", + "theta_prime_leading": "B * w^(-1/4)", + "T_leading": "-(A_4/B) * w^(-4) * (1 + lambda * log w)", + "N_leading": "e_p * T + (even 分量, 仅在不可分解参数非零时)", + "log_term": "是（若 level-2 null 被保留/嵌入 log 模）", + "free_amplitudes": "1（null 解耦 (i)）或 >=2（log / 保留 (ii)(iii)）", + "companions_below_21_4": "有：level 1,2,3 -> x=9/4,13/4,17/4, spin ±1,±2,±3（C4 禁止于标量观测量，可见于方向分辨观测量）", + "image_in_TN": "同一条射线 N = e_p T（纯 dual-odd 源），但 T(w) 带 log 因子；幅度数不同" + } + ], + "discriminators": [ + { + "id": "D1", + "observable": "C4 破缺（方向分辨）观测量，例如 <10> 缝 vs <11> 缝 / 格点各向异性", + "scalar_8arm": "无伴随修正；x<21/4 处只有它自己", + "thermal_spin4": "必须有伴随：x=9/4, 13/4, 17/4（spin ±1,±2,±3）", + "evidence": "模结构（我算）+ C4 选择定则（精确）。观测量是否可测未验证。" + }, + { + "id": "D2", + "observable": "根位移 T(w) 是否需要 log w 因子", + "scalar_8arm": "无首阶 log", + "thermal_spin4": "若 (iii) 成立则有 w^(-4)(1+lambda log w)", + "evidence": "条件性；lambda 的确切值未算。注意：h=21/8 在 c=0 扩展 Kac 表内，按 He(2024) 的一般陈述它同样可能是零范/log bottom，故 log 不是干净判据。" + }, + { + "id": "D3", + "observable": "(T,N) 平面上的可达像维数", + "scalar_8arm": "1 个自由幅度 A_S", + "thermal_spin4": "1（解耦）或 >=2（保留/log）", + "evidence": "模结构 + 我的等级计数。" + } + ] + } +} \ No newline at end of file diff --git a/results/research-dispatch/theory-768-level-dimensions-20260914.json b/results/research-dispatch/theory-768-level-dimensions-20260914.json new file mode 100644 index 000000000..ca2f6ad59 --- /dev/null +++ b/results/research-dispatch/theory-768-level-dimensions-20260914.json @@ -0,0 +1,533 @@ +{ + "marker": "PY39COMPAT_MARKER=1", + "source": "/workspace/theory768/out/g2_result.json", + "q_irreducible_quotient": [ + 1, + 0, + 0, + 1, + 1, + 1, + 2 + ], + "q_verma_retained": [ + 1, + 0, + 1, + 1, + 2, + 2, + 4 + ], + "q_generic_control": [ + 1, + 0, + 1, + 1, + 2, + 2, + 4 + ], + "note_q_definition": "q_n = dim(level n 商) - dim(L_{-1} 像) = 非导数手征类数", + "thermal_module_bilevel_irreducible_quotient": [ + { + "a": 0, + "b": 0, + "spin": 0, + "x": "5/4", + "x_float": 1.25, + "momentum_zero": true, + "non_deriv_dim": 1, + "c4_allowed": true, + "visible_in_momentum0_observable": true + }, + { + "a": 0, + "b": 1, + "spin": -1, + "x": "9/4", + "x_float": 2.25, + "momentum_zero": false, + "non_deriv_dim": 0, + "c4_allowed": false, + "visible_in_momentum0_observable": false + }, + { + "a": 0, + "b": 2, + "spin": -2, + "x": "13/4", + "x_float": 3.25, + "momentum_zero": false, + "non_deriv_dim": 0, + 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"visible_in_momentum0_observable": false + }, + { + "a": 1, + "b": 3, + "spin": -2, + "x": "21/4", + "x_float": 5.25, + "momentum_zero": false, + "non_deriv_dim": 0, + "c4_allowed": false, + "visible_in_momentum0_observable": false + }, + { + "a": 2, + "b": 0, + "spin": 2, + "x": "13/4", + "x_float": 3.25, + "momentum_zero": false, + "non_deriv_dim": 0, + "c4_allowed": false, + "visible_in_momentum0_observable": false + }, + { + "a": 2, + "b": 1, + "spin": 1, + "x": "17/4", + "x_float": 4.25, + "momentum_zero": false, + "non_deriv_dim": 0, + "c4_allowed": false, + "visible_in_momentum0_observable": false + }, + { + "a": 2, + "b": 2, + "spin": 0, + "x": "21/4", + "x_float": 5.25, + "momentum_zero": true, + "non_deriv_dim": 0, + "c4_allowed": true, + "visible_in_momentum0_observable": true + }, + { + "a": 3, + "b": 0, + "spin": 3, + "x": "17/4", + "x_float": 4.25, + "momentum_zero": false, + "non_deriv_dim": 1, + "c4_allowed": false, + "visible_in_momentum0_observable": false + }, + { + 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+ "torus_chiral_pi4_E4_coefficient": "1/24" + }, + "existing_data_diagnostics": [ + { + "L": 5, + "cardinality_sum_check": true, + "cylinder_root_input": "0.5922358232050258", + "normalization_only_not_rank_assignment_audit": true, + "ratio_pc_derivative": "-950.964558437368723291996", + "relative_deviation_from_E4_i": "0.0202268239781703722129468", + "shift_ratio": "1.4852083520445807045506", + "torus_root": "0.591988256518333844610968680211928879" + }, + { + "L": 6, + "cardinality_sum_check": true, + "cylinder_root_input": "0.592507356205638", + "normalization_only_not_rank_assignment_audit": true, + "ratio_pc_derivative": "-1970.77981232236608481856", + "relative_deviation_from_E4_i": "0.0100645373769641237477404", + "shift_ratio": "1.47041447230994514444914", + "torus_root": "0.592395070817704237693855807642505434" + } + ], + "existing_data_provenance": { + "cylinders": "results/geometric-consistency/fixed-width-charge-spectrum-derivatives-w4-w8-20260914.json; printed root values; no independent Perron rerun", + "pc_reference_not_certificate": "0.59274605079210", + "rank5": "results/geometric-consistency/rank-sector-C-L5-20260914.json; blob e3b03a7583b5346083907a4531ba51dc4f3d8819", + "rank6": "results/geometric-consistency/rank-sector-C-L6-20260914.json; blob 4146a7bd51ca8f2e5c964b5c157ce1b2c4bd1f2e" + }, + "modular_numerical": { + "E4_2i": "1.00083698843473765919291512747422264", + "E4_i": "1.45576289226870932246242200359886929", + "S_transform_error": "5.7273143e-71", + "T_transform_error": "1.8111358e-71", + "arithmetic_is_not_outward_interval": true, + "hex_E4_abs": "3.7061726e-71", + "hex_E4_derivative": "(-2.7163143322843540433439449752e-71 - 6.03508556884125381064036268813j)" + }, + "repo_snapshot": "8e1282f6d4397f03c80a2883b917c2d31a4b93a3", + "schema": "thermal-null-ward-modular-root-v1", + "scope": "Exact conditional Virasoro algebra; numerical modular functions; reanalysis of supplied rank counts. No site scaling theorem or new large enumeration." +} diff --git a/results/research-dispatch/thermal-window-clock-20260914.json b/results/research-dispatch/thermal-window-clock-20260914.json new file mode 100644 index 000000000..0ee992188 --- /dev/null +++ b/results/research-dispatch/thermal-window-clock-20260914.json @@ -0,0 +1,1567 @@ +{ + "schema": "matching-one.thermal-window-clock-and-effective-range.v1", + "date": "2026-09-14", + "author_agent": "bridge780b", + "machine": { + "alias": "DevEnvC_TVVfoB", + "account": 2, + "python": "3.9.9", + "numpy": "2.0.2", + "workdir": "/workspace/genealogy", + "note": "all numbers produced on this cloud machine; none on the local Mac" + }, + "scope": "Verification of the #780 companion derivation (thermal-window-no-merger-detailed-companion-20260914.md) plus the Q1 intensity-clock uniform error on an exact fixed-width component-retirement transfer. No exponential-window merger search, no Poisson-kernel/Laguerre/lineage-moment production.", + "inputs_read": [ + "/Users/lc/WorkBuddy/2026-09-13-12-33-57/ROUND-STATE-1745.md", + "issue #780 (body + 6 comments), #650, #801", + "branch analysis/geometric-balance-round2-integrated-20260914 : docs/WORK-NOW.md", + "dispatch-followthrough-20260914/{README-zh.md, EXECUTION.json, REMOTE_ACTIONS.json, repo/docs/manuscripts/geometric-balance/thermal-window-no-merger-detailed-companion-20260914.md, repo/scripts/essential_lineage_merger_control.py, repo/results/research-dispatch/essential-lineage-merger-control.json}", + "winding-intensity-20260913/repo/docs/manuscripts/geometric-balance/winding-intensity-and-prefactor.md (quoted closed forms nu_2, nu_3; NOT re-derived)" + ], + "A1_finite_check_independent_reproduction": { + "method": "own script a1_merger_control_independent.py (pure stdlib, nothing imported from the package). Two independent winding tests required to agree on all 512 masks: (T1) lift-potential BFS; (T2) explicit universal-cover exploration capped at |S|+1 lifted sites. Deliberately NOT the wrong 'm_wrap >= n_open' count.", + "reproduced": { + "physical_masks_crosschecked": 512, + "common_label_pairs": 19683, + "pairs_with_a_merger": 7, + "minimal_winding_witnesses": 3, + "explicit_merger": { + "early": 455, + "late": 463, + "ancestor_counts": [ + 2 + ], + "pairs": 1 + } + }, + "matches_package_json": true, + "own_derivation_of_721_125000000": { + "three_colour_weights": "(1/5)^e (1/20)^a (3/4)^c with e+a+c=9", + "integerisation": "(1/5)^e(1/20)^a(3/4)^c * 20^9 = 4^e 15^c (a cancels exactly)", + "identity": "sum over merging nested pairs of 4^e 15^c = 721*4096 = 2953216", + "computed": 2953216, + "other_two_pairs": { + "(1/4,1/3)": { + "integerisation": "(1/4)^e(1/12)^a(2/3)^c*12^9 = 2^(3c)3^e", + "identity": "= 217*729 = 158193", + "computed": 158193 + }, + "(2/5,3/5)": { + "integerisation": "(2/5)^e(1/5)^a(2/5)^c*5^9 = 2^(e+c)", + "identity": "= 1216", + "computed": 1216 + } + } + }, + "exact_probabilities": [ + { + "p_minus": "1/5", + "p_plus": "1/4", + "merger_probability": "721/125000000", + "factorial_pair_expectation": "721/125000000", + "lineage_loss_expectation": "721/125000000", + "one_witness_probability_at_p_plus": "12097/262144", + "two_disjoint_witnesses_probability_at_p_plus": "95/131072", + "BK_upper": "146337409/68719476736" + }, + { + "p_minus": "1/4", + "p_plus": "1/3", + "merger_probability": "217/7077888", + "factorial_pair_expectation": "217/7077888", + "lineage_loss_expectation": "217/7077888", + "one_witness_probability_at_p_plus": "2107/19683", + "two_disjoint_witnesses_probability_at_p_plus": "79/19683", + "BK_upper": "4439449/387420489" + }, + { + "p_minus": "2/5", + "p_plus": "3/5", + "merger_probability": "1216/1953125", + "factorial_pair_expectation": "1216/1953125", + "lineage_loss_expectation": "1216/1953125", + "one_witness_probability_at_p_plus": "1011933/1953125", + "two_disjoint_witnesses_probability_at_p_plus": "234009/1953125", + "BK_upper": "1024008396489/3814697265625" + } + ], + "per_pair_identities_asserted_on_all_19683_pairs": [ + "sum_C n_C == #early essential comps", + "0 <= loss <= mcount", + "mcount>0 implies two vertex-disjoint minimal witnesses inside the late mask", + "mcount <= C(h,2) = 3", + "sum event <= sum pairs <= C(h,2)*d", + "d <= q^2" + ], + "own_observation": "on the 3x3 free-vertical cylinder the merger probability, the factorial pair expectation and the lineage-loss expectation coincide exactly; i.e. every merger is one late essential comp containing exactly two early ones with loss 1. Minimal-width only; not extrapolated." + }, + "A2_step_by_step_verdicts": { + "1_merger_implies_two_disjoint_witnesses": { + "verdict": "holds (deterministic implication)", + "why": "two distinct essential clusters just before the merger have disjoint vertex sets and each contains a non-contractible cycle; the same uniform labels make occupancy monotone, so both cycles survive to p_+ and lie in one final cluster. Not an equivalence - the note says so explicitly.", + "extra_conditions_needed": [ + "same-label monotone coupling", + "'essential' means non-zero horizontal winding" + ] + }, + "2_Qw_bound_and_p_dependence": { + "verdict": "holds as an upper bound; p-dependence is self-consistent; two items must be flagged", + "cut_argument": "first reach of horizontal span w-1 forces the whole prefix into a window of w consecutive columns, so the site set injects into the plane; the simple subpath between the two extreme-x endpoints is a genuine plane connection. w = choice of starting column mod w; H^2 = the two endpoints' row indices (a deliberately loose union bound).", + "p_dependence": "single factor p, NOT p^w: the bound is on the two-point function t_p, which already sums over the interior; the w-site cost lives in e^{-(w-1)kappa(p)}. p^w would be a FALSE (too small) bound at fixed p_0, because kappa(p_0) < log(1/p_0); the two merge only in the sparse limit kappa(p) ~ log(1/p).", + "kappa_w_evidence_vs_log1over_p0": { + "p0=0.25": { + "log(1/p0)": 1.386, + "kappa_w": { + "3": 1.39575, + "4": 1.35488, + "8": 1.25299 + } + }, + "p0=0.40": { + "log(1/p0)": 0.916, + "kappa_w": { + "3": 0.96935, + "4": 0.8863, + "8": 0.71735 + } + }, + "p0=0.50": { + "log(1/p0)": 0.693, + "kappa_w": { + "3": 0.82099, + "4": 0.71153, + "8": 0.49629 + } + } + }, + "flagged": [ + "prefactor w*H^2 is a union-bound count, not tight: measured bound/Q grows like w^2.85", + "identification that the same kappa enters (2.1) and -(1/w)log nu_w -> kappa: an input, only internally cross-checked here (see B2 closure margins)" + ] + }, + "3_BK_on_occupied_witnesses": { + "verdict": "holds", + "why": "site BK (van den Berg-Kesten) gives Pr(A box B) <= Pr(A)Pr(B) for increasing events, which is an UPPER bound - the direction used. 'two vertex-disjoint occupied winding witnesses' is exactly E box E. BK is applied to the PLANE lift (product measure), not to cluster anchors with closed boundary conditions (those are non-monotone); the note makes this distinction correctly." + }, + "4_uniform_volume_tail_cycle_vs_plane_labels": { + "verdict": "holds", + "why": "a cylinder site has w plane translates; neither 'independent copies' nor 'one shared label' is right. The discovery-tree lift reads one fresh label per newly discovered cylinder site (distinct residues -> distinct plane sites, injective), giving |C_cyl| <= |C_plane| and hence the w-uniform plane volume tail [AV Thm 3].", + "flagged": [ + "[AV] Thm 3 is a quoted input, not re-proved here" + ] + }, + "5_mQw2_exponent_accounting": { + "verdict": "closes", + "why": "(1/w)log[(m+2H+2)Q_w^2] -> kappa(p_0) - 2 kappa(p_+) -> -kappa(p_0) < 0; the tail Cw(m+2H+2)e^{-cH} with H=w^2 is superexponentially dominated. The factor 2 (BK square) is essential: it is what permits p_+ - p_0 = o(1). For a fixed macroscopic interval the sufficient condition 2 kappa(p_b) > kappa(p_a) can genuinely fail." + } + }, + "A3_unclaimed_items": { + "affine_thermal_clock_p0_plus_x_over_vw_and_Lambda_exp_x": "not used by the theorem", + "uniformity_as_p0_to_pc": "not used (proof needs fixed compact subcritical p; the note flags the degeneration)", + "all_path_topologies_and_higher_moments": "not used (finite spatial windows x finite mark bins only)", + "actually_open_intermediate_steps": [ + "(a) uniformity of B_w/nu_0 -> 0 over the parameter window (the note itself asks for compact-uniform constants)", + "(b) the local log-intensity slope needed for p_0 + x/(v w) (the note flags this too)" + ], + "conclusion": "no unclaimed item is smuggled in; the soft spots are unclosed intermediate steps, not hidden assumptions" + }, + "B1_Qw_vs_bound": { + "method": "same exact transfer; Q_w(H;p)=Pr[a winding occupied closed path exists in an H-row free band] computed from the 'no winding so far' sub-chain", + "rows": [ + { + "w": 4, + "p": 0.25, + "H": 16, + "nu_w": 0.004429300954810734, + "kappa_w": 1.3548783763469074, + "Q": 0.06865019189930588, + "bound": 4.3953286892962184, + "bound_over_Q": 64.02500222786209, + "Q_over_H_nu": 0.968693940077946 + }, + { + "w": 4, + "p": 0.4, + "H": 16, + "nu_w": 0.02886241607310145, + "kappa_w": 0.886303753030951, + "Q": 0.3999532775997411, + "bound": 28.682001902345895, + "bound_over_Q": 71.71338130912835, + "Q_over_H_nu": 0.8660771775541009 + }, + { + "w": 4, + "p": 0.5, + "H": 16, + "nu_w": 0.05806909140463645, + "kappa_w": 0.7115304365685295, + "Q": 0.7291929753244208, + "bound": 60.56597794278336, + "bound_over_Q": 83.05891580460894, + "Q_over_H_nu": 0.7848333744401839 + }, + { + "w": 5, + "p": 0.25, + "H": 25, + "nu_w": 0.0013443169288498858, + "kappa_w": 1.3223738509047718, + "Q": 0.032786519632929, + "bound": 3.940863246593468, + "bound_over_Q": 120.19766936883045, + "Q_over_H_nu": 0.975559228015647 + }, + { + "w": 5, + "p": 0.4, + "H": 25, + "nu_w": 0.01601834274086165, + "kappa_w": 0.8268041584155194, + "Q": 0.3438839051527478, + "bound": 45.7724415270256, + "bound_over_Q": 133.10434376594102, + "Q_over_H_nu": 0.8587253019016118 + }, + { + "w": 5, + "p": 0.5, + "H": 25, + "nu_w": 0.04193460289723798, + "kappa_w": 0.6343287896146834, + "Q": 0.7459600288614474, + "bound": 123.56004387634196, + "bound_over_Q": 165.6389606624508, + "Q_over_H_nu": 0.7115460524945904 + }, + { + "w": 6, + "p": 0.25, + "H": 36, + "nu_w": 0.0004219779152266575, + "kappa_w": 1.295092929813972, + "Q": 0.014894672177433899, + "bound": 2.9952818558891496, + "bound_over_Q": 201.09753475656464, + "Q_over_H_nu": 0.9804799703685325 + }, + { + "w": 6, + "p": 0.4, + "H": 36, + "nu_w": 0.009202474183511567, + "kappa_w": 0.7813804830154581, + "Q": 0.28750199196112214, + "bound": 62.527358639118795, + "bound_over_Q": 217.48495797404473, + "Q_over_H_nu": 0.8678281823027149 + }, + { + "w": 6, + "p": 0.5, + "H": 36, + "nu_w": 0.03145597334932063, + "kappa_w": 0.5765277303757491, + "Q": 0.7543658307841303, + "bound": 217.67682608038396, + "bound_over_Q": 288.55605224605466, + "Q_over_H_nu": 0.6661566684956151 + }, + { + "w": 7, + "p": 0.25, + "H": 49, + "nu_w": 0.0001356156875547294, + "kappa_w": 1.2722407856025586, + "Q": 0.006535470565996682, + "bound": 2.03360823424258, + "bound_over_Q": 311.16477592649795, + "Q_over_H_nu": 0.9834920482384233 + }, + { + "w": 7, + "p": 0.4, + "H": 49, + "nu_w": 0.0054020251289817415, + "kappa_w": 0.7458544816900402, + "Q": 0.23392836888377644, + "bound": 76.56446940600564, + "bound_over_Q": 327.29877855919847, + "Q_over_H_nu": 0.8837515988132314 + }, + { + "w": 7, + "p": 0.5, + "H": 49, + "nu_w": 0.02416330970892604, + "kappa_w": 0.5318457034334962, + "Q": 0.7557152979244464, + "bound": 345.61634975012015, + "bound_over_Q": 457.3367122504295, + "Q_over_H_nu": 0.6382718827807896 + }, + { + "w": 8, + "p": 0.25, + "H": 64, + "nu_w": 4.432636548597084e-05, + "kappa_w": 1.2529913625315534, + "Q": 0.002795395582520954, + "bound": 1.271215856180789, + "bound_over_Q": 454.7534753683685, + "Q_over_H_nu": 0.9853741785058799 + }, + { + "w": 8, + "p": 0.4, + "H": 64, + "nu_w": 0.0032186746051071128, + "kappa_w": 0.717348202185326, + "Q": 0.18570664897112488, + "bound": 86.44250935410066, + "bound_over_Q": 465.478806671814, + "Q_over_H_nu": 0.9015097038916934 + }, + { + "w": 8, + "p": 0.5, + "H": 64, + "nu_w": 0.01886677707525762, + "kappa_w": 0.49629409130159696, + "Q": 0.7513761938354462, + "bound": 507.75644250872654, + "bound_over_Q": 675.7686052267006, + "Q_over_H_nu": 0.6222712539533484 + } + ], + "conclusions": [ + "bound holds at all 15 (w,p) points (Q >= 5e-6)", + "true Q ~ H*nu_w ~ w^2 A w^-beta e^{-kappa w}: same exponent as the bound", + "bound/Q grows 64 -> 455 (w=4 -> 8), i.e. ln(bound/Q) ~ 2.85 ln w: the polynomial prefactor is over-counted by ~w^2.9; harmless for (4.2), fatal for numeric substitution" + ] + }, + "B2_intensity_clock": { + "definition": "F_w(p)=log nu_w(p); v_w := -d kappa_w/dp at p_0 = F_w'(p_0)/w > 0; affine clock p_w(x)=p_0+x/(v_w w); phi_w(x)=F_w(p_w(x))-F_w(p_0)", + "exact_result": "E_w(X) := sup_{|x|<=X}|phi_w(x)-x| = |F_w''(p_0)| X^2 / (2 F_w'(p_0)^2) (1+o(1)) = |kappa_w''(p_0)| X^2 / (2 kappa_w'(p_0)^2 w) (1+o(1)) = O(X^2/w), the coefficient being computable at every (w,p_0)", + "table": [ + { + "p0": "0.25", + "w": 3, + "kappa_w": 1.3957493462042567, + "nu_w": 0.015188024270440522, + "v_w": 3.8666309405644355, + "F1": 11.599892821693306, + "F2": -52.26803907472839, + "Xmax_accessible": 1.3919871386031966, + "E_w_X1": 0.24685011522739142, + "predicted_E_at_X1": 0.19422192683624498, + "E_times_w_at_X1": 0.7405503456821743, + "E_window": { + "0.25": 0.012771190406966504, + "0.5": 0.05403986848185305, + "1.0": 0.24685011522739142, + "2.0": 0.5445024414175059 + } + }, + { + "p0": "0.25", + "w": 4, + "kappa_w": 1.3548783763469205, + "nu_w": 0.004429300954810502, + "v_w": 4.127718742886717, + "F1": 16.51087497154687, + "F2": -68.0308366812395, + "Xmax_accessible": 1.9813049965856242, + "E_w_X1": 0.1460749444676157, + "predicted_E_at_X1": 0.1247773158558752, + "E_times_w_at_X1": 0.5842997778704628, + "E_window": { + "0.25": 0.008077570960920433, + "0.5": 0.03355453189156421, + "1.0": 0.1460749444676157, + "2.0": 0.7100817653046312 + } + }, + { + "p0": "0.25", + "w": 5, + "kappa_w": 1.322373850904801, + "nu_w": 0.0013443169288496885, + "v_w": 4.3038580511923845, + "F1": 21.519290255961923, + "F2": -84.83046926399673, + "Xmax_accessible": 2.582314830715431, + "E_w_X1": 0.10286574373120327, + "predicted_E_at_X1": 0.0915937809563401, + "E_times_w_at_X1": 0.5143287186560164, + "E_window": { + "0.25": 0.005879345062556496, + "0.5": 0.024188922007236435, + "1.0": 0.10286574373120327, + "2.0": 0.47660918684386644 + } + }, + { + "p0": "0.25", + "w": 6, + "kappa_w": 1.29509292981526, + "nu_w": 0.0004219779152233961, + "v_w": 4.42983733070453, + "F1": 26.579023984227177, + "F2": -102.63847053700499, + "Xmax_accessible": 3.189482878107261, + "E_w_X1": 0.07959941837118123, + "predicted_E_at_X1": 0.07264439614085016, + "E_times_w_at_X1": 0.4775965102270874, + "E_window": { + "0.25": 0.004638568512795871, + "0.5": 0.018973350132196032, + "1.0": 0.07959941837118123, + "2.0": 0.3552759613018601 + } + }, + { + "p0": "0.25", + "w": 7, + "kappa_w": 1.272240785597338, + "nu_w": 0.0001356156875596853, + "v_w": 4.520987472688492, + "F1": 31.64691230881944, + "F2": -121.02396091817624, + "Xmax_accessible": 3.797629477058333, + "E_w_X1": 0.06516834271902638, + "predicted_E_at_X1": 0.06041971600718887, + "E_times_w_at_X1": 0.4561783990331847, + "E_window": { + "0.25": 0.0038446151693154462, + "0.5": 0.015666740933397705, + "1.0": 0.06516834271902638, + "2.0": 0.2844985998041558 + } + }, + { + "p0": "0.25", + "w": 8, + "kappa_w": 1.25299136256891, + "nu_w": 4.432636547272383e-05, + "v_w": 4.588614743600838, + "F1": 36.70891794880671, + "F2": -139.65818500090896, + "Xmax_accessible": 4.405070153856805, + "E_w_X1": 0.05528812134017791, + "predicted_E_at_X1": 0.051819501594368204, + "E_times_w_at_X1": 0.4423049707214233, + "E_window": { + "0.25": 0.003289311158413355, + "0.5": 0.0133688837353656, + "1.0": 0.05528812134017791, + "2.0": 0.23791767193016966 + } + }, + { + "p0": "0.4", + "w": 3, + "kappa_w": 0.9693481181602842, + "nu_w": 0.05458236932671215, + "v_w": 1.9902802783594329, + "F1": 5.970840835078299, + "F2": -30.47154504531857, + "Xmax_accessible": 0.7165009002093958, + "E_w_X1": 0.23679395114987756, + "predicted_E_at_X1": 0.42735962653221266, + "E_times_w_at_X1": 0.7103818534496327, + "E_window": { + "0.25": 0.02713911104806055, + "0.5": 0.11151313991888223, + "1.0": 0.23679395114987756, + "2.0": 0.23679395114987756 + } + }, + { + "p0": "0.4", + "w": 4, + "kappa_w": 0.8863037530309589, + "nu_w": 0.028862416073100538, + "v_w": 2.2778018467324395, + "F1": 9.111207386929758, + "F2": -41.29784199765248, + "Xmax_accessible": 1.093344886431571, + "E_w_X1": 0.26056467274220685, + "predicted_E_at_X1": 0.24873992474742762, + "E_times_w_at_X1": 1.0422586909688274, + "E_window": { + "0.25": 0.015597429950833241, + "0.5": 0.0629268586445968, + "1.0": 0.26056467274220685, + "2.0": 0.31438124111058086 + } + }, + { + "p0": "0.4", + "w": 5, + "kappa_w": 0.8268041584155086, + "nu_w": 0.01601834274086252, + "v_w": 2.46225972936025, + "F1": 12.31129864680125, + "F2": -51.49597779907644, + "Xmax_accessible": 1.47735583761615, + "E_w_X1": 0.17497136054487727, + "predicted_E_at_X1": 0.16987739011950187, + "E_times_w_at_X1": 0.8748568027243864, + "E_window": { + "0.25": 0.01064669487546599, + "0.5": 0.042840290022972294, + "1.0": 0.17497136054487727, + "2.0": 0.3933818877916062 + } + }, + { + "p0": "0.4", + "w": 6, + "kappa_w": 0.7813804830155014, + "nu_w": 0.009202474183509177, + "v_w": 2.584846376702433, + "F1": 15.509078260214599, + "F2": -61.22314784332931, + "Xmax_accessible": 1.8610893912257518, + "E_w_X1": 0.13009534261951172, + "predicted_E_at_X1": 0.12726637817608014, + "E_times_w_at_X1": 0.7805720557170703, + "E_window": { + "0.25": 0.007969598270841871, + "0.5": 0.032016054990367415, + "1.0": 0.13009534261951172, + "2.0": 0.4728103937835988 + } + }, + { + "p0": "0.4", + "w": 7, + "kappa_w": 0.7458544816900835, + "nu_w": 0.0054020251289801065, + "v_w": 2.670682107828763, + "F1": 18.69477475480134, + "F2": -70.56138472331479, + "Xmax_accessible": 2.2433729705761607, + "E_w_X1": 0.10239954406263507, + "predicted_E_at_X1": 0.10094774581756795, + "E_times_w_at_X1": 0.7167968084384455, + "E_window": { + "0.25": 0.006315092297057845, + "0.5": 0.02532684709589894, + "1.0": 0.10239954406263507, + "2.0": 0.428305276275768 + } + }, + { + "p0": "0.4", + "w": 8, + "kappa_w": 0.7173482021881835, + "nu_w": 0.0032186746050335354, + "v_w": 2.7337194402306717, + "F1": 21.869755521845374, + "F2": -79.67324761123014, + "Xmax_accessible": 2.6243706626214447, + "E_w_X1": 0.08441768969831731, + "predicted_E_at_X1": 0.08329034635982452, + "E_times_w_at_X1": 0.6753415175865385, + "E_window": { + "0.25": 0.005213365249175261, + "0.5": 0.020911119448146387, + "1.0": 0.08441768969831731, + "2.0": 0.3487821331940806 + } + }, + { + "p0": "0.5", + "w": 3, + "kappa_w": 0.8209905243072505, + "nu_w": 0.08518145161290265, + "v_w": 0.9673601832353445, + "F1": 2.9020805497060334, + "F2": -32.101990736545275, + "Xmax_accessible": 0.21111489907168043, + "E_w_X1": 0.08856588038029176, + "predicted_E_at_X1": 1.9058251091679497, + "E_times_w_at_X1": 0.26569764114087524, + "E_window": { + "0.25": 0.08856588038029176, + "0.5": 0.08856588038029176, + "1.0": 0.08856588038029176, + "2.0": 0.08856588038029176 + } + }, + { + "p0": "0.5", + "w": 4, + "kappa_w": 0.7115304365685393, + "nu_w": 0.058069091404634184, + "v_w": 1.1902372985437077, + "F1": 4.760949194174831, + "F2": -47.725668291299066, + "Xmax_accessible": 0.34634025189804957, + "E_w_X1": 0.13425326999088177, + "predicted_E_at_X1": 1.0527734618957065, + "E_times_w_at_X1": 0.5370130799635271, + "E_window": { + "0.25": 0.0687629578915816, + "0.5": 0.13425326999088177, + "1.0": 0.13425326999088177, + "2.0": 0.13425326999088177 + } + }, + { + "p0": "0.5", + "w": 5, + "kappa_w": 0.6343287896146892, + "nu_w": 0.04193460289723677, + "v_w": 1.3461782543244827, + "F1": 6.730891271622413, + "F2": -63.63441098977406, + "Xmax_accessible": 0.4896457583215463, + "E_w_X1": 0.1829056766884713, + "predicted_E_at_X1": 0.702290643700041, + "E_times_w_at_X1": 0.9145283834423565, + "E_window": { + "0.25": 0.04580183625653689, + "0.5": 0.1829056766884713, + "1.0": 0.1829056766884713, + "2.0": 0.1829056766884713 + } + }, + { + "p0": "0.5", + "w": 6, + "kappa_w": 0.5765277303757476, + "nu_w": 0.03145597334932093, + "v_w": 1.462149512171529, + "F1": 8.772897073029174, + "F2": -79.1511016270385, + "Xmax_accessible": 0.6381936160684453, + "E_w_X1": 0.23262520196150482, + "predicted_E_at_X1": 0.5142104519264287, + "E_times_w_at_X1": 1.395751211769029, + "E_window": { + "0.25": 0.03349677508550686, + "0.5": 0.13966573662347503, + "1.0": 0.23262520196150482, + "2.0": 0.23262520196150482 + } + }, + { + "p0": "0.5", + "w": 7, + "kappa_w": 0.5318457034334759, + "nu_w": 0.02416330970892949, + "v_w": 1.5540798017254198, + "F1": 10.878558612077939, + "F2": -93.98423815846388, + "Xmax_accessible": 0.7913721773390581, + "E_w_X1": 0.282485627654224, + "predicted_E_at_X1": 0.3970839356611675, + "E_times_w_at_X1": 1.9773993935795682, + "E_window": { + "0.25": 0.02583287398131251, + "0.5": 0.10765135745655874, + "1.0": 0.282485627654224, + "2.0": 0.282485627654224 + } + }, + { + "p0": "0.5", + "w": 8, + "kappa_w": 0.49629409130139324, + "nu_w": 0.01886677707528837, + "v_w": 1.6294041350962445, + "F1": 13.035233080769956, + "F2": -108.03799060086425, + "Xmax_accessible": 0.9482617277805527, + "E_w_X1": 0.33190869056133954, + "predicted_E_at_X1": 0.31791344867826615, + "E_times_w_at_X1": 2.6552695244907163, + "E_window": { + "0.25": 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"agrees to ~0.5-1%; the small deficit is the expected O(1/L) free-height boundary bias" + }, + "clock_deviation_phi_minus_x": { + "p0=0.40, delta=+0.01": { + "w=4": -0.0018, + "w=5": -0.0008, + "w=6": -0.002 + }, + "p0=0.40, delta=+0.02": { + "w=4": -0.007, + "w=5": -0.0103, + "w=6": -0.0125 + }, + "p0=0.40, delta=+0.04": { + "w=4": -0.032, + "w=5": -0.0441, + "w=6": -0.0447 + }, + "mc_se": { + "w=4": 0.003, + "w=5": 0.005, + "w=6": 0.006 + }, + "back_solved_C_w": { + "w=4": 0.249, + "w=5": 0.17, + "w=6": 0.122 + }, + "predicted_C_w": { + "w=4": 0.2487, + "w=5": 0.1699, + "w=6": 0.1273 + }, + "verdict": "common-label MC reproduces the exact affine clock within 1 s.e. for |x|<=0.15 and shows exactly the predicted -C_w x^2 curvature beyond" + } + } + }, + "B3_minimal_gaps": { + "1_clock_rate_not_certified": { + "v_w": { + "0.25": { + "v_inf_1overw": 5.0121204715262335, + "coef_1overw": -3.477481387992959, + "v_inf_1overw2": 5.1321525137904915, + "coefs": [ + -4.653454047028344, + 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Normalising Lambda to a prescribed shape (e.g. e^x) needs a proof that the local log-intensity slope converges (kappa_w - kappa = c/w + o(1/w))." + }, + "2_C2_local_convergence": "E_w = O(X^2/w) requires kappa_w', kappa_w'' -> kappa', kappa''; the companion only needs kappa continuous for (4.2). 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+ "p_plus": 0.625, + "kappa_w_p_plus": 0.42651945335847913, + "margin_2kplus_minus_k0": 0.356744815415565, + "ratio_to_kappa0": 0.718817373948783 + }, + { + "w": 8, + "p0": "0.5", + "C": "2", + "p_plus": 0.75, + "kappa_w_p_plus": 0.6680367204374497, + "margin_2kplus_minus_k0": 0.8397793495735061, + "ratio_to_kappa0": 1.6921002371223448 + }, + { + "w": 8, + "p0": "0.5", + "C": "4", + "p_plus": 1.0, + "skipped": "p_plus out of (0,1)" + } + ], + "comment": "2 kappa_w(p_+) - kappa_w(p_0) > 0 for C=1 everywhere; but C=2, p0=0.25, w=5..8 is NEGATIVE (w=8: -0.260) because p_+=0.50 and kappa_w(0.50)=0.496 < kappa_w(0.25)/2. The sufficient condition is not automatic and yields a concrete effective-range bound (a computable slice of Q3); it is not a phase line." + } + }, + "B4_fixed_location_protocol": { + "statement": "At fixed p_0>0 a fixed micro-site is black with probability exactly p_0 and, if white, may sit in a finite white island. No point is silently swapped: all numbers below are for a FIXED row / FIXED micro-site, including the black and finite-white cases.", + "exact_microsite_weights": [ + { + "w": 3, + "p0": 0.25, + "P_site_black": 0.25, + "P_site_white_in_winding_white_comp": 0.691019078641228, + "P_site_white_in_nonwinding_white_comp": 0.058980921358771976 + }, + { + "w": 3, + "p0": 0.4, + "P_site_black": 0.4, + "P_site_white_in_winding_white_comp": 0.40354798156167043, + "P_site_white_in_nonwinding_white_comp": 0.19645201843832955 + }, + { + "w": 3, + "p0": 0.5, + "P_site_black": 0.5, + "P_site_white_in_winding_white_comp": 0.22580645161290333, + "P_site_white_in_nonwinding_white_comp": 0.27419354838709664 + }, + { + "w": 4, + "p0": 0.25, + "P_site_black": 0.25, + "P_site_white_in_winding_white_comp": 0.6972268192121444, + "P_site_white_in_nonwinding_white_comp": 0.052773180787855645 + }, + { + "w": 4, + "p0": 0.4, + "P_site_black": 0.4, + "P_site_white_in_winding_white_comp": 0.3671523295117286, + "P_site_white_in_nonwinding_white_comp": 0.23284767048827137 + }, + { + "w": 4, + "p0": 0.5, + "P_site_black": 0.5, + "P_site_white_in_winding_white_comp": 0.16525131971540064, + "P_site_white_in_nonwinding_white_comp": 0.33474868028459936 + }, + { + "w": 5, + "p0": 0.25, + "P_site_black": 0.25, + "P_site_white_in_winding_white_comp": 0.7038004165698423, + "P_site_white_in_nonwinding_white_comp": 0.04619958343015773 + }, + { + "w": 5, + "p0": 0.4, + "P_site_black": 0.4, + "P_site_white_in_winding_white_comp": 0.3425856917046396, + "P_site_white_in_nonwinding_white_comp": 0.2574143082953604 + }, + { + "w": 5, + "p0": 0.5, + "P_site_black": 0.5, + "P_site_white_in_winding_white_comp": 0.12397869842320419, + "P_site_white_in_nonwinding_white_comp": 0.3760213015767958 + }, + { + "w": 6, + "p0": 0.25, + "P_site_black": 0.25, + "P_site_white_in_winding_white_comp": 0.7088599300666804, + "P_site_white_in_nonwinding_white_comp": 0.041140069933319645 + }, + { + "w": 6, + "p0": 0.4, + "P_site_black": 0.4, + "P_site_white_in_winding_white_comp": 0.32474108841852345, + "P_site_white_in_nonwinding_white_comp": 0.27525891158147653 + }, + { + "w": 6, + "p0": 0.5, + "P_site_black": 0.5, + "P_site_white_in_winding_white_comp": 0.0947694718666976, + "P_site_white_in_nonwinding_white_comp": 0.4052305281333024 + }, + { + "w": 7, + "p0": 0.25, + "P_site_black": 0.25, + "P_site_white_in_winding_white_comp": 0.7124800466978775, + "P_site_white_in_nonwinding_white_comp": 0.037519953302122544 + }, + { + "w": 7, + "p0": 0.4, + "P_site_black": 0.4, + "P_site_white_in_winding_white_comp": 0.31096199585845336, + "P_site_white_in_nonwinding_white_comp": 0.2890380041415466 + }, + { + "w": 7, + "p0": 0.5, + "P_site_black": 0.5, + "P_site_white_in_winding_white_comp": 0.07333901562126968, + "P_site_white_in_nonwinding_white_comp": 0.42666098437873035 + }, + { + "w": 8, + "p0": 0.25, + "P_site_black": 0.25, + "P_site_white_in_winding_white_comp": 0.7149852698339092, + "P_site_white_in_nonwinding_white_comp": 0.0350147301660908 + }, + { + "w": 8, + "p0": 0.4, + "P_site_black": 0.4, + "P_site_white_in_winding_white_comp": 0.29988928203346576, + "P_site_white_in_nonwinding_white_comp": 0.3001107179665342 + }, + { + "w": 8, + "p0": 0.5, + "P_site_black": 0.5, + "P_site_white_in_winding_white_comp": 0.057265537886500496, + "P_site_white_in_nonwinding_white_comp": 0.4427344621134995 + } + ], + "mc_fixed_row_u": { + "p0=0.25": { + "4": { + "P0": 0.0146484375, + "P1": 0.983154296875, + "P_ge2": 0.002197265625 + }, + "5": { + "P0": 0.005615234375, + "P1": 0.9892578125, + "P_ge2": 0.005126953125 + }, + "6": { + "P0": 0.00244140625, + "P1": 0.993408203125, + "P_ge2": 0.004150390625 + } + }, + "p0=0.40": { + "4": { + "P0": 0.24902248289345064, + "P1": 0.7482893450635386, + "P_ge2": 0.002688172043010753 + }, + "5": { + "P0": 0.229227761485826, + "P1": 0.7629521016617791, + "P_ge2": 0.007820136852394917 + }, + "6": { + "P0": 0.22116324535679374, + "P1": 0.7675953079178885, + "P_ge2": 0.011241446725317693 + } + }, + "comment": "p0=0.25 (white 0.75, far from pc): protocol usable, anomaly 0.2-0.5%. p0=0.40 (white 0.60, just above pc=0.592746): NOT in the asymptotic regime, P(u=0) ~ 0.22-0.25 and the anomaly even grows (0.27% -> 1.12% over w=4..6). p0 > 1-pc ~ 0.4073: white is subcritical and a fixed row usually carries NO winding white component (P0=0.877 at w=8, p0=0.5)." + }, + "extra_condition_needed": "the fixed-row protocol needs 1-p_0 strictly supercritical and in its winding regime; the companion's section 6 does not state this.", + "two_definitions_differ": "an EXACT transfer records only components that have ALREADY wound by the current row (P(u>=2)=0 exactly for w<=8); the protocol needs full-span winding components. An explicit witness (see below) shows the two differ." + }, + "B5_explicit_witness_two_winding_clusters_sharing_a_row": { + "found": true, + "w": 5, + "L": 14, + "popcount": 17, + "shared_first_row": 4, + "witness_mask": 26268577987584, + "component_A": [ + 10, + 11, + 12, + 13, + 14, + 17, + 22 + ], + "component_B": [ + 24, + 29, + 34, + 37, + 38, + 39, + 40, + 41, + 42, + 44 + ], + "picture": [ + ". . . . .", + ". . . . .", + "# # # # #", + ". . # . .", + ". . # . #", + ". . . . #", + ". . . . #", + ". . # # #", + "# # # . #", + ". . . . .", + ". . . . .", + ". . . . .", + ". . . . .", + ". . . . ." + ], + "meaning": "two distinct essential clusters CAN cover the same row, so 'the unique essential white component covering a fixed row' is not a one-sided Markov observable; computing the protocol's anomaly exactly needs a two-sided transfer or the full-cluster notion." + }, + "not_claimed": [ + "no statement that mergers are absent: small-width mergers are non-zero (721/125000000 reproduced)", + "B_w/nu_- -> 0 is the TARGET; only finite interfaces (3 widths, 2 parameter pairs) were computed as a regression", + "E_w(X)=O(X^2/w) is an exact finite-width quantity plus finite-w numerics, not a proved limit statement", + "the affine clock p_0+x/(vw), Lambda=e^x is NOT realised: only v_w and its drift are given", + "the (2.1) bound is not numerically tight (off by ~w^2.9)", + "the fixed-row protocol is not repaired: its anomaly does not decrease at p0=0.40 over w=4..6 and the protocol fails for p0 > 1-pc", + "quoted, not re-derived: [AV] volume tail / two-point bound, the seed argument for (4.1), the nu_2/nu_3 closed forms, pc ~ 0.592746" + ] +} diff --git a/results/research-dispatch/verify-768-20260914.json b/results/research-dispatch/verify-768-20260914.json new file mode 100644 index 000000000..afcd0777b --- /dev/null +++ b/results/research-dispatch/verify-768-20260914.json @@ -0,0 +1,110 @@ +{ + "verifier": "verify768", + "date": "2026-09-14", + "machine": "DevEnvC_NePnUn (account2)", + "cloud_workdir": "/workspace/verify768", + "target": { + "deliverable": "theory768-out/note-first-noncommon-correction.md", + "claim_under_test": "momentum-0 exclusion of thermal spin(+/-4) at x=21/4", + "issues_read_only": [768, 802] + }, + "method": { + "own_route": "independent Virasoro normal-ordering (greedy insert/push recursion), independent exact-rational linear algebra", + "reused_theory768_code": false, + "local_mac_computation": "none (all compute on cloud)", + "scripts": ["v768_v1_verma.py", "v768_v1b_diag.py", "v768_v2_classes.py"], + "controls": ["(c,h)=(1/2,3/7) generic", "(c,h)=(0,0) vacuum"], + "cross_check_routes_for_e_n": ["A complement+mod rad", "B full V_{n-1}+mod rad", "C explicit quotient coords", "D adjoint L_1 kernel"] + }, + "V1": { + "verdict": "all four sub-claims HOLD", + "p_n": [1, 1, 2, 3, 5, 7, 11], + "gram_rank": [1, 1, 1, 2, 3, 4, 6], + "gram_nullity": [0, 0, 1, 1, 2, 3, 5], + "rank_eq_p_minus_p_shift2_all": true, + "theory768_json_flag_rank_equals_p_shift2": false, + "theory768_json_flag_is_SCRIPT_BUG": true, + "singular_dims": {"2": 1, "others": 0}, + "chi_int": [-3, 2], + "chi_norm_gram": "0", + "chi_norm_direct": "0", + "chi_two_routes_agree": true, + "L1_chi_zero": true, + "L2_chi_zero": true, + "L3_chi_zero": true, + "intermediate": {"Lm2_norm_2x2": "5/2", "Lm1sq_norm": "45/8", "cross": "15/4", "detG2_c0_factor_4h2_8h_minus_5": "0"}, + "modLm1_irreducible_seq_1_4": [0, 0, 1, 1], + "modLm1_verma_seq_1_4": [0, 1, 1, 2], + "e_n_equals_d_prev_n_le_6": true, + "Lm1_injective_on_irreducible_quotient": true, + "is_0011_arithmetic_restatement": true, + "interpretation": "0,0,1,1 follows by PURE ARITHMETIC from rank G_n = p(n)-p(n-2) together with injectivity of L_{-1} on the irreducible quotient. The only non-arithmetic input is that injectivity (verified by 4 independent routes for n<=6). It is NOT an independent new fact." + }, + "V2": { + "headline": "one-order matrix-element statement holds; the 'different predicted images / feasible exclusion' does NOT hold as stated", + "V2_1_delta_h_hbar": { + "verdict": "HOLDS (exact, elementary)", + "derivation": "[P,phi]=(h-hbar)(2pi/L)phi, P|vac>=0 => (h-hbar)=0", + "conditions": ["translation invariance along compact direction (periodic BC)", "initial/final state momentum 0", "phi has definite (h,hbar)"], + "flaw_in_wording": "on an INFINITE periodic cylinder the one-point function of a PRIMARY vanishes for ALL h (conformal map from the plane); the delta_{h,hbar} statement is really about (a) the torus modular trace or (b) the matrix element . The note conflates one-point function / matrix element / per-row finite-size correction." + }, + "V2_2_observable_is_momentum0": { + "verdict": "HOLDS at first order; GAP on order (1st vs 2nd)", + "argument": "Theta_w / Delta_w = difference of per-row sector excitation energies; per-row energy = log of leading transfer-matrix eigenvalue per row; for a translation-invariant transfer matrix the leading eigenvalue is momentum 0 (Perron-Frobenius) => momentum-0 quantity.", + "gap": "#802 explicitly demands distinguishing first vs second order ('均值零列场在平移不变背景下一阶往往为零；需要先推导使用一阶还是二阶'). 'Matrix element = 0' is FIRST order. At SECOND order a spin-4 operator can shift a momentum-0 level via phi_4 phi_{-4} (spin 0) with a different exponent. The note never shows the w^{-17/4} term is first order.", + "note_self_admission": "note §3.6/§4.1 marks the Theta_w definition (momentum-0 vs direction-resolved) as UNDECIDED while §3.4 D1 is presented as a delivered criterion -- internal tension." + }, + "V2_3_distinguishing_power": { + "verdict": "8-arm weight HOLDS; but NOT unique", + "eight_arm": {"h_4_2": "21/8", "h_2_7": "21/8", "x": "21/4", "spin": 0}, + "thermal_s4": {"h": "37/8", "hbar": "5/8", "x": "21/4", "spin": 4}, + "c4_rules": {"C4_allowed_spins": [4, 8], "C3_allowed": [3, 6], "C6_allowed": [6]}, + "why_not_unique": "the momentum rule can only KILL spin!=0; it can never SELECT a scalar. Alive scalars at x<=21/4: k=2..7 scalar Kac arms (x=1/4,2/3,5/4,2,35/12,4) which the note itself calls '本轮最大的空白'; log partner psi-hat at x=5/4. Identity family is killed by parity (dual-even), not by momentum. Also the nonzero-ness of the 8-arm amplitude is NOT established by the momentum rule (and <0|8-arm|0>=0 for any primary with Delta!=0)." + }, + "V2_4_the_addendum": { + "verdict": "DOES NOT HOLD for scenario (ii); UNESTABLISHED for (iii). Contradicts the note's own §1.2.", + "arithmetic_confirmed": "independent recomputation: q_(ii)=[1,0,1,1,2,2,4]; at x=21/4 the class (a,b)=(2,2) has spin 0, dim 1, momentum 0, C4-ok. So the class EXISTS. In (i) q_2=0 so it does not.", + "why_the_inference_fails": [ + "(ii): the 'extra' level-2 non-derivative class IS the null direction itself. V_2/L_{-1}V_1 is 1-dimensional spanned by [L_{-2}|h>], and [chi] = -3[L_{-2}] != 0, so the whole class is the null direction. A null field is annihilated by all positive modes => its correlators vanish identically => the (2,2) class contributes NOTHING in (ii). This is consistent with the note's own §1.2 ('(i) and (ii) have identical correlators').", + "(iii): whether the (2,2) class is a genuine operator (non-zero matrix element) is NOT determined by the Virasoro algebra -- the note says so itself. => unestablished." + ], + "consequence": { + "exclusion_holds_in": ["(i) class projected out", "(ii) class is null => zero correlators"], + "exclusion_may_fail_in": "(iii) if the class is a genuine operator", + "but": "the note itself argues the physical module is (iii) while using the (ii) class counting to support a (iii) effect -- argument invalid", + "note_D3_claim_wrong_for_ii": "note §3.4 D3 says 'in (ii)/(iii) there are 2 independent momentum-0 amplitudes at x=21/4'; for (ii) this is WRONG (still 1, because the extra class is null)." + } + }, + "my_independent_class_tables": { + "q_irreducible_(i)": [1, 0, 0, 1, 1, 1, 2], + "q_verma_(ii)": [1, 0, 1, 1, 2, 2, 4], + "at_x_21_4_(i)": [{"ab": [0, 4], "spin": -4, "dim": 1, "momentum0": false, "C4": true}, {"ab": [4, 0], "spin": 4, "dim": 1, "momentum0": false, "C4": true}], + "momentum0_C4_at_x_21_4_(i)": [], + "at_x_21_4_(ii)": [{"ab": [0, 4], "spin": -4, "dim": 2, "momentum0": false, "C4": true}, {"ab": [2, 2], "spin": 0, "dim": 1, "momentum0": true, "C4": true}, {"ab": [4, 0], "spin": 4, "dim": 2, "momentum0": false, "C4": true}], + "momentum0_C4_at_x_21_4_(ii)": [{"ab": [2, 2], "spin": 0, "dim": 1}], + "matches_theory768_table": true + } + }, + "V3": { + "cannot_claim_level4_diff_matrix_element_nonzero": "respected in §1.5, but §3.3's prediction table implicitly assumes both amplitudes nonzero (mild leak)", + "cannot_claim_partial4_epsilon_insertion": "respected; and this is TRIVIALLY true (L_{-1}^4|h> lies in L_{-1}V_3 by definition)", + "cannot_declare_winner": "respected in §3.6, but §0#4/'feasible exclusion' and §3.4 D1 lean toward spin4 failing (tension)", + "cannot_claim_w_minus_17_4_explained": "respected" + }, + "issues_found": [ + {"id": "IF-1", "severity": "high", "what": "theory768's JSON flag rank_equals_p_shift2=false contradicts its own table and conclusion; script bug: compares rank G_n to p(n-2) instead of p(n)-p(n-2)", "repro": "g2_virasoro.py: pnm2 = len(B[n-2]); if rank[n] != pnm2: ok_char=False", "note_table_is_right": true}, + {"id": "IF-2", "severity": "high", "what": "note §0#4 addendum contradicts note §1.2: in scenario (ii) the extra (2,2) class is exactly the null direction => zero correlators => cannot enter momentum-0 observables", "minimal_repro": "V_2/L_{-1}V_1 is 1-dim spanned by [L_{-2}|h>]; chi=-3L_{-2}+2L_{-1}^2 => [chi]=-3[L_{-2}] != 0"}, + {"id": "IF-3", "severity": "medium", "what": "note conflates (ii) 'zero-norm state retained' with (iii) 'log module'"}, + {"id": "IF-4", "severity": "medium", "what": "note's 'one-point function ∝ delta_{h,hbar}' is a torus/matrix-element statement; on an infinite periodic cylinder primary one-point functions vanish for all h"}, + {"id": "IF-5", "severity": "medium", "what": "note uses the (ii) class counting to argue a (iii) effect without establishing nonvanishing"}, + {"id": "IF-6", "severity": "low", "what": "my own first implementation had a complement-selection bug (cur==rank[n-1] instead of cur==len(Bp)) giving 0,0,1,2; caught only by 4-route cross-check. Recorded to show a single 'independent' implementation can also be consistently wrong."} + ], + "cannot_claim": [ + "did not recompute #802's (T_g,N_g)", + "did not verify the numerical definition of Theta_w on the real sector engine", + "did not determine whether the (2,2) class is a genuine operator in (iii) (needs log-module construction; Virasoro algebra is insufficient)", + "did not establish the nonzero-ness of the 8-arm amplitude", + "all literature citations are 'quoted, not independently recomputed'" + ], + "bottom_line": "V1 holds throughout (chi norm is exactly 0; 0,0,1,1 holds but is an arithmetic consequence, not an independent fact). V2's momentum-0 exclusion of spin(+/-4) is correct as a FIRST-ORDER matrix-element statement and Theta_w is indeed a momentum-0 quantity, but (a) it is only first order (#802's second-order question unprocessed), (b) it only kills spins and cannot select a scalar (not unique), (c) its own acknowledged (2,2) addendum does NOT hold -- in (ii) that class is the null direction with identically zero correlators, contradicting the note's own §1.2; in (iii) it is unestablished. Hence the 'feasible exclusion' is CONDITIONAL: it holds in (i)/(ii) and may fail in (iii) if the class is a genuine operator -- yet the note itself argues the physical module is (iii)." +} diff --git a/results/research-dispatch/verify-768-v1-verma-20260914.json b/results/research-dispatch/verify-768-v1-verma-20260914.json new file mode 100644 index 000000000..02b6f2ecb --- /dev/null +++ b/results/research-dispatch/verify-768-v1-verma-20260914.json @@ -0,0 +1,355 @@ +{ + "marker": "PY39COMPAT_MARKER=1", + "nmax": 6, + "route": "independent greedy insert/push normal-ordering recursion", + "runs": { + "c0_h58": { + "label": "c=0,h=5/8 (thermal)", + "c": "0", + "h": "5/8", + "p": [ + 1, + 1, + 2, + 3, + 5, + 7, + 11 + ], + "verma_dims": { + "0": 1, + "1": 1, + "2": 2, + "3": 3, + "4": 5, + "5": 7, + "6": 11 + }, + "gram_rank": { + "0": 1, + "1": 1, + "2": 1, + "3": 2, + "4": 3, + "5": 4, + "6": 6 + }, + "gram_nullity": { + "0": 0, + "1": 0, + "2": 1, + "3": 1, + "4": 2, + "5": 3, + "6": 5 + }, + "rank_eq_p_minus_p_shift2": { + "0": true, + "1": true, + "2": true, + "3": true, + "4": true, + "5": true, + "6": true + }, + "rank_eq_p_minus_p_shift2_all": true, + "theory768_style_flag_p_shift2_only": false, + "singular_dims": { + "0": 0, + "1": 0, + "2": 1, + "3": 0, + "4": 0, + "5": 0, + "6": 0 + }, + "chi_int": [ + "-3", + "2" + ], + "chi_norm_gram": "0", + "chi_expansion": { + "(-2,)": "-3", + "(-1, -1)": "2" + }, + "chi_norm_direct": "0", + "chi_agrees": true, + "L1_chi": {}, + "L1_chi_zero": true, + "L2_chi": {}, + "L2_chi_zero": true, + "L3_chi": {}, + "L3_chi_zero": true, + "Lm2_norm": "5/2", + "Lm1sq_norm": "45/8", + "detG2_c0": null, + "detG2_c0_factor": "0", + "detG2_expected_4h2_8h_minus_5": "0", + "modLm1_irred": { + "0": 1, + "1": 0, + "2": 0, + "3": 1, + "4": 1, + "5": 1, + "6": 2 + }, + "modLm1_irred_seq_1_4": [ + 0, + 0, + 1, + 1 + ], + "modLm1_verma": { + "0": 1, + "1": 0, + "2": 1, + "3": 1, + "4": 2, + "5": 2, + "6": 4 + }, + "modLm1_verma_seq_1_4": [ + 0, + 1, + 1, + 2 + ], + "q_from_rank_minus_rank_prev": { + "0": 1, + "1": 0, + "2": 0, + "3": 1, + "4": 1, + "5": 1, + "6": 2 + }, + "q_from_d_seq_1_4": [ + 0, + 0, + 1, + 1 + ], + "is_0011_arithmetic_restatement": true, + "Lm1_injective_on_quotient": true + }, + "generic_c12_h37": { + "label": "generic control (1/2,3/7)", + "c": "1/2", + "h": "3/7", + "p": [ + 1, + 1, + 2, + 3, + 5, + 7, + 11 + ], + "verma_dims": { + "0": 1, + "1": 1, + "2": 2, + "3": 3, + "4": 5, + "5": 7, + "6": 11 + }, + "gram_rank": { + "0": 1, + "1": 1, + "2": 2, + "3": 3, + "4": 5, + "5": 7, + "6": 11 + }, + "gram_nullity": { + "0": 0, + "1": 0, + "2": 0, + "3": 0, + "4": 0, + "5": 0, + "6": 0 + }, + "rank_eq_p_minus_p_shift2": { + "0": true, + "1": true, + "2": false, + "3": false, + "4": false, + "5": false, + "6": false + }, + "rank_eq_p_minus_p_shift2_all": false, + "theory768_style_flag_p_shift2_only": false, + "singular_dims": { + "0": 0, + "1": 0, + "2": 0, + "3": 0, + "4": 0, + "5": 0, + "6": 0 + }, + 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"21/4", + "dim": 1, + "momentum0": true, + "C4_ok": true + } + ], + "chi_expansion_level2": { + "(-2,)": "-3", + "(-1, -1)": "2" + }, + "L_m2_expansion": { + "(-2,)": "1" + }, + "L_m1sq_expansion": { + "(-1, -1)": "1" + }, + "level2_quotient_class_is_chi_direction": true, + "chi_norm_via_gram": "0", + "Lm1sq_norm": "45/8", + "Lm2_norm": "5/2", + "momentum_rule_check": { + "5/8": { + "()": "5/8" + }, + "21/8": { + "()": "21/8" + }, + "37/8": { + "()": "37/8" + } + }, + "momentum_rule_statement": "[P,phi]=(h-hbar)phi, P|vac>=0 => =0 whenever h!=hbar (exact, 只要求沿紧致方向平移不变 + 初末态动量 0)", + "verdicts": { + "V2_1_delta_h_hbar": "成立（精确、初等）——条件是紧致方向平移不变(周期)、初末态动量 0、phi 有确定 (h,hbar)。开放边界下不成立。", + "V2_2_observable_is_momentum0": "Θ_w/Δ_w 作为「每行扇区能量差」是动量 0 的；但 note 把它当『一点函数』——无限周期圆柱上主场的一点函数恒为 0（无论 h 是否 = hbar），δ_{h,hbar} 其实是 torus/modular 迹的陈述。另：『矩阵元为 0』只在一阶成立（#802 明确要求区分一/二阶）。", + "V2_3_eight_arm_weight": "(21/8,21/8)、spin 0、x=21/4 成立；但动量规则只能杀 spin!=0，永远选不出标量；k=2..7 标量 arm 未排除 ⇒ 不唯一。", + "V2_4_third_scenario": "note 的附加结论（(2,2) 的 spin-0 类在 (ii)/(iii) 下进入动量 0）与它自己 §1.2『(i)(ii) 关联函数相同』矛盾：(ii) 下该额外类正是零范态 chi 的方向 ⇒ 关联函数恒为 0；(iii) 下该类是否为真算子不由 Virasoro 代数决定 ⇒ 未建立。" + } +} \ No newline at end of file diff --git a/results/topological-pivotal-pair/controls-20260914.json b/results/topological-pivotal-pair/controls-20260914.json new file mode 100644 index 000000000..4a245518e --- /dev/null +++ b/results/topological-pivotal-pair/controls-20260914.json @@ -0,0 +1,166 @@ +{ + "p_c": "0.59274605079210", + "note": "exact rational; floats are 20-digit displays", + "issue769_quoted_control_reproduction": { + "L3": { + "counts_match": true, + "deltas": { + "Ed0": "-5.551e-17", + "jump2": "+1.753e-11", + "A2": "+3.428e-12", + "Mpp": "+1.492e-13", + "cancel": "+3.605e-06" + } + }, + "L4": { + "counts_match": true, + "deltas": { + "Ed0": "+0.000e+00", + "jump2": "+1.404e-12", + "A2": "+4.412e-12", + "Mpp": "-4.352e-14", + "cancel": "-1.414e-07" + } + } + }, + "results": { + "L3": { + "L": 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+getcontext().prec = 50 + +d = json.load(open("/workspace/v802src/in/root-response.json")) +pc = d["provenance"]["p_c"] +print("p_c =", pc, " (repr)", repr(pc)) +print() + +BLACK = {4: (0.7685984887957081, -2.705373666212244, -0.3116848217980418, -1.350049753766846), + 5: (0.7156536558132538, -3.431122672615583, -0.3102994990346319, -1.7138942463383808), + 6: (0.6782986741556749, -4.1486566866355155, -0.3098063945349776, -2.0732497837477912), + 7: (0.6496075501952117, -4.861471334005446, -0.30962331020127565, -2.4300006319440906), + 8: (0.6263864413538887, -5.571598109893553, -0.30954105615205907, -2.7852763705546106)} +WHITE = {4: (-0.42195206568900034, -1.2132672401049405, 0.17111151832689642, -0.6080813474300464), + 5: (-0.3978349798390679, -1.571013991705516, 0.17249684109030636, -0.7864337384173812), + 6: (-0.3787489632422182, -1.9193898545520969, 0.17298994558996025, -0.9602971742425906), + 7: (-0.3633269977491772, -2.2622896281097638, 0.17317302992366124, -1.1315559208546921), + 8: (-0.35059892255526637, -2.6020940054447124, 0.17325528397288223, -1.3013395578810114)} + +print("=" * 78) +print("CHECK 1: T_W - T_B vs 2 p(1-p) （式 12）") +print("=" * 78) +P = Decimal(repr(pc)) +pred = 2 * P * (1 - P) +print(f" 2 p_c (1-p_c) = {pred}") +for w in (4, 5, 6, 7, 8): + Tb = Decimal(repr(BLACK[w][2])); Tw = Decimal(repr(WHITE[w][2])) + diff = Tw - Tb + print(f" w={w}: Tb={Tb:+.17f} Tw={Tw:+.17f} Tw-Tb={diff:+.17f} " + f"dev={diff-pred:+.3e}") + +print() +print("=" * 78) +print("CHECK 2: 修复 N̂ = raw + f_g, f_g = w p^2 (黑) / w(1-p)^2 (白)") +print("=" * 78) +fb = 8 * P * P # placeholder, per-w below +for w in (4, 5, 6, 7, 8): + fgb = w * P * P + fgw = w * (1 - P) ** 2 + nb = Decimal(repr(BLACK[w][3])) + fgb + nw = Decimal(repr(WHITE[w][3])) + fgw + d1 = Decimal(repr(BLACK[w][3])) - Decimal(repr(WHITE[w][3])) + print(f" w={w}: nb={nb:.17f} nw={nw:.17f} nb-nw={nb-nw:+.3e} w*nb={w*nb:.17f}") + if w == 8: + print(f" raw black - raw white = {d1:.17f} w(2p-1) = {w*(2*P-1):.17f}") + +print() +print("=" * 78) +print("CHECK 3: φ 修正后 (φ = S'_w Δ_g / (Δ'_w S_g), S_phys = S_raw + 2f)") +print("=" * 78) +# Δ'_w at pc (from JSON), S'_w values +Dp = {4: 2.465947762107346, 5: 2.3063319729477914, 6: 2.189427610665712, + 7: 2.0980576358185816, 8: 2.023597286707526} +Sp = {4: -0.01692144855859845, 5: -0.010745038097689896, 6: -0.006962797340485105, + 7: -0.00474793101433213, 8: -0.0033771571284497703} +for w in (4, 5, 6, 7, 8): + fgb = w * P * P; fgw = w * (1 - P) ** 2 + for nm, tab, fg in (("black", BLACK, fgb), ("white", WHITE, fgw)): + Dg = Decimal(repr(tab[w][0])) + Sg_raw = Decimal(repr(tab[w][1])) + Sg_phys = Sg_raw + 2 * fg + phi = (Decimal(repr(Sp[w])) * Dg) / (Decimal(repr(Dp[w])) * Sg_phys) + print(f" w={w} {nm:5s}: S_g_raw={Sg_raw:+.8f} S_g_phys={Sg_phys:+.8f} " + f"phi={phi:+.6e} 1/|phi|={abs(1/phi):.2f}") + +print() +print("=" * 78) +print("CHECK 4: f 的精确事实（Fraction, w=4..8, 三个有理 p）") +print("=" * 78) +def Zrow(p, g, w): + # Z_row = sum_m p^k (1-p)^(w-k) exp(g*h); h = #black adjacent pairs on cycle of w edges + tot = Fraction(0) + for mask in range(1 << w): + k = bin(mask).count("1") + h = sum(1 for j in range(w) if (mask >> j) & 1 and (mask >> ((j + 1) % w)) & 1) + term = (p ** k) * ((1 - p) ** (w - k)) + if g == 0: + tot += term + else: + tot += term * (Fraction(1) if h == 0 else None) if False else term + return tot +# 只验证 g=0 恒等式与 f_g（f_g 直接用组合期望，不建指数） +for w in (4, 5, 6, 7, 8): + for p in (Fraction(1, 2), Fraction(3, 5), Fraction(7, 13)): + tot = sum((p ** bin(m).count("1")) * ((1 - p) ** (w - bin(m).count("1"))) + for m in range(1 << w)) + assert tot == 1, (w, p, tot) + # f_g = w p^2 (black), w(1-p)^2 (white) — 期望相邻黑对数 + for p in (Fraction(1, 2), Fraction(3, 5), Fraction(7, 13)): + Eblack = sum(w * p * p for _ in (0,)) + assert Eblack == w * p * p +print(" Z_row(p,0)=1 全部精确成立 ✓ f_g=w p² / w(1-p)² 形式成立 ✓") +print(" （注：完整 Fraction 期望另见 verify802src-out/evidence.json 的 v1_exact.json）") +print() +print("ALL CHECKS DONE") diff --git a/scripts/audit_802_source_v1_exact.py b/scripts/audit_802_source_v1_exact.py new file mode 100644 index 000000000..97038e37a --- /dev/null +++ b/scripts/audit_802_source_v1_exact.py @@ -0,0 +1,514 @@ +#!/usr/bin/env python3 +"""verify802src -- C1 / C2 / C3 independent verification (exact + full precision). + +Nothing here re-runs the w=4..8 transfer engines; C2 only post-processes the +team's own printed JSON values (its stated error class). All exact claims use +Fraction arithmetic and full enumeration of small tori. + +Outputs: /workspace/v802src/out/v1_exact.json +""" +from __future__ import annotations +import json +import math +import itertools +from fractions import Fraction as F + +IN = "/workspace/v802src/in" +OUT = "/workspace/v802src/out" +PC_S = "0.5927460507921" + + +# ------------------------------------------------------------------ helpers +def torus_configs(w, m): + """Yield (K, H_B, H_W) for every configuration of a w x m torus. + + H_B / H_W = total number of horizontally (cyclic, within-row) adjacent + BLACK / WHITE occupied pairs, summed over rows. Same convention as the + team's s4_torus_bruteforce.py. + """ + N = w * m + for omega in range(1 << N): + K = 0 + HB = 0 + HW = 0 + for y in range(m): + base = y * w + row = [(omega >> (base + i)) & 1 for i in range(w)] + K += sum(row) + for i in range(w): + a = row[i] + b = row[(i + 1) % w] + if a and b: + HB += 1 + elif (not a) and (not b): + HW += 1 + yield K, HB, HW + + +def row_marks(w): + """All single-row masks with (K, H_B, H_W) for the row of length w.""" + for mask in range(1 << w): + K = bin(mask).count("1") + HB = 0 + HW = 0 + for j in range(w): + a = (mask >> j) & 1 + b = (mask >> ((j + 1) % w)) & 1 + if a and b: + HB += 1 + elif (not a) and (not b): + HW += 1 + yield K, HB, HW + + +def logit_shift_peff(p, Y): + """Exact p_eff with logit(p_eff) = logit(p) - 2*log(Y) (Y = exp(g)). + + p_eff = p / (p + (1-p) Y^2). + """ + return p / (p + (1 - p) * Y * Y) + + +# --------------------------------------------------------------- part C1a +def c1_identity(): + out = {} + for (w, m) in [(3, 3), (4, 4), (3, 4)]: + N = w * m + viol = 0 + worst = 0 + n = 0 + for K, HB, HW in torus_configs(w, m): + n += 1 + rhs = N - 2 * K + HB + d = abs(HW - rhs) + if d != 0: + viol += 1 + worst = max(worst, d) + out["%dx%d" % (w, m)] = {"n_configs": n, "N": N, + "identity_HW_eq_N_minus_2K_plus_HB": + "violations=%d max_abs_dev=%d" % (viol, worst), + "violations": viol} + print("C1a %dx%d: %d configs, violations=%d" % (w, m, n, viol), flush=True) + # per-row proof-level table (all row masks, w=3,4,5). Note cyclic pairs. + rowtbl = {} + for w in (3, 4, 5, 6): + bad = 0 + for K, HB, HW in row_marks(w): + if HW != w - 2 * K + HB: + bad += 1 + rowtbl[str(w)] = {"n_row_masks": 1 << w, "violations": bad} + print("C1a row w=%d: %d masks, violations=%d" % (w, 1 << w, bad), flush=True) + out["per_row_all_masks"] = rowtbl + return out + + +# --------------------------------------------------------------- part C1b +def c1_measure_equivalence(): + """Exact check of (11): mu_(p,g H_W) == mu_(p_eff,g H_B), logit shift -2g. + + Take Y = exp(g) rational, p rational. Claim: for every configuration the + ratio + + W(cfg)/B(cfg) = [p^K(1-p)^(N-K) Y^H_W] / [p_eff^K(1-p_eff)^(N-K) Y^H_B] + + is independent of cfg (so the two normalised measures coincide), with + p_eff/(1-p_eff) = (p/(1-p)) Y^{-2}. + """ + cases = [] + for (w, m) in [(3, 3), (4, 3)]: + N = w * m + for p in [F(1, 2), F(3, 5), F(2, 3)]: + for Y in [F(2), F(3, 2), F(5)]: + pe = logit_shift_peff(p, Y) + # exact logit identity check + lhs = pe / (1 - pe) + rhs = (p / (1 - p)) / (Y * Y) + logit_ok = (lhs == rhs) + ratios = set() + allpos = True + for K, HB, HW in torus_configs(w, m): + Wn = p ** K * (1 - p) ** (N - K) * Y ** HW + Bn = pe ** K * (1 - pe) ** (N - K) * Y ** HB + if Bn == 0: + allpos = False + continue + ratios.add(Wn / Bn) + cases.append({ + "w": w, "m": m, "p": str(p), "Y_exp_g": str(Y), + "p_eff": str(pe), + "logit_identity_exact": logit_ok, + "distinct_ratios": len(ratios), + "common_ratio": str(sorted(ratios)[0]) if ratios else None, + "measure_equiv_exact": (len(ratios) == 1 and allpos and logit_ok), + }) + print("C1b w=%d m=%d p=%s Y=%s: ratios=%d logit_ok=%s" + % (w, m, p, Y, len(ratios), logit_ok), flush=True) + return {"cases": cases} + + +# --------------------------------------------------------------- part C1c +def c1_T_identity(): + with open(IN + "/root-response.json") as f: + d = json.load(f) + pc = float(PC_S) + pred = 2 * pc * (1 - pc) + rows = [] + for w in ["4", "5", "6", "7", "8"]: + tb = d["sources"]["black"]["at_pc"][w]["T"] + tw = d["sources"]["white"]["at_pc"][w]["T"] + diff = tw - tb + rows.append({"w": int(w), "T_black": tb, "T_white": tw, + "T_W_minus_T_B": diff, "two_p_one_minus_p": pred, + "abs_dev": abs(diff - pred), + "rel_dev": abs(diff - pred) / abs(pred)}) + print("C1c w=%s T_W-T_B=%.17g 2p(1-p)=%.17g dev=%.3e" + % (w, diff, pred, abs(diff - pred)), flush=True) + # the exact decimal subtraction quoted in the task spec + spec_meas = 0.17325528 - (-0.30954106) + return {"rows": rows, "two_p_one_minus_p_full": pred, + "spec_8dec_measurement": spec_meas} + + +# --------------------------------------------------------------- part C1d +def c1d_finite_torus_TN(w=3, m=3, p0=0.4, h=1e-5): + """Independent finite-torus check of (12): N_W == N_B and T_W - T_B = 2p(1-p). + + Uses ONLY full enumeration + the rank function: builds the rank law (P0,P1,P2) + as a function of (p, gamma) for both sources (black pairs / white pairs) and + differentiates numerically. No transfer engine, no asymptotic identification. + """ + N = w * m + rankcache = {} + HB = {} + HW = {} + Kc = {} + for omega in range(1 << N): + Kc[omega] = bin(omega).count("1") + rankcache[omega] = _rank(None, omega, w, m) + hb = hw = 0 + for y in range(m): + for i in range(w): + a = (omega >> (y * w + i)) & 1 + b = (omega >> (y * w + (i + 1) % w)) & 1 + if a and b: + hb += 1 + elif (not a) and (not b): + hw += 1 + HB[omega] = hb + HW[omega] = hw + # index configs by rank once + byrank = {0: [], 1: [], 2: []} + for omega in range(1 << N): + byrank[rankcache[omega]].append(omega) + + def law(p, g, src): + Z = {0: 0.0, 1: 0.0, 2: 0.0} + for r in (0, 1, 2): + s = 0.0 + for omega in byrank[r]: + K = Kc[omega] + Hv = HB[omega] if src == "B" else HW[omega] + s += math.exp(K * math.log(p) + (N - K) * math.log(1 - p) + g * Hv) + Z[r] = s + return Z + + def bd(p, g, src): + Z = law(p, g, src) + bb = 0.5 * math.log(Z[0] / Z[2]) + dd = math.log(Z[1] / math.sqrt(Z[0] * Z[2])) + return bb, dd + + out = {"w": w, "m": m, "p0": p0, "h": h, "sources": {}} + for src in ("B", "W"): + bgp = (bd(p0 + h, 0.0, src)[0] - bd(p0 - h, 0.0, src)[0]) / (2 * h) + dgp = (bd(p0 + h, 0.0, src)[1] - bd(p0 - h, 0.0, src)[1]) / (2 * h) + bgg = (bd(p0, h, src)[0] - bd(p0, -h, src)[0]) / (2 * h) + dgg = (bd(p0, h, src)[1] - bd(p0, -h, src)[1]) / (2 * h) + T = -bgg / bgp + Nn = dgg - (dgp / bgp) * bgg + out["sources"][src] = {"b_p": bgp, "d_p": dgp, "b_g": bgg, "d_g": dgg, + "T": T, "N": Nn} + print("C1d src=%s T=%.12f N=%.6e" % (src, T, Nn), flush=True) + TW = out["sources"]["W"]["T"] + TB = out["sources"]["B"]["T"] + out["T_W_minus_T_B"] = TW - TB + out["two_p_one_minus_p"] = 2 * p0 * (1 - p0) + out["T_identity_residual"] = (TW - TB) - 2 * p0 * (1 - p0) + out["N_W_minus_N_B"] = out["sources"]["W"]["N"] - out["sources"]["B"]["N"] + print("C1d: T_W-T_B=%.12f (2p(1-p)=%.12f, resid=%.2e) N_W-N_B=%.2e" + % (out["T_W_minus_T_B"], out["two_p_one_minus_p"], + out["T_identity_residual"], out["N_W_minus_N_B"]), flush=True) + return out + + +# --------------------------------------------------------------- part C2f +def c2_f_identities(): + """Exact (Fraction) single-row partition function facts, w = 4..8. + + Z_row(p,g) = sum_m p^K (1-p)^(w-K) exp(g H(m)), f = log Z_row. + Verify at g=0: Z_row = 1, f_p = 0, + d_g f = E[H] = w p^2 (black pairs) / w (1-p)^2 (white pairs). + """ + res = {} + for w in (4, 5, 6, 7, 8): + marks = list(row_marks(w)) + per = {} + for p in [F(1, 2), F(3, 5), F(7, 13)]: + Z = sum((p ** K) * ((1 - p) ** (w - K)) for (K, HB, HW) in marks) + # f_p at g=0 : sum (K/p - (w-K)/(1-p)) * weight == 0 + fp = sum((F(K)/p - F(w - K) / (1 - p)) * (p ** K) * ((1 - p) ** (w - K)) + for (K, HB, HW) in marks) + # d_g f at g=0 = E[H] = sum H * weight (f_g, not raw normalised) + fgB = sum(HB * (p ** K) * ((1 - p) ** (w - K)) for (K, HB, HW) in marks) + fgW = sum(HW * (p ** K) * ((1 - p) ** (w - K)) for (K, HB, HW) in marks) + per[str(p)] = { + "Z_row_at_g0": str(Z), + "Z_row_is_one": (Z == 1), + "f_p_at_g0": str(fp), + "f_p_is_zero": (fp == 0), + "f_g_black": str(fgB), + "f_g_black_eq_w_p2": (fgB == w * p * p), + "f_g_white": str(fgW), + "f_g_white_eq_w_1mp2": (fgW == w * (1 - p) ** 2), + "w_p2": str(w * p * p), + "w_1mp2": str(w * (1 - p) ** 2), + } + print("C2f w=%d p=%s: Z=1? %s f_p=0? %s f_gB=wp^2? %s f_gW=w(1-p)^2? %s" + % (w, p, Z == 1, fp == 0, fgB == w * p * p, + fgW == w * (1 - p) ** 2), flush=True) + res[str(w)] = per + return res + + +# --------------------------------------------------------------- part C2 +def c2_recompute(): + with open(IN + "/root-response.json") as f: + d = json.load(f) + pc = float(PC_S) + fb = lambda w: w * pc * pc + fw = lambda w: w * (1 - pc) ** 2 + rows = [] + diffs = [] + wNhat = [] + for w in [4, 5, 6, 7, 8]: + s = d["sources"] + nb = s["black"]["at_pc"][str(w)]["Nhat_per_row"] + nw = s["white"]["at_pc"][str(w)]["Nhat_per_row"] + cb = nb + fb(w) + cw = nw + fw(w) + rows.append({"w": w, "raw_black": nb, "raw_white": nw, + "f_g_black": fb(w), "f_g_white": fw(w), + "corr_black": cb, "corr_white": cw, + "diff_corr": cb - cw, + "raw_white_minus_black": nw - nb, + "w_times_2p_minus_1": w * (2 * pc - 1), + "w_times_corr_black": w * cb, + "w_times_corr_white": w * cw}) + diffs.append(abs(cb - cw)) + wNhat.append(w * cb) + print("C2 w=%d rawB=%.11f rawW=%.11f corrB=%.11f corrW=%.11f " + "diff=%.3e w*corr=%.8f" + % (w, nb, nw, cb, cw, cb - cw, w * cb), flush=True) + ref = {4: 0.05534176915, 5: 0.04284515731, 6: 0.03483750063, + 7: 0.02943453316, 8: 0.02550667528} + refw = [0.22137, 0.21423, 0.20903, 0.20604, 0.20405] + cmp_rows = [] + for i, w in enumerate([4, 5, 6, 7, 8]): + cmp_rows.append({"w": w, "my_corr_black": rows[i]["corr_black"], + "analysis_table": ref[w], + "abs_dev_vs_analysis": abs(rows[i]["corr_black"] - ref[w]), + "my_w_times_corr": wNhat[i], + "analysis_w_times": refw[i], + "abs_dev_w_times": abs(wNhat[i] - refw[i])}) + print(" vs analysis: dev=%.3e (w*Nhat dev=%.3e)" + % (abs(rows[i]["corr_black"] - ref[w]), abs(wNhat[i] - refw[i])), flush=True) + return {"rows": rows, "max_abs_diff_corrected": max(diffs), + "max_rel_diff_corrected": max(diffs) / max(abs(r["corr_black"]) for r in rows), + "w_times_Nhat": wNhat, "vs_analysis": cmp_rows, + "analysis_claim_1p5e-15": 1.5e-15} + + +# --------------------------------------------------------------- part C3 +def c3_counterexample(w=3, m=3): + """Exact: popcount source exp(g K) is a pure logit shift. + + It is a source whose rank-law tangent is parallel to d/dp (so N == 0) yet + whose T equals -p(1-p), not -1. Verified by exact evaluation of the + finite-torus b(p) curve together with the exact identification + b(p, g) = b_0(p_eff(p,g)). + """ + N = w * m + # rank of a config on the torus (ambient homology rank of the black set) + def rank_of(omega): + return _rank(None, omega, w, m) + + # collect P0,P2 as exact Fraction polynomials in p for every rank class + # (weights are p^K (1-p)^(N-K); accumulate exact integer coefficient lists) + # coefficient[K] = #configs of rank r with K black sites + from collections import defaultdict + coeff = {0: defaultdict(int), 1: defaultdict(int), 2: defaultdict(int)} + for omega in range(1 << N): + K = bin(omega).count("1") + coeff[rank_of(omega)][K] += 1 + + def P(r, p): + return sum(c * (p ** K) * ((1 - p) ** (N - K)) for K, c in coeff[r].items()) + + def coords(p): + P0, P1, P2 = P(0, p), P(1, p), P(2, p) + b = F(0) if (P0 == 0 or P2 == 0) else (F(1) / 2) * _log_ratio(P0, P2) + d = None + return b, P0, P1, P2 + + # b is an irrational (log) of a rational -> evaluate numerically in high + # precision using Fraction -> float is enough for a counterexample of O(1); + # we also give the exact p-derivative identity at the level of the measure. + def bn(pf): + p = float(pf) + P0 = sum(c * p ** K * (1 - p) ** (N - K) for K, c in coeff[0].items()) + P2 = sum(c * p ** K * (1 - p) ** (N - K) for K, c in coeff[2].items()) + return 0.5 * math.log(P0 / P2) + + def dn(pf): + p = float(pf) + P0 = sum(c * p ** K * (1 - p) ** (N - K) for K, c in coeff[0].items()) + P1 = sum(c * p ** K * (1 - p) ** (N - K) for K, c in coeff[1].items()) + P2 = sum(c * p ** K * (1 - p) ** (N - K) for K, c in coeff[2].items()) + return math.log(P1 / math.sqrt(P0 * P2)) + + p0 = 0.4 + out = {"w": w, "m": m, "p0": p0, "rank_class_counts_by_K": + {str(r): dict(sorted(coeff[r].items())) for r in (0, 1, 2)}} + + # (i) exact measure identification: weight of config under popcount source + # at coupling g equals Bernoulli weight at p_eff with + # logit(p_eff) = logit(p) + g (exact for rational Y=exp(g)) + tests = [] + for pr in [F(1, 3), F(2, 5)]: + for Y in [F(2), F(3, 2)]: + r0 = pr / (1 - pr) * Y # odds ratio of p_eff + pe = r0 / (1 + r0) + ok = True + eK = defaultdict(int) + for omega in range(1 << N): + K = bin(omega).count("1") + eK[K] += 1 + # ratio of full weights per config with same K must be constant + ratios = set() + for K in range(N + 1): + Wn = pr ** K * (1 - pr) ** (N - K) * Y ** K + Bn = pe ** K * (1 - pe) ** (N - K) + if Bn == 0: + if Wn != 0: + ok = False + continue + ratios.add(Wn / Bn) + tests.append({"p": str(pr), "Y_exp_g": str(Y), "p_eff": str(pe), + "distinct_ratios": len(ratios), "exact": ok and len(ratios) == 1}) + out["popcount_is_exact_logit_shift"] = tests + + # (ii) numeric: b(g) from the source reweighting == b_0(p_eff(g)) + def b_of_g(g, p): + # explicit sum over configs (small torus) + Z0 = Z1 = Z2 = 0.0 + for omega in range(1 << N): + K = bin(omega).count("1") + w0 = math.exp(K * math.log(p) + (N - K) * math.log(1 - p)) + r = rank_of(omega) + t = w0 * math.exp(g * K) + if r == 0: + Z0 += t + elif r == 1: + Z1 += t + else: + Z2 += t + bb = 0.5 * math.log(Z0 / Z2) + dd = math.log(Z1 / math.sqrt(Z0 * Z2)) + return bb, dd + + h = 1e-5 + gs = [-2 * h, -h, h, 2 * h] + bg = [b_of_g(g, p0) for g in gs] + db_dg = (-bg[3][0] + 8 * bg[2][0] - 8 * bg[1][0] + bg[0][0]) / (12 * h) + dd_dg = (-bg[3][1] + 8 * bg[2][1] - 8 * bg[1][1] + bg[0][1]) / (12 * h) + # p-derivatives of b,d at p0 + ph = 1e-5 + def bd(p): + return bn(p), dn(p) + bp = (bd(p0 + ph)[0] - bd(p0 - ph)[0]) / (2 * ph) + dp = (bd(p0 + ph)[1] - bd(p0 - ph)[1]) / (2 * ph) + T = -db_dg / bp + N = dd_dg - (dp / bp) * db_dg + out["numeric"] = {"p0": p0, "b_g": db_dg, "d_g": dd_dg, "b_p": bp, "d_p": dp, + "T": T, "N": N, + "p(1-p)": p0 * (1 - p0), + "T_equals_minus_p_1mp": abs(T + p0 * (1 - p0)), + "N_is_zero_abs": abs(N), + "minus_one_is_the_truth_abs": abs(T + 1.0)} + print("C3 counterexample (popcount source, %dx%d, p0=%.3f): T=%.10f N=%.3e " + "(p(1-p)=%.6f)" % (w, m, p0, T, N, p0 * (1 - p0)), flush=True) + return out + + +def _log_ratio(a, b): + # exact log not needed; only used for display of exact Fractions + return math.log(float(a) / float(b)) + + +def _rank(_, omega, w, m): + """Ambient homology rank in {0,1,2} of the black set (4-neighbour torus).""" + N = w * m + occupied = set(i for i in range(N) if (omega >> i) & 1) + pot = {} + cyc = [] + for start in occupied: + if start in pot: + continue + pot[start] = (0, 0) + stack = [start] + while stack: + u = stack.pop() + i, y = u % w, u // w + pu = pot[u] + for (j, yy, di, dy) in (((i + 1) % w, y, 1, 0), + ((i - 1) % w, y, -1, 0), + (i, (y + 1) % m, 0, 1), + (i, (y - 1) % m, 0, -1)): + v = yy * w + j + if v not in occupied: + continue + cand = (pu[0] + di, pu[1] + dy) + if v not in pot: + pot[v] = cand + stack.append(v) + else: + dv = pot[v] + cyc.append((cand[0] - dv[0], cand[1] - dv[1])) + nz = [c for c in cyc if c != (0, 0)] + if not nz: + return 0 + a, b = nz[0] + for c, d in nz[1:]: + if a * d - b * c != 0: + return 2 + return 1 + + +def main(): + import os + os.makedirs(OUT, exist_ok=True) + res = {} + res["C1a_identity"] = c1_identity() + res["C1b_measure_equivalence"] = c1_measure_equivalence() + res["C1c_T_identity"] = c1_T_identity() + res["C1d_finite_torus"] = c1d_finite_torus_TN() + res["C2f_row_partition_exact"] = c2_f_identities() + res["C2_recompute"] = c2_recompute() + res["C3_counterexample"] = c3_counterexample() + with open(OUT + "/v1_exact.json", "w") as f: + json.dump(res, f, indent=1, default=str) + print("-> %s/v1_exact.json" % OUT) + + +if __name__ == "__main__": + main() diff --git a/scripts/audit_802_source_v2_c4.py b/scripts/audit_802_source_v2_c4.py new file mode 100644 index 000000000..15918d206 --- /dev/null +++ b/scripts/audit_802_source_v2_c4.py @@ -0,0 +1,158 @@ +#!/usr/bin/env python3 +"""verify802src -- C4: why does the staggered row field have a first-order +response only on ODD torus length? + +Everything is exact full enumeration on small tori. For each configuration we +compute + * K_y = #black sites in row y, H_s = sum_y s_y K_y with s_y = (+1)^y + (the team's / note's staggering; sign +1 on even rows, -1 on odd rows), + * the ambient homology rank r in {0,1,2} of the black 4-neighbour subgraph, + * the Bernoulli weight w0 = p^K (1-p)^(N-K), +and then report + * _{w0} (the global mean) + * E[H_s | r] for r=0,1,2 (the rank-conditional means) + * b_g = (E[H_s|0]-E[H_s|2])/2 (the conditional source scoring used) + * the "seam defect" H_s(T cfg) + H_s(cfg) under the y-shift T, + which measures how far H_s is from being shift-ODD. + +Outputs /workspace/v802src/out/v2_c4.json +""" +from __future__ import annotations +import json +import math +import os +from collections import defaultdict + +OUT = "/workspace/v802src/out" + + +def rank_of(omega, w, m): + N = w * m + occupied = [i for i in range(N) if (omega >> i) & 1] + occ = set(occupied) + pot = {} + cyc = [] + for start in occupied: + if start in pot: + continue + pot[start] = (0, 0) + stack = [start] + while stack: + u = stack.pop() + i, y = u % w, u // w + pu = pot[u] + for (j, yy, di, dy) in (((i + 1) % w, y, 1, 0), + ((i - 1) % w, y, -1, 0), + (i, (y + 1) % m, 0, 1), + (i, (y - 1) % m, 0, -1)): + v = yy * w + j + if v not in occ: + continue + cand = (pu[0] + di, pu[1] + dy) + if v not in pot: + pot[v] = cand + stack.append(v) + else: + dv = pot[v] + cyc.append((cand[0] - dv[0], cand[1] - dv[1])) + nz = [c for c in cyc if c != (0, 0)] + if not nz: + return 0 + a, b = nz[0] + for c, d in nz[1:]: + if a * d - b * c != 0: + return 2 + return 1 + + +def shift_y(omega, w, m): + """Cyclic shift of the configuration by one row (y -> y+1).""" + out = 0 + for y in range(m): + for i in range(w): + if (omega >> (y * w + i)) & 1: + out |= 1 << (((y + 1) % m) * w + i) + return out + + +def Hk(omega, w, m): + """K_y per row (list) and H_s = sum_y (-1)^y K_y.""" + Ky = [] + for y in range(m): + Ky.append(sum((omega >> (y * w + i)) & 1 for i in range(w))) + Hs = sum((1 if y % 2 == 0 else -1) * Ky[y] for y in range(m)) + return Ky, Hs + + +def run_case(w, m, p): + N = w * m + n = 1 << N + Z = defaultdict(float) # rank -> sum w0 + SH = defaultdict(float) # rank -> sum w0 * H_s + SH2 = defaultdict(float) # rank -> sum w0 * H_s^2 + tot = 0.0 + seam_nonzero = 0 # configs with H_s(T cfg) != -H_s(cfg) + seam_absmax = 0 + shift_invariance_viol = 0 # w0(T cfg) != w0(cfg)? + for omega in range(n): + K = bin(omega).count("1") + w0 = p ** K * (1 - p) ** (N - K) + r = rank_of(omega, w, m) + _, Hs = Hk(omega, w, m) + Z[r] += w0 + SH[r] += w0 * Hs + SH2[r] += w0 * Hs * Hs + tot += w0 + o2 = shift_y(omega, w, m) + K2 = bin(o2).count("1") + _, Hs2 = Hk(o2, w, m) + if K2 != K: + shift_invariance_viol += 1 + d = Hs2 + Hs + if d != 0: + seam_nonzero += 1 + seam_absmax = max(seam_absmax, abs(d)) + P = {r: Z[r] / tot for r in Z} + EH = {r: (SH[r] / Z[r]) if Z[r] > 0 else float("nan") for r in Z} + VarH = {r: (SH2[r] / Z[r] - EH[r] ** 2) if Z[r] > 0 else float("nan") for r in Z} + Hbar = sum(SH[r] for r in SH) / tot + # theory: = w p * sum_y s_y = w p * (1 if m odd else 0) + summ = sum(1 if y % 2 == 0 else -1 for y in range(m)) + rec = { + "w": w, "m": m, "p": p, "N": N, "n_configs": n, + "period_parity": "odd" if m % 2 else "even", + "sum_y_pm1": summ, + "theory_Hbar": w * p * summ, + "Hbar": Hbar, + "Hbar_minus_theory": Hbar - w * p * summ, + "P_r": {str(r): P[r] for r in sorted(P)}, + "EH_by_rank": {str(r): EH[r] for r in sorted(EH)}, + "VarH_by_rank": {str(r): VarH[r] for r in sorted(VarH)}, + "b_g_cond_score": 0.5 * (EH.get(0, float("nan")) - EH.get(2, float("nan"))), + "base_translation_invariance_violations": shift_invariance_viol, + "n_configs_where_Hs_T_is_not_minus_Hs": seam_nonzero, + "max_abs_seam_defect_Hs_T_plus_Hs": seam_absmax, + } + print("C4 w=%d m=%d (%s) p=%.4f: =%.10f (theory %.10f) " + "E[H|0]=%.6f E[H|2]=%.6f b_g=%+.6e seam_nonzero=%d/%d " + "w0_T_inv_viol=%d" + % (w, m, rec["period_parity"], p, Hbar, w * p * summ, + EH.get(0, float("nan")), EH.get(2, float("nan")), + rec["b_g_cond_score"], seam_nonzero, n, shift_invariance_viol), + flush=True) + return rec + + +def main(): + os.makedirs(OUT, exist_ok=True) + cases = [] + for (w, m) in [(3, 3), (4, 4), (3, 4), (4, 3), (3, 5)]: + for p in [0.4, 0.59274605079210]: + cases.append(run_case(w, m, p)) + with open(OUT + "/v2_c4.json", "w") as f: + json.dump({"cases": cases}, f, indent=1) + print("-> %s/v2_c4.json" % OUT) + + +if __name__ == "__main__": + main() diff --git a/scripts/audit_802_source_v3_evidence.py b/scripts/audit_802_source_v3_evidence.py new file mode 100644 index 000000000..3500efbd3 --- /dev/null +++ b/scripts/audit_802_source_v3_evidence.py @@ -0,0 +1,85 @@ +#!/usr/bin/env python3 +"""verify802src -- assemble evidence.json on the cloud (no local arithmetic). + +Merges v1_exact.json + v2_c4.json and adds the derived quantities that the +erratum needs (corrected phi, corrected Nhat/S_g, raw-vs-corrected comparison). + +Outputs /workspace/v802src/out/evidence.json +""" +from __future__ import annotations +import json + +IN = "/workspace/v802src/out" +PC = 0.5927460507921 + + +def main(): + v1 = json.load(open(IN + "/v1_exact.json")) + v2 = json.load(open(IN + "/v2_c4.json")) + resp = json.load(open("/workspace/v802src/in/root-response.json")) + at = resp["sources"] + + derived = [] + for w in [4, 5, 6, 7, 8]: + w = str(w) + fB = int(w) * PC * PC + fW = int(w) * (1 - PC) ** 2 + row = {} + for src, fg in (("black", fB), ("white", fW)): + e = at[src]["at_pc"][w] + Dg = e["Delta_g"] + Sg_raw = e["S_g"] + Sg_phys = Sg_raw + 2 * fg + Sp = resp["delta_table"][w]["0.5927460507921"]["S_p"] + Dp = resp["delta_table"][w]["0.5927460507921"]["Delta_p"] + row[src] = { + "f_g": fg, + "S_g_unrenormalised": Sg_raw, + "S_g_physical": Sg_phys, + "Nhat_raw": e["Nhat_per_row"], + "Nhat_physical": e["Nhat_per_row"] + fg, + "phi_raw": (Sp * Dg) / (Dp * Sg_raw), + "phi_physical": (Sp * Dg) / (Dp * Sg_phys), + "one_over_abs_phi_raw": abs(1.0 / ((Sp * Dg) / (Dp * Sg_raw))), + "one_over_abs_phi_physical": abs(1.0 / ((Sp * Dg) / (Dp * Sg_phys))), + } + row["w"] = int(w) + row["T_black"] = at["black"]["at_pc"][w]["T"] + row["T_white"] = at["white"]["at_pc"][w]["T"] + row["T_white_minus_T_black"] = row["T_white"] - row["T_black"] + row["two_p_one_minus_p"] = 2 * PC * (1 - PC) + derived.append(row) + print("w=%s phi_black_raw=%.6e phi_black_phys=%.6e " + "1/|phi| raw=%.1f -> phys=%.1f" + % (w, row["black"]["phi_raw"], row["black"]["phi_physical"], + row["black"]["one_over_abs_phi_raw"], + row["black"]["one_over_abs_phi_physical"]), flush=True) + + out = { + "meta": { + "p_c": PC, + "team_json": "sector-root-response-20260914.json (commit 723f629)", + "analysis_note": "docs/research-bridges-post-compass-20260914.md / " + "/tmp/concurrent-out/two-observable-response-and-angular-alias-20260914.md " + "(§7)", + "engine_weight_reading": "sector802_lib.build_matrices: R_ij = sum p^k (1-p)^(w-k) " + "exp(gamma h) -> UNNORMALISED row weight (no /Z_row)", + "local_computation": "none; every number produced on DevEnvC_NePnUn", + }, + "C1a_identity_full_enumeration": v1["C1a_identity"], + "C1b_measure_equivalence_exact": v1["C1b_measure_equivalence"], + "C1c_T_identity_on_team_data": v1["C1c_T_identity"], + "C1d_finite_torus_independent": v1["C1d_finite_torus"], + "C2f_row_partition_exact": v1["C2f_row_partition_exact"], + "C2_recompute": v1["C2_recompute"], + "C3_counterexample": v1["C3_counterexample"], + "derived_phi_and_Sg": derived, + "C4_torus_cases": v2["cases"], + } + with open(IN + "/evidence.json", "w") as f: + json.dump(out, f, indent=1, default=str) + print("-> %s/evidence.json" % IN) + + +if __name__ == "__main__": + main() diff --git a/scripts/coalescent_bulk_clock.py b/scripts/coalescent_bulk_clock.py new file mode 100644 index 000000000..cb1aeb607 --- /dev/null +++ b/scripts/coalescent_bulk_clock.py @@ -0,0 +1,363 @@ +#!/usr/bin/env python3 +"""Exact finite checks for circular cut genealogies and a shared-label noise clock. + +The cycle enumeration is an exact finite combinatorial calculation, NOT a +new percolation simulation. The conditional Gaussian/Poisson formulas below +belong to the explicitly specified limiting process. No asymptotic claim is +inferred by fitting these controls. +""" +from __future__ import annotations + +import argparse +from collections import Counter +from fractions import Fraction as F +from functools import lru_cache +from itertools import combinations, permutations +import json +from math import comb, factorial, log, sqrt +from pathlib import Path + + +def partition_key(blocks): + return tuple(sorted(tuple(sorted(b)) for b in blocks)) + + +def cycle_history(cycle, deletion_order): + """Vertices are terminal cells; edges are the n distinct surviving cuts. + + n=2 has two parallel edges; deleting the last edge changes topology but + not the one-block partition. Rotations are removed by fixing vertex 0. + """ + n = len(cycle) + blocks = [{j} for j in range(n)] + out = [partition_key(blocks)] + for edge in deletion_order: + x, y = cycle[edge], cycle[(edge + 1) % n] + i = next(i for i, b in enumerate(blocks) if x in b) + j = next(i for i, b in enumerate(blocks) if y in b) + if i != j: + blocks[i] |= blocks[j] + blocks.pop(j) + out.append(partition_key(blocks)) + assert len(out) == n + return tuple(out) + + +@lru_cache(None) +def enumerate_histories(n): + if not 2 <= n <= 6: + raise ValueError('exact history enumeration supports 2 <= n <= 6') + histories = Counter() + orders = tuple(permutations(range(n))) + for tail in permutations(range(1, n)): + cycle = (0,) + tail + for order in orders: + histories[cycle_history(cycle, order)] += 1 + total = factorial(n - 1) * factorial(n) + assert sum(histories.values()) == total + expected_histories = 1 + for k in range(2, n + 1): + expected_histories *= comb(k, 2) + assert len(histories) == expected_histories + assert set(histories.values()) == {2 ** (n - 1)} + levels = [Counter() for _ in range(n)] + prefixes = {} + for hist, count in histories.items(): + for depth, part in enumerate(hist): + levels[depth][part] += count + for depth in range(n - 1): + prefix = hist[:depth + 1] + prefixes.setdefault(prefix, Counter())[hist[depth + 1]] += count + transition_checks = 0 + for prefix, future in prefixes.items(): + k = len(prefix[-1]) + assert len(future) == comb(k, 2) + assert len(set(future.values())) == 1 + transition_checks += 1 + eppf_checks = 0 + mass_checks = 0 + for depth, law in enumerate(levels): + k = n - depth + second_mass = F(0) + for part, count in law.items(): + prob = F(factorial(n-k) * factorial(k) * factorial(k-1), + factorial(n) * factorial(n-1)) + for block in part: + prob *= factorial(len(block)) + assert F(count, total) == prob + eppf_checks += 1 + second_mass += F(count, total) * F( + sum(len(block)*(len(block)+1) for block in part), n*(n+1)) + assert second_mass == F(2, k + 1) + mass_checks += 1 + return {'n': n, 'cyclic_orders': factorial(n-1), + 'edge_orders': factorial(n), 'realizations': total, + 'ranked_histories': len(histories), + 'multiplicity_per_history': 2**(n-1), + 'full_history_transition_checks': transition_checks, + 'eppf_equalities': eppf_checks, + 'dirichlet_second_mass_equalities': mass_checks} + + +def pair_separated_by_edges(n, survival): + """Two uniformly chosen terminal cell labels; average cyclic separation.""" + survival = F(survival) + if not 0 <= survival <= 1 or n < 2: + raise ValueError('bad parameters') + gone = 1 - survival + return sum((1-gone**j)*(1-gone**(n-j)) for j in range(1,n))/F(n-1) + + +def pair_separated_by_levels(n, survival): + survival = F(survival) + ans = F(0) + for k in range(2,n+1): + prob = comb(n,k) * survival**k * (1-survival)**(n-k) + same = F(2*(n-k), (k+1)*(n-1)) + ans += prob * (1-same) + return ans + + +# Small bivariate rational jets. Keys are powers of the two Fourier variables. +DEGREE = 4 + + +def add(*xs): + ans = {} + for x in xs: + for m,v in x.items(): + ans[m] = ans.get(m,F(0)) + v + return {m:v for m,v in ans.items() if v} + + +def scale(x,a): + return {m:v*F(a) for m,v in x.items() if v*F(a)} + + +def mul(x,y): + ans = {} + for (i,j),a in x.items(): + for (k,l),b in y.items(): + if i+j+k+l <= DEGREE: + key=(i+k,j+l) + ans[key]=ans.get(key,F(0))+a*b + return {m:v for m,v in ans.items() if v} + + +def inv(x): + c=x.get((0,0),F(0)) + if not c: + raise ZeroDivisionError('zero jet constant') + tail=scale(add(x,{(0,0):-c}),-1/c) + ans={(0,0):F(1)}; power=ans.copy() + for _ in range(DEGREE): + power=mul(power,tail); ans=add(ans,power) + return scale(ans,1/c) + + +def score_cf_jet(k,a): + k,a=F(k),F(a) + A={(0,0):F(1),(2,0):F(1,2)} + numerator=add(A,{(0,0):k-1}) + denominator2=add(numerator,{(0,2):k/2,(1,1):k*a}) + one=mul(numerator,mul(inv(A),inv(denominator2))) + return mul(one,one) + + +def score_moments(k,a): + k,a=F(k),F(a) + if k < 1 or a*a*k > 1: + raise ValueError('conditional Gaussian covariance is not admissible') + jet=score_cf_jet(k,a) + cov=-jet.get((1,1),F(0)) + var1=-2*jet.get((2,0),F(0)); var2=-2*jet.get((0,2),F(0)) + fourth1=24*jet.get((4,0),F(0)); fourth2=24*jet.get((0,4),F(0)) + mixed=4*jet.get((2,2),F(0)) + assert cov==2*a and var1==var2==2 + assert fourth1==fourth2==18 + assert mixed==4+F(2,k)+12*a*a + return {'k':k,'a':a,'variance':var1,'covariance':cov, + 'fourth':fourth1,'mixed_fourth':mixed, + 'raw_score_correlation':a, + 'squared_score_correlation':(F(1,k)+6*a*a)/7} + + +def six_variable_transform(k,a,r,s,t,z,v,u1,u2): + """Laplace(Y1,Y2), PGF(H1,H2), Fourier(Q1,Q2), all scalar inputs. + + Formula is real even when a cross term makes B negative. The complete + quadratic form in its denominator is positive semidefinite for k*a^2<=1. + """ + k,a,r,s,t,z,v,u1,u2=map(F,(k,a,r,s,t,z,v,u1,u2)) + if k<1 or a*a*k>1 or min(r,s,t)<0 or not (0<=z<=1 and 0<=v<=1): + raise ValueError('parameters outside stated transform domain') + A=1+s+r*(1-z)+u1*u1/2 + B=k*(t+u2*u2/2+a*u1*u2)+r*(1-v)*(z+k-1) + assert A>0 and A+k-1+B>0 + return ((A+k-1)/(A*(A+k-1+B)))**2 + + +def retention_mean(k): + k=float(k) + if k<1: + raise ValueError('k must be at least one') + if k==1: + return 1.0 + return 2*((k-1)-log(k))/(k-1)**2 + + +def retention_density(x,k): + a=float(k)-1 + if not 050: + break + return ans + + +def simpson(f,n=20000): + if n%2: + raise ValueError('even n required') + total=f(0)+f(1) + for j in range(1,n): + total+=(4 if j%2 else 2)*f(j/n) + return total/(3*n) + + +def label_covariance_checks(): + ans=[] + for e in (F(1,100),F(1,1000),F(1,10000)): + for t1,t2 in ((F(1),F(2)),(F(1),F(4)),(F(2),F(5))): + u,v=e*t1,e*t2 + outcomes=((0,0,1-v),(0,1,v-u),(1,1,u)) + covariance=sum(prob*(x-u)*(y-v) for x,y,prob in outcomes) + assert covariance==u*(1-v) + # Covariance of the rescaled empirical sheet per unit length. + assert covariance/e==t1*(1-e*t2) + ans.append({'epsilon':e,'t1':t1,'t2':t2, + 'scaled_covariance':covariance/e}) + return ans + + +def gaussian_even_moment(m): + if m % 2: + return 0 + ans = 1 + for j in range(1,m,2): + ans *= j + return ans + + +def xy_moment(a,b,h): + """E[X^a Y^b (X+Y)^h] for independent rate-one exponentials.""" + return sum(comb(h,j)*factorial(a+j)*factorial(b+h-j) for j in range(h+1)) + + +def closure_stationarity_checks(): + """Exact invariant-generator identities for 4-coordinate closure. + + Test functions: x_minus^a*x_plus^b*Q^m*(H falling n). + eta is the temporal label-refresh coefficient; dilute models have eta=1/w. + All expectations below use the CONDITIONAL Gaussian and Poisson marks. + """ + checked=0 + for eta in (F(0),F(1,8),F(1,4),F(1)): + for a in range(4): + for b in range(4): + for m in (0,2,4): + for n in range(4): + r=F(3,7) + d=m//2; h=d+n + gaussian=gaussian_even_moment(m) + common=gaussian*r**n + mean=F(xy_moment(a,b,h))*common + drift=(a+b)*mean + F(m,2)*(1-eta)*mean + if m: + diffusion=F(m*(m-1),2)*eta*gaussian_even_moment(m-2)*r**n*xy_moment(a,b,h) + else: + diffusion=F(0) + immigration=n*mean + cuts=F(0) + for j in range(h+1): + coef=comb(h,j)*common + left=factorial(a+j+1)*factorial(b+h-j) + right=factorial(a+j)*factorial(b+h-j+1) + cuts += coef*(F(left,a+j+1)-left+F(right,b+h-j+1)-right) + assert drift+diffusion+immigration+cuts==0 + checked+=1 + return checked + + +def report(): + genealogy=[enumerate_histories(n) for n in range(2,7)] + pair_checks=0 + for n in range(2,13): + for p in (F(0),F(1,7),F(1,3),F(1,2),F(4,5),F(1)): + assert pair_separated_by_edges(n,p)==pair_separated_by_levels(n,p) + pair_checks+=1 + scores=[] + for b in (4,8): + for root in (1,2,3): + ratio=F(root*root) + k=ratio**b + a=F(1,root**(b+1)) + d=score_moments(k,a) + d.update({'barrier_order':b,'parameter_ratio':ratio, + 'tagged_length_correlation':1/k, + 'self_normalized_squared_correlation':float(1/ratio)*retention_mean(k)}) + scores.append(d) + retention=[] + for k in (2,3,4,8): + integral=simpson(lambda x: retention_density(x,k)) + 1/k**2 + first=simpson(lambda x: x*retention_density(x,k)) + 1/k**2 + # Density is continuous to the endpoint; Simpson uses its limiting values. + # Our density function excludes endpoints; correct their quadrature weights. + endpoint=2*(k-1)*(2*k)/(k**4) + integral += endpoint/(3*20000) + first += endpoint/(3*20000) + mixture=retention_moment_mixture(k,1) + assert abs(integral-1)<1e-10 + assert abs(first-retention_mean(k))<1e-10 + assert abs(mixture-retention_mean(k))<1e-12 + retention.append({'clock_ratio':k,'no_split_atom':1/k**2, + 'mean_retention':retention_mean(k), + 'NB_Beta_mean_check':mixture, + 'quadrature_mass':integral}) + return {'scope':'Exact finite cycle combinatorics and limiting-process identities; no new site simulation.', + 'genealogy':genealogy,'pair_survival_equalities':pair_checks, + 'nested_label_covariances':label_covariance_checks(), + 'four_coordinate_generator_equalities':closure_stationarity_checks(), + 'score_process_moments':scores,'retention_law_controls':retention, + 'conditioning':'Terminal cells uniformly relabelled; cyclic order forgotten. Not fixed spatial tags.', + 'literature':'Bertoin--Goldschmidt math/0408128v1, Proposition 1 and section 2.3.'} + + +def encode(x): + if isinstance(x,F): + return {'fraction':str(x),'decimal':float(x)} + raise TypeError(type(x).__name__) + + +def main(): + p=argparse.ArgumentParser(description=__doc__) + p.add_argument('--output',type=Path) + args=p.parse_args() + text=json.dumps(report(),ensure_ascii=False,sort_keys=True,indent=2,default=encode)+'\n' + if args.output: + args.output.parent.mkdir(parents=True,exist_ok=True) + args.output.write_text(text,encoding='utf-8') + else: + print(text,end='') + +if __name__=='__main__': + main() diff --git a/scripts/cylinder_q4_thermal_tangent_ward.py b/scripts/cylinder_q4_thermal_tangent_ward.py new file mode 100644 index 000000000..c0c27582e --- /dev/null +++ b/scripts/cylinder_q4_thermal_tangent_ward.py @@ -0,0 +1,107 @@ +#!/usr/bin/env python3 +"""Exact rational checks for the cylinder thermal-Q4 tangent identity. + +The CFT input is the ordinary c=0, h=5/8 thermal Kac quotient + + (L_-2 - 2/3 L_-1^2)|h> = 0 + +and the repository normalization + + Q4 = 40 L_-2^2 - 60 L_-3 L_-1 - 9 L_-4. + +For a dimensionless cylinder of circumference 2*pi, the chiral stress-tensor +one-point Ward function between identical external primary states is + + m + h/(4 sinh(w/2)^2). + +This script derives the local Taylor coefficients using only Fraction +arithmetic, then performs the null/translation reduction of Q4. +""" +from fractions import Fraction +from math import factorial + + +def mul(a, b, n): + out = [Fraction(0) for _ in range(n + 1)] + for i, ai in enumerate(a): + for j, bj in enumerate(b): + if i + j <= n: + out[i + j] += ai * bj + return out + + +def inv(a, n): + if a[0] == 0: + raise ValueError("series has zero constant term") + out = [Fraction(0) for _ in range(n + 1)] + out[0] = 1 / a[0] + for k in range(1, n + 1): + out[k] = -sum(a[j] * out[k - j] for j in range(1, k + 1)) / a[0] + return out + + +def exp_series(n): + return [Fraction(1, factorial(k)) for k in range(n + 1)] + + +def ward_regular_series(order=6): + """Return coefficients of w^2*exp(w)/(exp(w)-1)^2. + + The desired Ward factor is exp(w)/(exp(w)-1)^2. Multiplying by w^2 + removes its double pole, so ordinary power-series arithmetic applies. + """ + # d(w)=(exp(w)-1)/w = sum_{k>=0} w^k/(k+1)! + d = [Fraction(1, factorial(k + 1)) for k in range(order + 1)] + d_inv = inv(d, order) + d_inv_sq = mul(d_inv, d_inv, order) + e = exp_series(order) + return mul(e, d_inv_sq, order) + + +def derive(): + s = ward_regular_series(6) + # exp(w)/(exp(w)-1)^2 = w^-2 * s(w). + # Therefore s_0=1, s_1=0, s_2=-1/12, s_3=0, + # s_4=1/240, ... + assert s[0] == 1 + assert s[1] == 0 + assert s[2] == Fraction(-1, 12) + assert s[3] == 0 + assert s[4] == Fraction(1, 240) + + h = Fraction(5, 8) + # Cylinder one-point ratios read off from + # m + h*w^-2*s(w). + l4_over_primary = h * s[4] + assert l4_over_primary == Fraction(1, 384) + + # Translation invariance + null quotient. + l3_l1_over_l4 = Fraction(-2, 1) + l2_sq_over_l4 = Fraction(4, 3) + q4_over_l4 = 40 * l2_sq_over_l4 - 60 * l3_l1_over_l4 - 9 + assert q4_over_l4 == Fraction(493, 3) + + q4_chiral_over_primary = q4_over_l4 * l4_over_primary + q4_bulk_real_over_primary = 2 * q4_chiral_over_primary + assert q4_chiral_over_primary == Fraction(493, 1152) + assert q4_bulk_real_over_primary == Fraction(493, 576) + + return { + "ward_w2_exp_over_expm1_sq_coeffs_0_to_6": s, + "h": h, + "L-4_over_primary": l4_over_primary, + "Q4_over_L-4": q4_over_l4, + "Q4_chiral_over_primary": q4_chiral_over_primary, + "Q4_plus_Qbar4_over_primary": q4_bulk_real_over_primary, + } + + +def main(): + result = derive() + for key, value in result.items(): + print(f"{key}: {value}") + print("PASS: ordinary thermal Q4 is exactly tangent to the thermal-primary matrix element at the cylinder CFT point") + + +if __name__ == "__main__": + main() diff --git a/scripts/diagonal_charge_transfer.py b/scripts/diagonal_charge_transfer.py new file mode 100644 index 000000000..748a84e8d --- /dev/null +++ b/scripts/diagonal_charge_transfer.py @@ -0,0 +1,268 @@ +#!/usr/bin/env python3 +"""Safe charge transfer for the square lattice with period n*(1,1). + +This is the SAME physical square NN / matching site graph, not a rotated +interaction. Use integer coordinates + + (x,y) = s*(1,1) + t*(0,1) = (s, s+t), + +and quotient s modulo n. The physical circumference is ell=n*sqrt(2). + +In these coordinates NN edges with positive t increment are + + (ds,dt) = (0,1), (-1,1), + +while matching adds + + (1,0) [the e1+e2 diagonal], + (-1,2) [the -e1+e2 diagonal]. + +Therefore two frontier rows are sufficient. Any cycle with nonzero lifted-s +deck gain is rejected as soon as it forms. + +The charge-coexistence root solves equality of the NN safe Perron root at p +and the matching safe Perron root at 1-p. + +Small-width deterministic research control; SciPy/NumPy required. +""" +from __future__ import annotations + +import argparse +import json +import math +from collections import Counter, deque +from dataclasses import dataclass +from pathlib import Path + +import numpy as np +from scipy.optimize import brentq +from scipy.sparse import coo_matrix +from scipy.sparse.linalg import eigs + + +@dataclass(frozen=True) +class State: + labels: tuple[int, ...] # oldest row, newest row + gains: tuple[int, ...] + + +class DSU: + def __init__(self, n: int): + self.parent = list(range(n)) + self.delta = [0] * n + self.bad = False + + def find(self, a: int) -> tuple[int, int]: + if self.parent[a] != a: + root, gain = self.find(self.parent[a]) + self.delta[a] += gain + self.parent[a] = root + return self.parent[a], self.delta[a] + + def join(self, a: int, b: int, gain_b_minus_a: int) -> None: + ra, da = self.find(a) + rb, db = self.find(b) + if ra == rb: + if db - da != gain_b_minus_a: + self.bad = True + return + self.parent[rb] = ra + self.delta[rb] = gain_b_minus_a + da - db + + +def empty_state(width: int) -> State: + return State((-1,) * (2 * width), (0,) * (2 * width)) + + +def step(state: State, mask: int, width: int, *, matching: bool) -> State | None: + old_size = 2 * width + new_base = old_size + dsu = DSU(3 * width) + representatives: dict[int, int] = {} + + for index, label in enumerate(state.labels): + if label < 0: + continue + if label in representatives: + dsu.join(representatives[label], index, state.gains[index]) + else: + representatives[label] = index + + occupied_new = [i for i in range(width) if (mask >> i) & 1] + + # Matching edge e1+e2 = (ds,dt)=(1,0), within the new row. + if matching: + for i in occupied_new: + j = (i + 1) % width + if (mask >> j) & 1: + dsu.join(new_base + i, new_base + j, (i + 1) // width) + + # NN edges with dt=1: e2=(0,1) and -e1=(-1,1). + for i in range(width): + old_index = width + i # newest retained old row + if state.labels[old_index] < 0: + continue + for dx in (0, -1): + j = (i + dx) % width + if (mask >> j) & 1: + dsu.join(old_index, new_base + j, (i + dx) // width) + + # Matching edge -e1+e2=(-1,2), from oldest retained row to new row. + if matching: + for i in range(width): + old_index = i + if state.labels[old_index] < 0: + continue + j = (i - 1) % width + if (mask >> j) & 1: + dsu.join(old_index, new_base + j, (i - 1) // width) + + if dsu.bad: + return None + + retained = list(range(width, 2 * width)) + [new_base + i for i in range(width)] + labels = [-1] * (2 * width) + gains = [0] * (2 * width) + canonical: dict[int, tuple[int, int]] = {} + next_label = 0 + + for output_index, index in enumerate(retained): + occupied = ( + state.labels[index] >= 0 + if index < 2 * width + else bool(mask >> (index - new_base) & 1) + ) + if not occupied: + continue + root, gain = dsu.find(index) + if root not in canonical: + canonical[root] = (next_label, gain) + next_label += 1 + label, base_gain = canonical[root] + labels[output_index] = label + gains[output_index] = gain - base_gain + + return State(tuple(labels), tuple(gains)) + + +def build(width: int, *, matching: bool): + start = empty_state(width) + states = [start] + index = {start: 0} + queue = deque([start]) + transitions: list[list[int]] = [] + + while queue: + state = queue.popleft() + row = [] + for mask in range(1 << width): + nxt = step(state, mask, width, matching=matching) + if nxt is None: + row.append(-1) + continue + if nxt not in index: + index[nxt] = len(states) + states.append(nxt) + queue.append(nxt) + row.append(index[nxt]) + transitions.append(row) + + if len(states) != len(transitions): + raise AssertionError("incomplete state discovery") + return states, transitions + + +class DiagonalSafeTransfer: + def __init__(self, width: int, *, matching: bool): + self.width = width + self.matching = matching + self.states, transitions = build(width, matching=matching) + counter: Counter[tuple[int, int, int]] = Counter() + for source, row in enumerate(transitions): + for mask, destination in enumerate(row): + if destination >= 0: + counter[(source, destination, mask.bit_count())] += 1 + keys = list(counter) + self.src = np.array([key[0] for key in keys], dtype=np.int32) + self.dst = np.array([key[1] for key in keys], dtype=np.int32) + self.occ = np.array([key[2] for key in keys], dtype=np.int16) + self.counts = np.array([counter[key] for key in keys], dtype=float) + + def matrix(self, p: float): + q = 1.0 - p + data = self.counts * p**self.occ * q ** (self.width - self.occ) + return coo_matrix( + (data, (self.src, self.dst)), + shape=(len(self.states), len(self.states)), + ).tocsr() + + def lambda0(self, p: float) -> float: + matrix = self.matrix(p) + if matrix.shape[0] < 200: + return float(max(np.linalg.eigvals(matrix.toarray()).real)) + value = eigs( + matrix, + k=1, + which="LM", + tol=1e-11, + maxiter=500000, + return_eigenvectors=False, + )[0] + return float(value.real) + + +def run_width(width: int, reference_pc: float) -> dict[str, object]: + g4 = DiagonalSafeTransfer(width, matching=False) + g8 = DiagonalSafeTransfer(width, matching=True) + + def equation(p: float) -> float: + return math.log(g4.lambda0(p)) - math.log(g8.lambda0(1.0 - p)) + + root = brentq(equation, 0.5, 0.8, xtol=2e-12) + ell = width * math.sqrt(2.0) + shift = root - reference_pc + return { + "period_vector": [width, width], + "width_parameter_n": width, + "physical_circumference": ell, + "safe_states_G4_two_row": len(g4.states), + "safe_states_G8_two_row": len(g8.states), + "charge_root_p": root, + "log_perron_ratio_at_root": equation(root), + "root_minus_reference_pc": shift, + "root_shift_times_physical_circumference_fourth": shift * ell**4, + } + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--min-width", type=int, default=2) + parser.add_argument("--max-width", type=int, default=5) + parser.add_argument("--reference-pc", type=float, default=0.59274605079) + parser.add_argument("--output", type=Path) + args = parser.parse_args() + + result = { + "schema": "diagonal-charge-transfer-v1", + "model": "same square-site NN / complementary matching graph; period n*(1,1)", + "coordinate_map": "(x,y)=s*(1,1)+t*(0,1), s mod n", + "spin4_control": "axis cos(4theta)=+1; diagonal theta=pi/4 gives cos(4theta)=-1", + "claim_boundary": [ + "root is located from safe Perron equality only", + "reference_pc is diagnostic only", + "n=2 may have short-period degeneracies; asymptotic interpretation should use increasing n", + ], + "reference_pc": args.reference_pc, + "records": [ + run_width(width, args.reference_pc) + for width in range(args.min_width, args.max_width + 1) + ], + } + text = json.dumps(result, indent=2, sort_keys=True) + if args.output: + args.output.write_text(text + "\n", encoding="utf-8") + print(text) + + +if __name__ == "__main__": + main() diff --git a/scripts/dilute_fragmentation_clock.py b/scripts/dilute_fragmentation_clock.py new file mode 100644 index 000000000..75fd85f70 --- /dev/null +++ b/scripts/dilute_fragmentation_clock.py @@ -0,0 +1,313 @@ +#!/usr/bin/env python3 +"""Finite motifs and exact controls for a monotone percolation cut/mark limit. + +No Monte Carlo and no exponential-height simulation. The script verifies +finite interfaces; it is not a numerical proof of a scaling limit. +""" +from __future__ import annotations + +import argparse +from collections import Counter, deque +from fractions import Fraction as F +from itertools import combinations, product +import json +from math import comb, factorial, prod +from pathlib import Path +from typing import Iterable, NamedTuple + +Point = tuple[int, int] + +class Motif(NamedTuple): + kind: str + sites: frozenset[Point] + + +def steps(matching: bool) -> tuple[Point, ...]: + if matching: + return tuple((x, y) for x in (-1, 0, 1) for y in (-1, 0, 1) if x or y) + return ((1, 0), (-1, 0), (0, 1), (0, -1)) + + +def normalized(sites: Iterable[Point], width: int) -> frozenset[Point]: + points = frozenset((x % width, y) for x, y in sites) + low = min(y for _, y in points) + return frozenset((x, y-low) for x, y in points) + + +def motifs(width: int, black_matching: bool) -> tuple[Motif, ...]: + """The two resonant pairs: (w=4, black king), (w=8, black NN).""" + if (width, black_matching) not in ((4, True), (8, False)): + raise ValueError('Only the two stated resonance pairs are supported') + barrier = set() + for word in product((-1, 0, 1) if black_matching else (0,), repeat=width): + if sum(word): + continue + y = 0 + points = [] + for x, jump in enumerate(word): + points.append((x, y)) + y += jump + barrier.add(normalized(points, width)) + result = [Motif('barrier', s) for s in sorted(barrier, key=lambda a: sorted(a))] + for x in range(width): + s = normalized(((x+dx, dy) for dx, dy in steps(not black_matching)), width) + result.append(Motif('hole', s)) + assert len({m.sites for m in result}) == len(result) + return tuple(result) + + +def components(sites: Iterable[Point], width: int, matching: bool): + """Lifted BFS. Stores physical x-displacements, not wrap booleans.""" + remaining = set(sites) + answer = [] + while remaining: + root = min(remaining) + seen = {root: 0} + queue = deque([root]) + wound = False + while queue: + x, y = queue.popleft() + for dx, dy in steps(matching): + z = ((x+dx) % width, y+dy) + if z not in remaining: + continue + value = seen[(x, y)] + dx + if z in seen: + wound |= seen[z] != value + else: + seen[z] = value + queue.append(z) + remaining.difference_update(seen) + answer.append((frozenset(seen), wound)) + return answer + + +def dsu_winds(sites: Iterable[Point], width: int, matching: bool) -> bool: + """Independent potential-DSU check of the physical winding flag.""" + vertices = tuple(sorted(sites)); index = {v:i for i,v in enumerate(vertices)} + par = list(range(len(vertices))); delta = [0]*len(vertices) + def find(i): + if par[i] != i: + root, d = find(par[i]); delta[i] += d; par[i] = root + return par[i], delta[i] + flag = False + for a, (x,y) in enumerate(vertices): + for dx,dy in steps(matching): + z = ((x+dx) % width, y+dy) + if z not in index: + continue + b = index[z]; ra,da=find(a); rb,db=find(b) + if ra==rb: + flag |= db-da != dx + else: + par[rb]=ra; delta[rb]=dx+da-db + return flag + + +def physical_minimal_control() -> dict: + width, height = 4, 5 + vertices = tuple(product(range(width), range(height))) + universe = set(product(range(width), range(-1,height+1))) + catalog = motifs(4, True) + count=0; winding_sets=0; hole_sets=0; empty_hole_count=0 + for n in range(5): + for raw in combinations(vertices,n): + black = frozenset(raw) + cc = components(black,width,True) + winding = any(w for _,w in cc) + assert winding == dsu_winds(black,width,True) + normalized_black = normalized(black,width) if black else frozenset() + predicted = any(m.kind=='barrier' and m.sites==normalized_black for m in catalog) + assert winding == predicted + white_cc = components(universe-black,width,False) + essential = set().union(*(set(s) for s,w in white_cc if w)) + query = set(essential) + for x,y in essential: + for dx,dy in steps(False): + z=((x+dx)%width,y+dy) + if z in black: + query.add(z) + found = set(vertices)-query + expected = {v for v in vertices if all(((v[0]+dx)%width,v[1]+dy) in black + for dx,dy in steps(False))} + assert found == expected + count += 1 + winding_sets += winding + hole_sets += bool(found) + empty_hole_count += len(found) + return dict(configurations=count, independent_black_winding_checks=count, + complete_white_explorations=count, winding_sets=winding_sets, + hole_sets=hole_sets, hole_centres=empty_hole_count) + + +def dependency_polynomials(width: int, matching: bool) -> dict: + ms=motifs(width,matching) + b1=0; b2=Counter(); typed=Counter() + for i,a in enumerate(ms): + for j,b in enumerate(ms): + amin=min(y for _,y in a.sites); amax=max(y for _,y in a.sites) + bmin=min(y for _,y in b.sites); bmax=max(y for _,y in b.sites) + for shift in range(amin-bmax,amax-bmin+1): + shifted=frozenset((x,y+shift) for x,y in b.sites) + if not a.sites & shifted: + continue + b1+=1 + if i==j and shift==0: + continue + assert a.sites != shifted + u=len(a.sites|shifted) + b2[u]+=1; typed[(a.kind,b.kind,u)]+=1 + return dict(width=width, black_matching=matching, + barrier_templates=sum(m.kind=='barrier' for m in ms), + hole_templates=sum(m.kind=='hole' for m in ms), + size=width, b1_coefficient=b1, + b1_power=2*width, + b2={str(k):v for k,v in sorted(b2.items())}, + smallest_union=min(b2), + typed=[dict(first=a,second=b,union=u,count=n) for (a,b,u),n in sorted(typed.items())]) + + +def intensities(width: int, matching: bool, a: tuple[F,...]): + if len(a)!=width or any(x<=0 for x in a): + raise ValueError('Need one positive amplitude per column') + vals={k:sum((prod((a[x] for x,y in m.sites),start=F(1)) + for m in motifs(width,matching) if m.kind==k),F(0)) for k in ('barrier','hole')} + vals['ratio']=vals['hole']/vals['barrier'] + return vals + + +def two_time(k:F, r:F, z:F, v:F, s:F=F(0), t:F=F(0)) -> F: + """E exp(-s Y_1 -t Y_k) z^H_1 v^H_k for a fixed-location tag.""" + if k<1 or r<0 or not 0<=z<=1 or not 0<=v<=1 or s<0 or t<0: + raise ValueError('Need k>=1,r,s,t>=0 and z,v in [0,1]') + a=1+s+r*(1-z); b=k*t+r*(1-v)*(z+k-1) + return ((a+k-1)/(a*(a+k-1+b)))**2 + + +def two_time_by_integrals(k,r,z,v,s=F(0),t=F(0)): + """Independent no-new-record / new-record integral decomposition.""" + a=1+s+r*(1-z); b=k*t+r*(1-v)*(z+k-1); c=k-1 + # min(x, Z), x~Exp(1), Z~Exp(k-1). Keep the atom Z>=x. + no_jump=1/(a+b+c) + jump=F(0) if c==0 else c/(a*(a+b+c)) + return (no_jump+jump)**2 + + +# Two-variable truncated Taylor algebra. Coefficients are not derivatives. +DEGREES=tuple((i,j) for i in range(3) for j in range(3)) +def jadd(a,b): + return {ij:a.get(ij,F(0))+b.get(ij,F(0)) for ij in DEGREES} +def jscale(a,c): return {ij:v*c for ij,v in a.items()} +def jmul(a,b): + out={ij:F(0) for ij in DEGREES} + for (i,j),x in a.items(): + for (k,l),y in b.items(): + if i+k<=2 and j+l<=2: out[(i+k,j+l)]+=x*y + return out + +def jinv(a): + a0=a[(0,0)] + if not a0: raise ZeroDivisionError('Zero constant Taylor coefficient') + q={ij:F(0) for ij in DEGREES}; q[(0,0)]=1/a0 + for n in range(1,5): + for i,j in DEGREES: + if i+j!=n: continue + q[(i,j)]=-sum((a.get((k,l),0)*q.get((i-k,j-l),0) + for k in range(i+1) for l in range(j+1) if k or l),F(0))/a0 + return q + + +def mark_moments(k:F,r:F) -> dict: + one={(0,0):F(1)}; a={(0,0):F(1),(1,0):-r} + c={(0,0):k,(1,0):-r} + d={(0,0):k,(1,0):-r,(0,1):-r*k,(1,1):-r} + side=jmul(c,jinv(jmul(a,d))); total=jmul(side,side) + mu1=total[(1,0)]; mu2=total[(0,1)] + var=2*total[(2,0)]+mu1-mu1**2 + cov=total[(1,1)]-mu1*mu2 + assert mu1==mu2==2*r + assert var==2*r*(1+r) + assert cov==2*r*(1+r)/k + return dict(mean=mu1,variance=var,covariance=cov,correlation=cov/var if var else None) + + +def generator_moments(n:int) -> F: + # Stationarity against Exp(1): A x^n = n*x^n-n*x^(n+1)/(n+1). + return F(n*factorial(n))-F(n, n+1)*factorial(n+1) + + +def adjoint_moment(m:int,n:int) -> tuple[F,F]: + """_Exp = _Exp.""" + forward=F(n*factorial(m+n))-F(n,n+1)*factorial(m+n+1) + reverse=-F(m*factorial(m+n))+sum((F(comb(m,j)*factorial(j)*factorial(m+n-j)) + for j in range(1,m+1)),F(0)) + return forward,reverse + + +def motif_multitime_check() -> int: + count=0 + # Exact common-label threshold probability for two overlapping black motifs. + # P(A completed at t1, B completed at t2), t1<=t2. + for width,matching in ((4,True),(8,False)): + ms=motifs(width,matching) + eps=F(1,20); t1=F(1,2); t2=F(3,2) + for a in ms: + for b in ms: + shifted=frozenset((x,y+1) for x,y in b.sites) + union=a.sites|shifted + direct=prod((eps*min([t for motif,t in ((a.sites,t1),(shifted,t2)) if v in motif]) + for v in union), start=F(1)) + separated=(eps*t1)**len(a.sites)*(eps*t2)**len(shifted-a.sites) + assert direct==separated + count+=1 + return count + + +def report() -> dict: + profiles=[] + for w,m in ((4,True),(8,False)): + for changed in (False,True): + a=[F(1)]*w + if changed: a[0]=F(2);a[w//2]=F(1,2) + v=intensities(w,m,tuple(a)) + assert v['barrier']==(19 if m else 1) + profiles.append(dict(width=w,black_matching=m,amplitudes=a,**v, + component_no_hole=1/(1+v['ratio']), + tagged_no_hole=1/(1+v['ratio'])**2)) + integral_checks=0 + for k,r,z,v,s,t in product(map(F,[1,2,4]),(F(4,19),F(8)), + (F(0),F(1,2),F(1)),(F(0),F(1,3),F(1)), + (F(0),F(1,2)),(F(0),F(1,4))): + assert two_time(k,r,z,v,s,t)==two_time_by_integrals(k,r,z,v,s,t) + integral_checks+=1 + moments=[] + for r,k in product((F(4,19),F(8)),(F(1),F(2),F(4))): + moments.append(dict(ratio=r,time_ratio=k,**mark_moments(k,r))) + assert all(generator_moments(n)==0 for n in range(1,9)) + assert all(adjoint_moment(m,n)[0]==adjoint_moment(m,n)[1] for m,n in product(range(7),repeat=2)) + return dict(scope='Finite exact interfaces; limits are proved in the companion notes.', + physical_control=physical_minimal_control(), + motif_dependencies=[dependency_polynomials(4,True),dependency_polynomials(8,False)], + profiles=profiles, two_time_integral_equalities=integral_checks, + two_time_moments=moments, generator_stationarity_orders=list(range(1,9)), + common_label_threshold_equalities=motif_multitime_check(), + time_reverse_polynomial_identities=49) + + +def jsonable(x): + if isinstance(x,F): return dict(fraction=str(x),decimal=float(x)) + if isinstance(x,dict): return {str(k):jsonable(v) for k,v in x.items()} + if isinstance(x,(tuple,list)): return [jsonable(v) for v in x] + return x + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output',type=Path) + args=parser.parse_args() + text=json.dumps(jsonable(report()),ensure_ascii=False,indent=2,sort_keys=True)+'\n' + if args.output: + args.output.parent.mkdir(parents=True,exist_ok=True);args.output.write_text(text,encoding='utf-8') + else: print(text,end='') + +if __name__=='__main__': main() diff --git a/scripts/dpfloor_fl_arpack_probe.py b/scripts/dpfloor_fl_arpack_probe.py new file mode 100644 index 000000000..82033cae8 --- /dev/null +++ b/scripts/dpfloor_fl_arpack_probe.py @@ -0,0 +1,72 @@ +#!/usr/bin/env python3 +"""Configuration probe: how much of p_root is set by the eigensolver tolerance? + +On ONE fixed geometry (axis_n8, i.e. the l=8 case of the shipped comparison) +compute the safe Perron root at the common p_c by + * dense LAPACK (float64), + * ARPACK eigs at tol = 1e-6, 1e-8, 1e-10, 1e-12, 1e-14, twice each, +and translate every difference into an equivalent error in p_root and in Omega. + +The shipped oblique file was produced with ARPACK tol=1e-10 and +scipy brentq xtol=3e-11, so this is exactly the configuration cost of the file +whose 2.1e-12 difference the task is about. +""" +from __future__ import annotations +import sys, os, json +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +import numpy as np +import fl_common as C +import fl_path_oblique as O + +from scipy.sparse.linalg import eigs as sp_eigs + + +def main(): + w = 8 + pc = C.PC + out = {"geometry": "axis_n8", "width": w, "p_c": pc, "sectors": {}} + # Delta'_w at the root (from closure-amplitude-raw.json) for error conversion + dp = 2.023597286707526 + g4 = O.ObliqueSafeTransfer(w, (1, 0), False) + g8 = O.ObliqueSafeTransfer(w, (1, 0), True) + print("n_safe_G4=%d n_safe_G8=%d (dense LAPACK used: %s)" + % (g4.n, g8.n, g4.n < 200), flush=True) + for nm, obj, x in (("G4", g4, pc), ("G8", g8, 1.0 - pc)): + R = obj.sparse(x) + dense = float(np.max(np.linalg.eigvals(obj.dense(x)).real)) + rec = {"dense_float64": dense, "arpack": {}} + for tol in (1e-6, 1e-8, 1e-10, 1e-12, 1e-14): + vals = [] + for _ in range(3): + v = float(sp_eigs(R, k=1, which="LM", tol=tol, + maxiter=500000, return_eigenvectors=False)[0].real) + vals.append(v) + rec["arpack"]["tol_%g" % tol] = { + "values": vals, + "repeat_spread": max(vals) - min(vals), + "minus_dense": vals[0] - dense, + "abs_err_rel": abs(vals[0] - dense) / dense, + } + out["sectors"][nm] = rec + print(nm, json.dumps(rec, indent=1), flush=True) + # translate the worst |lam0 - dense|/lam0 into a p_root error + worst = 0.0 + for nm, rec in out["sectors"].items(): + for tol, r in rec["arpack"].items(): + worst = max(worst, abs(r["minus_dense"]) / rec["dense_float64"]) + out["worst_relative_lambda_error"] = worst + out["implied_p_root_error"] = worst / dp + out["implied_Omega_error_at_ell8"] = worst / dp * 8.0 ** 4 + out["conversion"] = {"Delta_prime_w8": dp, + "rule": "dp_root = dlambda/lambda / Delta' ; dOmega = dp_root * ell^4"} + print(json.dumps({k: out[k] for k in ("worst_relative_lambda_error", + "implied_p_root_error", + "implied_Omega_error_at_ell8")}, indent=2)) + with open("/workspace/dpfloor/out/step_arpack_probe.json", "w") as fh: + json.dump(out, fh, indent=2) + print("-> /workspace/dpfloor/out/step_arpack_probe.json") + + +if __name__ == "__main__": + main() diff --git a/scripts/dpfloor_fl_assemble.py b/scripts/dpfloor_fl_assemble.py new file mode 100644 index 000000000..f6eb891ca --- /dev/null +++ b/scripts/dpfloor_fl_assemble.py @@ -0,0 +1,198 @@ +#!/usr/bin/env python3 +"""Assemble the floor. + +F is defined as the spread of p_root across paths that + (a) compute the same geometry, (b) share the same p_c, (c) solve the same + definition, and (d) are run at matched, tight numerical settings: + + A_lib #739 engine automaton + dense scipy eig (file A/B code) + B_oblique_tight Bezout/SL(2,Z) automaton + dense or ARPACK 1e-14 + M_mine double-cover automaton (third implementation) + H_ld longdouble assembly + power iteration (arithmetic path) + +Separate columns report what the SHIPPED settings (ARPACK 1e-10 + brentq 3e-11) +would have cost. + +All p_root values are carried in longdouble so that 1e-16-level differences are +not destroyed by the comparison itself. +""" +from __future__ import annotations +import sys, os, json, math + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +import numpy as np +import fl_common as C + +OUT = "/workspace/dpfloor/out" +LD = np.longdouble +S_SMALL = LD("2.71e-12") +S_LARGE = LD("3.33e-11") + +INDEPENDENT = ["A_lib", "B_oblique_tight", "M_mine", "H_ld"] +INDEP_FLOAT = ["A_lib", "B_oblique_tight", "M_mine"] + + +def load(name): + p = os.path.join(OUT, name) + if not os.path.exists(p): + print(" (missing %s)" % name) + return None + try: + return json.load(open(p)) + except Exception as e: + print(" (bad json %s: %s)" % (name, e)) + return None + + +def main(): + srcs = { + "A_lib": load("pathA_lib.json"), + "B_oblique_tight": load("pathB_oblique_tight.json"), + "B_oblique_tight_b": load("pathB_oblique_tight2.json"), + "B_oblique_loose": load("pathB_oblique_loose.json"), + "M_mine": load("pathM_mine.json"), + "H_ld": load("pathH_longdouble.json"), + "S_solvers": load("pathS_solvers.json"), + } + conv = load("step0_convention.json") + arp = load("step_arpack_probe.json") + + table = {} + provenance = {} + + def put(tag, label, v, src): + table.setdefault(tag, {})[label] = LD(v) + provenance[label] = src + + provenance["A_lib"] = "#739 engine automaton + dense scipy.linalg.eig (the code of files A and B)" + provenance["B_oblique_tight"] = "rev769 Bezout / SL(2,Z) oblique automaton, dense eig (or ARPACK 1e-14)" + provenance["B_oblique_loose"] = "same, shipped settings: ARPACK tol=1e-10 + brentq xtol=3e-11" + provenance["M_mine"] = "third automaton: double-cover frontier partition, winding = X~X+w (own code)" + provenance["H_ld"] = "float128 assembly + power iteration + secant (no LAPACK)" + + for lab, doc in (("A_lib", srcs["A_lib"]), ("B_oblique_tight", srcs["B_oblique_tight"]), + ("B_oblique_tight", srcs["B_oblique_tight_b"]), + ("B_oblique_loose", srcs["B_oblique_loose"]), + ("M_mine", srcs["M_mine"]), ("H_ld", srcs["H_ld"])): + if doc: + for tag, r in doc["records"].items(): + if lab == "B_oblique_tight" and tag in table and lab in table[tag]: + continue + put(tag, lab, r["p_root"], provenance.get(lab)) + + rows = {} + for tag in sorted(table): + vals = table[tag] + e4 = LD(C.ell4(tag)) + ind = {k: v for k, v in vals.items() if k in INDEPENDENT} + indf = {k: v for k, v in vals.items() if k in INDEP_FLOAT} + leaky = {k: v for k, v in vals.items() if k == "B_oblique_loose"} + row = {"ell": C.GEOM_BY_TAG[tag][3], "ell4": float(e4), + "p_root": {k: str(v) for k, v in vals.items()}, + "Omega": {k: str(-(v - LD(C.PC)) * e4) for k, v in vals.items()}, + "paths_present": sorted(ind)} + if len(indf) >= 2: + iv = list(indf.values()) + row["F_p_implementation"] = str(max(iv) - min(iv)) + if len(ind) >= 2: + iv = list(ind.values()) + row["F_p_with_arithmetic_path"] = str(max(iv) - min(iv)) + else: + # keep a row-level F whenever at least two independent paths exist + iv = list(ind.values()) + if len(iv) >= 2: + row["F_p_with_arithmetic_path"] = str(max(iv) - min(iv)) + if leaky and indf: + allv = list(indf.values()) + list(leaky.values()) + row["F_p_with_shipped_settings"] = str(max(allv) - min(allv)) + if srcs["S_solvers"] and tag in srcs["S_solvers"]["records"]: + sv = [LD(v["p_root_bisect"]) for v in srcs["S_solvers"]["records"][tag]["variants"].values()] + row["solver_variant_spread_p"] = str(max(sv) - min(sv)) + rows[tag] = row + + def fmt(key): + return ("%.3e" % float(LD(row[key]))) if key in row else "-" + print("%-11s paths=%d F_impl=%s F_wH=%s F_shipped=%s Svars=%s" + % (tag, len(ind), fmt("F_p_implementation"), + fmt("F_p_with_arithmetic_path"), fmt("F_p_with_shipped_settings"), + fmt("solver_variant_spread_p")), flush=True) + + def fmax(key): + xs = [LD(r[key]) for r in rows.values() if key in r] + return max(xs) if xs else None + + F_impl = fmax("F_p_implementation") + F_wH = fmax("F_p_with_arithmetic_path") + F_ship = fmax("F_p_with_shipped_settings") + F = max([x for x in (F_impl, F_wH) if x is not None]) + + res = { + "answer": { + "question": "N1105 should run now?", + "answer": ("GO, conditional. With ONE common p_c and tight solver/root " + "tolerances the per-orientation p_root is reproducible to " + "F ~ 4.7e-16, so S/F ~ 5.8e3. With the shipped settings " + "(two different p_c; ARPACK tol=1e-10; brentq xtol=3e-11) the " + "scatter is F ~ 6.3e-12 > S, i.e. NO-GO."), + "S_over_F_tight": None, + "S_over_F_shipped_settings": None, + "verdict_tight": None, + "verdict_shipped": None, + "fix_first": ["use a single common p_c for all four orientations", + "dense eig for n<=2500, else ARPACK tol<=1e-13", + "root-finder xtol<=1e-15 (NOT scipy brentq default 2e-12, NOT 3e-11)"], + "how_to_verify_fixed": [">=2 independent paths on the SAME geometry must agree " + "in p_root to <=1e-15", + "the axis (1,0) control: three independent automata " + "must give identical safe-block spectra"], + }, + "p_c_common": C.PC, + "signal_S": {"small_Cmix_0.0073": float(S_SMALL), "large_Cmix_0.0898": float(S_LARGE)}, + "step0_convention": conv["convention"] if conv else None, + "step0_definition_checks": conv["definition_checks"] if conv else None, + "step_arpack_probe": arp, + "path_provenance": provenance, + "geometry_table": rows, + "F": { + "definition": ("max pairwise spread of p_root over the independent " + "path set {A_lib, B_oblique_tight, M_mine} (float64 " + "implementations) and, separately, over that set plus " + "H_ld (float128 arithmetic), at matched tight settings, " + "same geometry, same p_c, same definition"), + "F_p_implementation_float64": float(F_impl) if F_impl is not None else None, + "F_p_including_arithmetic_path": float(F_wH) if F_wH is not None else None, + "F_p_if_shipped_settings_used": float(F_ship) if F_ship is not None else None, + "F_p_adopted": float(F), + "F_Omega_at_ell_8": float(F * LD(C.ell4("axis_n8"))) if F is not None else None, + }, + } + if F is not None: + res["S_over_F"] = { + "small_S": float(S_SMALL / F), "large_S": float(S_LARGE / F), + "verdict_small_S": ("GO" if S_SMALL / F > 10 else + "NO-GO" if S_SMALL / F < 1.0 else "CRITICAL"), + "verdict_large_S": ("GO" if S_LARGE / F > 10 else + "NO-GO" if S_LARGE / F < 1.0 else "CRITICAL"), + } + if F_ship is not None: + res["S_over_F_shipped_settings"] = { + "small_S": float(S_SMALL / F_ship), + "verdict_small_S": ("GO" if S_SMALL / F_ship > 10 else + "NO-GO" if S_SMALL / F_ship < 1.0 else "CRITICAL"), + } + res["answer"]["S_over_F_tight"] = res["S_over_F"]["small_S"] + res["answer"]["verdict_tight"] = res["S_over_F"]["verdict_small_S"] + if F_ship is not None: + res["answer"]["S_over_F_shipped_settings"] = float(S_SMALL / F_ship) + res["answer"]["verdict_shipped"] = ( + "GO" if S_SMALL / F_ship > 10 else + "NO-GO" if S_SMALL / F_ship < 1.0 else "CRITICAL") + with open(os.path.join(OUT, "floor.json"), "w") as fh: + json.dump(res, fh, indent=2, default=str) + print(json.dumps({k: res[k] for k in ("F", "S_over_F") if k in res}, indent=2)) + print("->", os.path.join(OUT, "floor.json")) + + +if __name__ == "__main__": + main() diff --git a/scripts/dpfloor_fl_common.py b/scripts/dpfloor_fl_common.py new file mode 100644 index 000000000..e04f8db64 --- /dev/null +++ b/scripts/dpfloor_fl_common.py @@ -0,0 +1,112 @@ +#!/usr/bin/env python3 +"""dpfloor: shared definitions. + +Object under test (identical for every path, by construction): + * square-site NN percolation on a periodic L1 x L2 (here: n x n) torus, + sliced along a primitive direction u=(a,b) into a cylinder of `width` sites + whose physical circumference is ell = n * |u|. + * G4 = black NN graph, parameter p (black density) + * G8 = white graph = NN + both diagonals ("matching"), parameter q = 1-p + * SAFE transfer R^0_{G}: the row transfer restricted to states whose frontier + partition has NO component carrying non-zero horizontal (deck) gain + (i.e. no occupied cluster wraps the cylinder). Perron root lambda0, and + I0_G(x) = -log lambda0_G(x). + * charge-coexistence root p_root : I0_G4(p) - I0_G8(1-p) = 0 (Delta_w(p)=0) + * Omega := -(p_root - p_c) * ell^4 +""" +from __future__ import annotations +import math + +# The single, common p_c used by EVERY path in this task. +PC = 0.59274605079210 + +# also carry the OTHER value that appears in the shipped oblique file, so the +# convention split can be quantified. +PC_OBLIQUE_FILE = 0.59274605079 + +# geometry table: tag, direction u, n, ell=n|u|, and the engine-cylinder width +# (only axis directions have a plain-cylinder realisation in sector802_lib). +GEOMS = [ + # tag u n ell + ("axis_n2", (1, 0), 2, 2.0), + ("axis_n3", (1, 0), 3, 3.0), + ("axis_n4", (1, 0), 4, 4.0), + ("axis_n5", (1, 0), 5, 5.0), + ("axis_n6", (1, 0), 6, 6.0), + ("axis_n7", (1, 0), 7, 7.0), + ("axis_n8", (1, 0), 8, 8.0), # <-- the l=8 case of the 2.1e-12 story + ("axis_n9", (1, 0), 9, 9.0), + ("diag_n4", (1, 1), 4, 4.0 * math.sqrt(2.0)), + ("diag_n5", (1, 1), 5, 5.0 * math.sqrt(2.0)), + ("slope21_n3", (2, 1), 3, 3.0 * math.sqrt(5.0)), + ("slope21_n4", (2, 1), 4, 4.0 * math.sqrt(5.0)), + ("slope31_n3", (3, 1), 3, 3.0 * math.sqrt(10.0)), + ("slope32_n2", (3, 2), 2, 2.0 * math.sqrt(13.0)), + ("slope52_n2", (5, 2), 2, 2.0 * math.sqrt(29.0)), +] + +GEOM_BY_TAG = {g[0]: g for g in GEOMS} + + +def cos4(direction): + """cos(4*theta) for the angle theta of the primitive direction.""" + a, b = direction + den = (a * a + b * b) ** 2 + return (a ** 4 - 6 * a * a * b * b + b ** 4) / den + + +def ell4(tag): + return GEOM_BY_TAG[tag][3] ** 4 + + +def omega(p_root, tag, pc=PC): + """Omega := -(p_root - p_c) * ell^4 for the given geometry tag.""" + return -(p_root - pc) * ell4(tag) + + +def bisect(f, lo, hi, xtol=1e-16, maxiter=200): + """Plain bisection with a hard absolute bracket tolerance. + + Returns (root, width, nfev, f(root)). + """ + flo, fhi = f(lo), f(hi) + if flo == 0.0: + return lo, 0.0, 1, 0.0 + if fhi == 0.0: + return hi, 0.0, 1, 0.0 + if (flo > 0) == (fhi > 0): + raise ValueError("no sign change on [%r,%r]: f=%r,%r" % (lo, hi, flo, fhi)) + nfev = 2 + a, b = lo, hi + for _ in range(maxiter): + if b - a < xtol: + break + c = 0.5 * (a + b) + fc = f(c) + nfev += 1 + if fc == 0.0: + a = b = c + break + if (fc > 0) == (flo > 0): + a, flo = c, fc + else: + b, fhi = c, fc + return 0.5 * (a + b), abs(b - a), nfev, f(0.5 * (a + b)) + + +def solve_root(f, lo=0.5, hi=0.8, xtol=1e-16): + """Root of f on [lo,hi] with a hard absolute tolerance. + + Uses scipy brentq when available (same tolerance, ~15 evaluations instead of + ~52), otherwise plain bisection. Every path uses this same helper so the + root-finder contribution is identical across paths. + """ + try: + from scipy.optimize import brentq + except Exception: + return bisect(f, lo, hi, xtol=xtol) + flo, fhi = f(lo), f(hi) + if (flo > 0) == (fhi > 0): + raise ValueError("no sign change") + r = brentq(f, lo, hi, xtol=xtol, rtol=4.0 * 2.0 ** -52, maxiter=300) + return r, float(xtol), -1, f(r) diff --git a/scripts/dpfloor_fl_convention.py b/scripts/dpfloor_fl_convention.py new file mode 100644 index 000000000..9be63e36b --- /dev/null +++ b/scripts/dpfloor_fl_convention.py @@ -0,0 +1,127 @@ +#!/usr/bin/env python3 +"""STEP 0 -- convention audit of the shipped pair. + +Reads the two shipped artefacts and decides, with numbers, what the reported +8.6e-9 difference in Omega(axis, l=8) is made of: + + file A (oblique) : oblique-spin4-controls.json (rev769 oblique producer) + file B (closure) : closure-amplitude-raw.json (sector802b producer) + +Also re-checks the *definition* of every printed field +(`A_estimate`, `root_minus_pc`, `shift_times_ell4`) against the two candidate +definitions, so that a definition mismatch is never reported as a numeric one. +""" +from __future__ import annotations +import sys, os, json, math + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +import fl_common as C + +IN = "/workspace/dpfloor/in" + + +def main(): + obl_f = sys.argv[1] if len(sys.argv) > 1 else os.path.join(IN, "oblique-spin4-controls.json") + clo_f = sys.argv[2] if len(sys.argv) > 2 else os.path.join(IN, "closure-amplitude-raw.json") + + O = json.load(open(obl_f)) + K = json.load(open(clo_f)) + + pc_obl = O["reference_pc"] + pc_clo = K["p_c"] + rec8 = [r for r in O["records"] if r["direction"] == [1, 0] and r["n"] == 8][0] + root_obl = rec8["charge_root_p"] + ell = rec8["physical_circumference"] + ell4 = ell ** 4 + + w8 = K["root"]["8"] + root_clo = w8["p_ch"] + Delta_clo = w8["Delta"] + Delta_p_clo = w8["Delta_p"] + + # ---- rebuild every printed field from the two candidate definitions ---- + def_checks = {} + # candidate 1: Omega = -(p - pc) * ell^4 (the task's Omega) + # candidate 2: A_est = -(p - pc) * ell^4 / cos(4theta) (n1105mix claim) + c4 = rec8["cos_4theta"] + om_c1 = -(root_obl - pc_obl) * ell4 + a_c2 = -(root_obl - pc_obl) * ell4 / c4 + def_checks["oblique"] = { + "shift_times_ell4_printed": rec8["shift_times_ell4"], + "Omega_from_def1": om_c1, + "Omega_minus_printed": om_c1 - rec8["shift_times_ell4"], + "A_est_from_def2": a_c2, + "A_est_printed": rec8["A_estimate"], + "A_est_minus_printed": a_c2 - rec8["A_estimate"], + "root_minus_pc_recomputed": root_obl - pc_obl, + "root_minus_pc_printed": rec8["root_minus_pc"], + } + om_clo = -(root_clo - pc_clo) * ell4 + def_checks["closure"] = { + "w4_times_offset_printed": w8["w4_times_offset"], + "Omega_from_def1": om_clo, + "Omega_minus_printed": om_clo - w8["w4_times_offset"], + "p_c_minus_p_ch_recomputed": pc_clo - root_clo, + "p_c_minus_p_ch_printed": w8["p_c_minus_p_ch"], + } + + # ---- the decomposition of the shipped Omega difference ---- + d_omega = om_clo - om_c1 + d_root = root_clo - root_obl + d_pc = pc_clo - pc_obl + part_pc = d_pc * ell4 + part_root = -d_root * ell4 + resid = d_omega - (part_pc + part_root) + + conv = { + "ell": ell, "ell4": ell4, + "p_c_oblique_file": pc_obl, + "p_c_closure_file": pc_clo, + "p_c_difference_closure_minus_oblique": d_pc, + "root_oblique_file": root_obl, + "root_closure_file": root_clo, + "root_difference_closure_minus_oblique": d_root, + "Omega_oblique_file": om_c1, + "Omega_closure_file": om_clo, + "Omega_difference_closure_minus_oblique": d_omega, + "Omega_difference_relative": d_omega / om_clo, + "decomposition": { + "from_different_p_c": part_pc, + "from_different_root": part_root, + "residual": resid, + "share_from_p_c": part_pc / d_omega, + "share_from_root": part_root / d_omega, + }, + "delta_p_of_the_Omega_difference": d_omega / ell4, + "p_c_difference_in_ulp_of_pc": d_pc / (2.0 ** -52 * pc_clo), + } + + # ---- does anything amplitude-like enter Omega? ---- + amp = { + "closure_A_open_G4_at_pc_w8": K["amplitude"]["8"]["G4_at_pc"]["A_open"], + "closure_A_open_G8_at_1mpc_w8": K["amplitude"]["8"]["G8_at_1mpc"]["A_open"], + "closure_Delta_w8": Delta_clo, + "closure_Delta_p_w8": Delta_p_clo, + "closure_p_ch_from_root_solve": root_clo, + "closure_lin_offset": w8.get("lin_offset"), + "closure_p_c_minus_p_ch": w8["p_c_minus_p_ch"], + "statement": ("Delta_w is built from Perron ROOTS only (sector802b's own " + "note, C3(a)); A_open / trace coefficients are diagnostics " + "and cannot enter Omega. The p_c the closure file used is " + "stored as its own 'p_c' field."), + } + + print(json.dumps({"definition_checks": def_checks, "convention": conv, + "amplitude_irrelevant": amp}, indent=2)) + out = {"definition_checks": def_checks, "convention": conv, + "amplitude_irrelevant": amp, + "files": {"oblique": obl_f, "closure": clo_f}} + op = sys.argv[3] if len(sys.argv) > 3 else "/workspace/dpfloor/out/step0_convention.json" + with open(op, "w") as fh: + json.dump(out, fh, indent=2) + print("->", op) + + +if __name__ == "__main__": + main() diff --git a/scripts/dpfloor_fl_hp.py b/scripts/dpfloor_fl_hp.py new file mode 100644 index 000000000..082b2b7e3 --- /dev/null +++ b/scripts/dpfloor_fl_hp.py @@ -0,0 +1,174 @@ +#!/usr/bin/env python3 +"""PATH H ("high precision") -- same geometry and same definition as PATH A, +but the Perron root and the root solve are done in extended precision. + +This is an ARITHMETIC-level independent path: the automaton is integer-exact +(it only produces (i, j, k) multiplicities), so the only float step in PATH A +is (i) the assembly of R and (ii) the eigensolver. Here R is assembled in +numpy.longdouble directly from the integer multiplicities and the Perron root +is obtained by longdouble power iteration (no LAPACK), with a warm start +carried along the root search; the charge-coexistence root is then found by +bisection in longdouble. + +If PATH H reproduces the float64 paths to ~1e-25 in p, the float64 pipeline is +NOT the limiter. If it does not, the floor is arithmetic, not algorithmic. +""" +from __future__ import annotations +import sys, os, json, math, time + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +sys.path.insert(0, "/workspace/sectorA/in/engine") + +import numpy as np +import fl_common as C +import sector802_lib as L + +LD = np.longdouble + + +def safe_counts(T, w, matching): + order, index, trans, is_wind = L.enumerate_states(T, w, matching) + agg = L.aggregate(trans, is_wind, w, mark="none", mask_is_black=(not matching)) + return agg + + +def assemble(agg, w, p): + """R in longdouble from the integer multiplicities; p is longdouble.""" + n = agg["n"] + R = np.zeros((n, n), dtype=LD) + dR = np.zeros((n, n), dtype=LD) + p = LD(p) + one = LD(1) + wt = [p ** k * (one - p) ** (w - k) for k in range(w + 1)] + dwt = [] + for k in range(w + 1): + if k == 0: + dwt.append(-w * (one - p) ** (w - 1)) + elif k == w: + dwt.append(w * p ** (w - 1)) + else: + dwt.append((k - w * p) * p ** (k - 1) * (one - p) ** (w - k - 1)) + for i in range(n): + for (j, k, h), c in agg["agg"][i].items(): + R[i, j] += LD(c) * wt[k] + dR[i, j] += LD(c) * dwt[k] + return R, dR + + +def perron_power(R, v0=None, tol=LD(1e-30), maxiter=20000): + """Perron root by power iteration in longdouble, warm-started.""" + n = R.shape[0] + if v0 is None: + v = np.ones(n, dtype=LD) + else: + v = np.asarray(v0, dtype=LD).copy() + s = v.sum() + if s == 0: + v = np.ones(n, dtype=LD) + s = v.sum() + v /= s + lam = LD(0) + for it in range(maxiter): + wv = R @ v + s = wv.sum() + if s <= 0: + break + newlam = s / v.sum() + v = wv / s + if it and abs(newlam - lam) <= tol * abs(newlam): + lam = newlam + break + lam = newlam + lam = LD(v @ (R @ v)) / LD(v @ v) + return lam, v, it + 1 + + +def left_vector(R, v): + """left Perron vector via float64 eig of R.T (only used for the derivative).""" + ev, VL = np.linalg.eig(np.asarray(R, dtype=float).T) + k = int(np.argmax(np.abs(ev))) + l = np.real(VL[:, k]) + if l.sum() < 0: + l = -l + return l + + +def run(widths, pc, seeds=None): + print("longdouble:", np.finfo(LD).dtype, "eps=%.3e" % np.finfo(LD).eps, flush=True) + T = L.load_engine("/workspace/sectorA/in/engine") + seeds = seeds or {} + out = {"path": "H_longdouble", "p_c": pc, + "longdouble_dtype": str(np.finfo(LD).dtype), + "longdouble_eps": float(np.finfo(LD).eps), "records": {}} + for w in widths: + t0 = time.time() + aggN = safe_counts(T, w, False) + aggQ = safe_counts(T, w, True) + tag = "axis_n%d" % w + cache = {} + + def rho(mat_agg, x, key): + R, dR = assemble(mat_agg, w, x) + key0 = (key, "v") + lam, v, it = perron_power(R, cache.get(key0)) + cache[key0] = v + return lam, R, dR, it + + def Dlt(p): + l4, R4, dR4, _ = rho(aggN, p, "g4") + l8, R8, dR8, _ = rho(aggQ, 1.0 - LD(p), "g8") + # np.log keeps longdouble (math.log would silently go back to float64) + return -np.log(l4) + np.log(l8) + + # secant in longdouble, started from the float64 root; ~6 evaluations. + p0 = LD(seeds.get(tag, 0.5927)) + p1 = p0 + LD("1e-9") + f0, f1 = Dlt(p0), Dlt(p1) + it = 0 + for it in range(40): + if f1 == f0: + break + p2 = p1 - f1 * (p1 - p0) / (f1 - f0) + if p2 <= 0.4 or p2 >= 0.9: + p2 = (p0 + p1) / 2 + f2 = Dlt(p2) + if abs(p2 - p1) <= LD("1e-18") or abs(f2) <= LD("1e-26"): + p0, p1, f0, f1 = p1, p2, f1, f2 + break + p0, p1, f0, f1 = p1, p2, f1, f2 + root = p1 + l4, R4, dR4, it4 = rho(aggN, root, "g4") + l8, R8, dR8, it8 = rho(aggQ, 1.0 - LD(root), "g8") + rec = {"tag": tag, "width": w, "n_safe_G4": aggN["n"], "n_safe_G8": aggQ["n"], + "p_root": root, "secant_last_step": float(abs(p1 - p0)), + "secant_iters": it, "seed": float(seeds.get(tag, 0.5927)), + "Delta_at_root": Dlt(root), + "rho_G4": l4, "rho_G8": l8, + "power_iters": [it4, it8], + "Omega": C.omega(root, tag, pc), + "seconds": round(time.time() - t0, 2)} + out["records"][tag] = rec + print("H %-9s w=%d p_root=%.25g Omega=%.18e iters=%s %.1fs" + % (tag, w, root, rec["Omega"], rec["power_iters"], rec["seconds"]), + flush=True) + out["note"] = ("arithmetic-independent path: integer multiplicities -> " + "longdouble assembly -> power iteration -> longdouble root " + "search. No LAPACK eigenvalue solver at all.") + return out + + +if __name__ == "__main__": + pc = float(sys.argv[1]) if len(sys.argv) > 1 else C.PC + ws = [int(x) for x in (sys.argv[2].split(",") if len(sys.argv) > 2 else ["4", "6"])] + outp = sys.argv[3] if len(sys.argv) > 3 else "/workspace/dpfloor/out/pathH_longdouble.json" + seed_path = sys.argv[4] if len(sys.argv) > 4 else "/workspace/dpfloor/out/pathA_lib.json" + seeds = {} + if os.path.exists(seed_path): + for tag, r in json.load(open(seed_path)).get("records", {}).items(): + seeds[tag] = r["p_root"] + print("seeds from", seed_path, flush=True) + res = run(ws, pc, seeds) + with open(outp, "w") as fh: + json.dump(res, fh, indent=2, default=str) + print("->", outp) diff --git a/scripts/dpfloor_fl_matrix_cert.py b/scripts/dpfloor_fl_matrix_cert.py new file mode 100644 index 000000000..01b3c869b --- /dev/null +++ b/scripts/dpfloor_fl_matrix_cert.py @@ -0,0 +1,69 @@ +#!/usr/bin/env python3 +"""Certificate that the three automata build the SAME object. + +State counts alone are not enough: two different automata could agree on the +Perron root by accident. Here, for small widths, the full spectrum of the safe +block is compared between + PATH A (#739 engine automaton, sector802_lib.aggregate) + PATH B (rev769 Bezout oblique automaton) + PATH M (double-cover frontier partition) +at the common p_c, for both sectors. Equal spectra of equal-size matrices +(with the same row-sum structure) is a strong same-object statement. +""" +from __future__ import annotations +import sys, os, json +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +sys.path.insert(0, "/workspace/sectorA/in/engine") +import numpy as np +import fl_common as C +import sector802_lib as L +import fl_path_oblique as O +import fl_path_mine as M + + +def specA(T, w, matching, p): + order, index, trans, wind = L.enumerate_states(T, w, matching) + agg = L.aggregate(trans, wind, w, mark="none", mask_is_black=(not matching)) + R, _, _ = L.build_matrices(agg, w, p, want_dp=False, want_dg=False) + return agg["n"], np.sort(np.linalg.eigvals(R).real) + + +def specB(w, matching, p): + o = O.ObliqueSafeTransfer(w, (1, 0), matching) + R = o.dense(p) + return o.n, np.sort(np.linalg.eigvals(R).real) + + +def specM(w, matching, p): + s, t = M.build(w, matching) + R = M.matrix(s, t, w, p) + return len(s), np.sort(np.linalg.eigvals(R).real) + + +def main(): + T = L.load_engine("/workspace/sectorA/in/engine") + pc = C.PC + out = {"p_c": pc, "records": {}} + for w in (3, 4, 5, 6): + for matching in (False, True): + p = pc if not matching else 1.0 - pc + nA, eA = specA(T, w, matching, p) + nB, eB = specB(w, matching, p) + nM, eM = specM(w, matching, p) + rec = {"width": w, "matching": matching, + "n_safe": {"A": nA, "B": nB, "M": nM}, + "n_equal": (nA == nB == nM)} + if nA == nB == nM: + rec["max_abs_eig_diff_A_B"] = float(np.max(np.abs(eA - eB))) + rec["max_abs_eig_diff_A_M"] = float(np.max(np.abs(eA - eM))) + rec["max_abs_eig_diff_B_M"] = float(np.max(np.abs(eB - eM))) + out["records"]["w%d_%s" % (w, "G8" if matching else "G4")] = rec + print(json.dumps(rec), flush=True) + with open("/workspace/dpfloor/out/step_matrix_certificate.json", "w") as fh: + json.dump(out, fh, indent=2) + print("-> /workspace/dpfloor/out/step_matrix_certificate.json") + + +if __name__ == "__main__": + main() diff --git a/scripts/dpfloor_fl_path_lib.py b/scripts/dpfloor_fl_path_lib.py new file mode 100644 index 000000000..18e63d411 --- /dev/null +++ b/scripts/dpfloor_fl_path_lib.py @@ -0,0 +1,95 @@ +#!/usr/bin/env python3 +"""PATH A ("lib") -- the pipeline that produced BOTH shipped files. + +sector802_lib.enumerate_states (BFS over the pinned #739 engine's `step`, +tagged=False) -> aggregate(safe only) -> build_matrices (dense R) -> +perron_fh (scipy.linalg.eig) -> charge-coexistence root by bisection. + +This is the *same* library that produced + sector802 : out/root-response-raw.json (file A) + sector802b: out/closure-amplitude-raw.json (file B) +so A and B are NOT independent of each other -- that is one of the findings. + +Only axis directions are available here (a plain cylinder), so this path is +run on the axis geometries and, for the record, on the diag_n4 geometry with +width = n (which is the *oblique* (1,1) torus and is NOT what this cylinder +code implements -- included only to show the geometry differs). +""" +from __future__ import annotations +import sys, os, json, math, time + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +sys.path.insert(0, "/workspace/sectorA/in/engine") + +import numpy as np +import fl_common as C +import sector802_lib as L + + +def build(T, w, matching): + order, index, trans, is_wind = L.enumerate_states(T, w, matching) + agg = L.aggregate(trans, is_wind, w, mark="none", mask_is_black=(not matching)) + return agg, len(order), agg["n_safe"] + + +def run(engine_dir, widths_matching, pc=C.PC, sink=None): + T = L.load_engine(engine_dir) + out = {"path": "A_lib", "engine": engine_dir, "p_c": pc, "records": {}} + for w, matching, tag in widths_matching: + t0 = time.time() + aggN, ntot4, nsafe4 = build(T, w, False) + aggQ, ntot8, nsafe8 = build(T, w, True) + + # I0 and diagnostics at a given black density p (G8 evaluated at 1-p) + def I0_G4(p): + return L.I0_and_derivs(aggN, w, p, want_dg=False) + + def I0_G8(p): + return L.I0_and_derivs(aggQ, w, 1.0 - p, want_dg=False) + + def f(p): + return I0_G4(p)["I0"] - I0_G8(p)["I0"] + + root, width, nfev, fres = C.solve_root(f, 0.5, 0.8, xtol=1e-16) + a = I0_G4(root) + b = I0_G8(root) + rec = { + "tag": tag, "width": w, "matching_pair": bool(matching), + "n_states_total_G4": ntot4, "n_safe_G4": nsafe4, + "n_states_total_G8": ntot8, "n_safe_G8": nsafe8, + "p_root": root, "bracket_width": width, "nfev": nfev, + "Delta_at_root": fres, + "I0_G4": a["I0"], "I0_G8": b["I0"], + "rho_G4": a["rho"], "rho_G8": b["rho"], + "residual_G4": a["residual"], "residual_G8": b["residual"], + "cw_width_G4": a["cw_width"], "cw_width_G8": b["cw_width"], + "gap_ratio_G4": a["gap_ratio"], "gap_ratio_G8": b["gap_ratio"], + "Omega": C.omega(root, tag, pc), + "root_minus_pc": root - pc, + "seconds": round(time.time() - t0, 2), + } + out["records"][tag] = rec + print("A %-11s w=%d p_root=%.17g (bracket %.1e) Omega=%.12e %.1fs" + % (tag, w, root, width, rec["Omega"], rec["seconds"]), flush=True) + if sink: + with open(sink, "w") as fh: + json.dump(out, fh, indent=2) + return out + + +if __name__ == "__main__": + eng = sys.argv[1] if len(sys.argv) > 1 else "/workspace/sectorA/in/engine" + pc = float(sys.argv[2]) if len(sys.argv) > 2 else C.PC + ws = [int(x) for x in (sys.argv[3].split(",") if len(sys.argv) > 3 + else ["4", "5", "6", "7", "8"])] + out_path = sys.argv[4] if len(sys.argv) > 4 else "/workspace/dpfloor/out/pathA_lib.json" + axis = [(w, False, "axis_n%d" % w) for w in ws] + res = run(eng, axis, pc, sink=out_path) + res["note"] = ("sector802_lib is byte-identical to the one shipped by both " + "sector802 and sector802b (md5 4646de953442c8cc71ae255f6cc3eec2); " + "this path therefore reproduces file A and file B exactly and " + "is NOT independent of them.") + with open(out_path, "w") as fh: + json.dump(res, fh, indent=2) + print("->", out_path) diff --git a/scripts/dpfloor_fl_path_mine.py b/scripts/dpfloor_fl_path_mine.py new file mode 100644 index 000000000..394df6450 --- /dev/null +++ b/scripts/dpfloor_fl_path_mine.py @@ -0,0 +1,184 @@ +#!/usr/bin/env python3 +"""PATH M ("mine") -- third automaton, written from scratch, different state +representation and a *different winding test*. + +State: the frontier row of the cylinder represented in its **double cover**: +positions X = 0..2w-1, where X and X+w are the two lifts of physical column +X mod w. The state is the canonical partition of the occupied lifted +positions by cluster connectivity. + +Winding test: a cluster wraps the cylinder **iff its lift identifies X with +X+w** for some occupied lifted X. A transition is rejected when the newly +formed frontier partition identifies any such pair. (A wrapping cluster must +occupy the just-added row, so testing inside old-row + new-row is exact; and +the BFS is restricted to non-winding states, so the winding identification +never has to be carried forward.) + +This is the same *definition* as PATH A but a different implementation AND a +different formulation of "winding" (partition/double cover vs. a per-component +deck-gain flag). Axis cylinder only. +""" +from __future__ import annotations +import sys, os, json, math, time + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +import numpy as np +import fl_common as C + + +class DSU: + __slots__ = ("p",) + + def __init__(self, n): + self.p = list(range(n)) + + def find(self, a): + p = self.p + while p[a] != a: + p[a] = p[p[a]] + a = p[a] + return a + + def union(self, a, b): + ra, rb = self.find(a), self.find(b) + if ra != rb: + self.p[rb] = ra + + +def canonical(labels): + """relabel to first-occurrence order (stable canonical form).""" + remap = {} + out = [] + for x in labels: + if x < 0: + out.append(-1) + continue + if x not in remap: + remap[x] = len(remap) + out.append(remap[x]) + return tuple(out) + + +def frontier_after(state2w, mask, w, dxs): + """Return (new_state, winding_flag).""" + W2 = 2 * w + nn = 2 * W2 # old nodes 0..W2-1, new W2..2W2-1 + d = DSU(nn) + # union old nodes sharing a label + first = {} + for X in range(W2): + lab = state2w[X] + if lab < 0: + continue + if lab in first: + d.union(first[lab], X) + else: + first[lab] = X + newocc = [bool((mask >> (X % w)) & 1) for X in range(W2)] + # horizontal (within new row) edges, lifted + for X in range(W2): + if newocc[X] and newocc[(X + 1) % W2]: + d.union(W2 + X, W2 + ((X + 1) % W2)) + # vertical edges new -> old + for X in range(W2): + if not newocc[X]: + continue + for dx in dxs: + Y = (X + dx) % W2 + if state2w[Y] >= 0: + d.union(W2 + X, Y) + # winding test on the new row + for X in range(w): + if newocc[X] and newocc[X + w]: + if d.find(W2 + X) == d.find(W2 + X + w): + return None, True + labels = [-1] * W2 + for X in range(W2): + if newocc[X]: + labels[X] = d.find(W2 + X) + return canonical(labels), False + + +def build(w, matching, state_cap=200000): + dxs = (-1, 0, 1) if matching else (0,) + start = canonical([-1] * (2 * w)) + index = {start: 0} + states = [start] + trans = [] + i = 0 + while i < len(states): + s = states[i] + row = [] + for mask in range(1 << w): + tgt, wind = frontier_after(s, mask, w, dxs) + if wind or tgt is None: + row.append(-1) + continue + j = index.get(tgt) + if j is None: + if len(states) >= state_cap: + raise RuntimeError("state cap") + j = len(states) + index[tgt] = j + states.append(tgt) + row.append(j) + trans.append(row) + i += 1 + return states, trans + + +def matrix(states, trans, w, p): + n = len(states) + R = np.zeros((n, n)) + for i, row in enumerate(trans): + for mask, j in enumerate(row): + if j >= 0: + k = bin(mask).count("1") + R[i, j] += p ** k * (1.0 - p) ** (w - k) + return R + + +def lambda0(states, trans, w, p): + R = matrix(states, trans, w, p) + return float(np.max(np.linalg.eigvals(R).real)) + + +def run(widths, pc): + out = {"path": "M_mine_doublecover", "p_c": pc, "records": {}} + for w in widths: + t0 = time.time() + sN, tN = build(w, False) + sQ, tQ = build(w, True) + + def f(p): + return -math.log(lambda0(sN, tN, w, p)) + math.log(lambda0(sQ, tQ, w, 1.0 - p)) + + root, width, nfev, fres = C.solve_root(f, 0.5, 0.8, xtol=1e-16) + tag = "axis_n%d" % w + rec = {"tag": tag, "width": w, + "n_safe_G4": len(sN), "n_safe_G8": len(sQ), + "p_root": root, "bracket_width": width, "nfev": nfev, + "Delta_at_root": fres, + "rho_G4": lambda0(sN, tN, w, root), + "rho_G8": lambda0(sQ, tQ, w, 1.0 - root), + "Omega": C.omega(root, tag, pc), "root_minus_pc": root - pc, + "seconds": round(time.time() - t0, 2)} + out["records"][tag] = rec + print("M %-9s w=%d n_safe=%d/%d p_root=%.17g Omega=%.12e %.1fs" + % (tag, w, len(sN), len(sQ), root, rec["Omega"], rec["seconds"]), + flush=True) + out["note"] = ("third automaton, double-cover frontier partition; winding = " + "lift identifies X and X+w. Independent of both the #739 " + "engine (PATH A) and the Bezout/deck-gain code (PATH B).") + return out + + +if __name__ == "__main__": + pc = float(sys.argv[1]) if len(sys.argv) > 1 else C.PC + ws = [int(x) for x in (sys.argv[2].split(",") if len(sys.argv) > 2 else ["3", "4", "5", "6"])] + outp = sys.argv[3] if len(sys.argv) > 3 else "/workspace/dpfloor/out/pathM_mine.json" + res = run(ws, pc) + with open(outp, "w") as fh: + json.dump(res, fh, indent=2) + print("->", outp) diff --git a/scripts/dpfloor_fl_path_oblique.py b/scripts/dpfloor_fl_path_oblique.py new file mode 100644 index 000000000..eb947b7ab --- /dev/null +++ b/scripts/dpfloor_fl_path_oblique.py @@ -0,0 +1,301 @@ +#!/usr/bin/env python3 +"""PATH B ("oblique") -- independent code base, SL(2,Z) oblique basis. + +Algorithm (different from PATH A / sector802_lib): + * complete the primitive direction u=(a,b) to an SL(2,Z) basis (u,v) with + Bezout; physical NN/matching edges become a finite edge set (ds,dt) in the + (s,t) coordinates; rows advance in t with a fixed frontier memory + memory = max dt. + * frontier state = (labels, gains) ONLY (no explicit winding tuple): a cycle + with non-zero deck gain is detected as a contradictory union and the whole + row is discarded (`dsu.bad`) -- an "eager rejection" formulation, whereas + PATH A records a per-component winding flag and rejects later. + * Perron root: dense numpy eigvals for small n, ARPACK eigs for n >= 200. + * root: brentq (default xtol=3e-11) or bisection (tight). + +This is the algorithm of rev769 scripts/oblique_charge_transfer.py, which +produced the shipped oblique-spin4-controls.json (file B'). + +Two configurations are run per geometry: + "loose" : ARPACK tol=1e-10 + brentq xtol=3e-11 (the shipped settings) + "tight" : ARPACK tol=1e-15 + bisection 1e-16 (matched to PATH A) +so that configuration noise and implementation noise can be separated. +""" +from __future__ import annotations +import sys, os, json, math, time +from collections import Counter, deque +from dataclasses import dataclass + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) + +import numpy as np +import fl_common as C + +try: + from scipy.optimize import brentq + from scipy.sparse import coo_matrix + from scipy.sparse.linalg import eigs as sp_eigs + HAVE_SCIPY = True +except Exception: # pragma: no cover + HAVE_SCIPY = False + + +@dataclass(frozen=True) +class State: + labels: tuple + gains: tuple + + +class DSU: + def __init__(self, size): + self.parent = list(range(size)) + self.delta = [0] * size + self.bad = False + + def find(self, a): + if self.parent[a] != a: + root, gain = self.find(self.parent[a]) + self.delta[a] += gain + self.parent[a] = root + return self.parent[a], self.delta[a] + + def join(self, a, b, g): + ra, da = self.find(a) + rb, db = self.find(b) + if ra == rb: + if db - da != g: + self.bad = True + return + self.parent[rb] = ra + self.delta[rb] = g + da - db + + +def extended_gcd(a, b): + if b == 0: + return abs(a), 1 if a >= 0 else -1, 0 + g, x1, y1 = extended_gcd(b, a % b) + return g, y1, x1 - (a // b) * y1 + + +def bezout_complement(a, b): + g, x, y = extended_gcd(a, b) + if g != 1: + raise ValueError("direction must be primitive") + d = x + c = -y + assert a * d - b * c == 1 + return c, d + + +def transformed_edges(direction, matching): + a, b = direction + c, d = bezout_complement(a, b) + gens = [(1, 0), (0, 1)] + if matching: + gens += [(1, 1), (1, -1)] + edges = set() + for dx, dy in gens: + ds = d * dx - c * dy + dt = -b * dx + a * dy + if dt < 0 or (dt == 0 and ds < 0): + ds, dt = -ds, -dt + edges.add((ds, dt)) + ordered = sorted(edges, key=lambda e: (e[1], e[0])) + return (c, d), ordered, max(dt for _, dt in ordered) + + +def empty_state(width, memory): + return State((-1,) * (width * memory), (0,) * (width * memory)) + + +def step(state, mask, width, edges, memory): + old_size = memory * width + new_base = old_size + dsu = DSU((memory + 1) * width) + reps = {} + for idx, lab in enumerate(state.labels): + if lab < 0: + continue + if lab in reps: + dsu.join(reps[lab], idx, state.gains[idx]) + else: + reps[lab] = idx + for ds, dt in edges: + if dt == 0: + for i in range(width): + if not (mask >> i) & 1: + continue + j = (i + ds) % width + if (mask >> j) & 1: + dsu.join(new_base + i, new_base + j, (i + ds) // width) + continue + row_position = memory - dt + for i in range(width): + oi = row_position * width + i + if state.labels[oi] < 0: + continue + j = (i + ds) % width + if (mask >> j) & 1: + dsu.join(oi, new_base + j, (i + ds) // width) + if dsu.bad: + return None + retained = list(range(width, old_size)) + [new_base + i for i in range(width)] + labels = [-1] * old_size + gains = [0] * old_size + canon = {} + nxt = 0 + for out_i, idx in enumerate(retained): + occ = state.labels[idx] >= 0 if idx < old_size else bool(mask >> (idx - new_base) & 1) + if not occ: + continue + root, gain = dsu.find(idx) + if root not in canon: + canon[root] = (nxt, gain) + nxt += 1 + lab, base = canon[root] + labels[out_i] = lab + gains[out_i] = gain - base + return State(tuple(labels), tuple(gains)) + + +def build(width, direction, matching, state_cap=400000): + comp, edges, memory = transformed_edges(direction, matching) + start = empty_state(width, memory) + states = [start] + index = {start: 0} + queue = deque([start]) + transitions = [] + while queue: + state = queue.popleft() + row = [] + for mask in range(1 << width): + nxt = step(state, mask, width, edges, memory) + if nxt is None: + row.append(-1) + continue + if nxt not in index: + if len(states) >= state_cap: + raise RuntimeError("state cap reached") + index[nxt] = len(states) + states.append(nxt) + queue.append(nxt) + row.append(index[nxt]) + transitions.append(row) + return states, transitions, comp, edges, memory + + +class ObliqueSafeTransfer: + def __init__(self, width, direction, matching, state_cap=400000): + self.width = width + self.direction = direction + self.states, transitions, self.complement, self.edges, self.memory = \ + build(width, direction, matching, state_cap) + counter = Counter() + for src, row in enumerate(transitions): + for mask, dst in enumerate(row): + if dst >= 0: + counter[(src, dst, mask.bit_count())] += 1 + keys = list(counter) + self.src = np.array([k[0] for k in keys], dtype=np.int32) + self.dst = np.array([k[1] for k in keys], dtype=np.int32) + self.occ = np.array([k[2] for k in keys], dtype=np.int16) + self.counts = np.array([counter[k] for k in keys], dtype=float) + self.n = len(self.states) + + def dense(self, p): + q = 1.0 - p + R = np.zeros((self.n, self.n)) + data = self.counts * p ** self.occ * q ** (self.width - self.occ) + np.add.at(R, (self.src, self.dst), data) + return R + + def sparse(self, p): + q = 1.0 - p + data = self.counts * p ** self.occ * q ** (self.width - self.occ) + return coo_matrix((data, (self.src, self.dst)), + shape=(self.n, self.n)).tocsr() + + def lambda0(self, p, mode="auto", arpack_tol=1e-10): + if mode == "auto": + mode = "dense" if self.n < 200 else "arpack" + if mode == "dense": + return float(np.max(np.linalg.eigvals(self.dense(p)).real)), "dense" + sp = self.sparse(p) + val = sp_eigs(sp, k=1, which="LM", tol=arpack_tol, maxiter=500000, + return_eigenvectors=False)[0] + return float(val.real), "arpack(%g)" % arpack_tol + + +def run(engine_note, spec, pc, config, sink=None): + """spec: list of (tag, direction, n). config: 'loose' | 'tight'.""" + out = {"path": "B_oblique", "config": config, "p_c": pc, "records": {}} + for tag, direction, n in spec: + t0 = time.time() + g4 = ObliqueSafeTransfer(n, direction, False) + g8 = ObliqueSafeTransfer(n, direction, True) + tol = 1e-10 if config == "loose" else 1e-15 + # tight config: dense (full-accuracy LAPACK) whenever the matrix is + # small enough for a dense eig; ARPACK with tol=1e-14 above that + # (empirically ~1e-16 in p, same as dense -- see the ARPACK probe). + MODE_CAP = 2500 + if config == "loose": + mode = "auto" + else: + mode = "dense" if g4.n <= MODE_CAP and g8.n <= MODE_CAP else "arpack" + + def equation(p): + l4, _ = g4.lambda0(p, mode=mode, arpack_tol=tol) + l8, _ = g8.lambda0(1.0 - p, mode=mode, arpack_tol=tol) + return math.log(l4) - math.log(l8) + + if config == "loose": + if HAVE_SCIPY: + root = brentq(equation, 0.5, 0.8, xtol=3e-11) + width = float("nan") + else: + root, width, _, _ = C.solve_root(equation, 0.5, 0.8, xtol=1e-16) + else: + root, width, _, _ = C.solve_root(equation, 0.5, 0.8, xtol=1e-16) + l4, m4 = g4.lambda0(root, mode=mode, arpack_tol=tol) + l8, m8 = g8.lambda0(1.0 - root, mode=mode, arpack_tol=tol) + ell = n * math.hypot(*direction) + rec = {"tag": tag, "direction": list(direction), "n": n, + "ell": ell, "cos4": C.cos4(direction), + "n_safe_G4": g4.n, "n_safe_G8": g8.n, + "row_memory_G4": g4.memory, "row_memory_G8": g8.memory, + "edges_G4": [list(e) for e in g4.edges], + "edges_G8": [list(e) for e in g8.edges], + "lambda0_mode_G4": m4, "lambda0_mode_G8": m8, + "p_root": float(root), "bracket_width": width, + "Delta_at_root": math.log(l4) - math.log(l8), + "rho_G4": l4, "rho_G8": l8, + "Omega_ell_common": -(root - pc) * ell ** 4, + "root_minus_pc": root - pc, + "seconds": round(time.time() - t0, 2)} + out["records"][tag] = rec + print("B/%-5s %-11s n=%d p_root=%.17g Omega=%.12e modes=%s/%s %.1fs" + % (config, tag, n, root, rec["Omega_ell_common"], m4, m8, + rec["seconds"]), flush=True) + if sink: + with open(sink, "w") as fh: + json.dump(out, fh, indent=2) + out["note"] = engine_note + return out + + +if __name__ == "__main__": + pc = float(sys.argv[1]) if len(sys.argv) > 1 else C.PC + cfg = sys.argv[2] if len(sys.argv) > 2 else "tight" + out_path = sys.argv[3] if len(sys.argv) > 3 else \ + "/workspace/dpfloor/out/pathB_oblique_%s.json" % cfg + only = sys.argv[4].split(",") if len(sys.argv) > 4 else None + spec = [(t, u, n) for (t, u, n, _e) in C.GEOMS] + if only: + spec = [s for s in spec if s[0] in only] + print("restricted to", [s[0] for s in spec], flush=True) + res = run("implementation of rev769 scripts/oblique_charge_transfer.py", + spec, pc, cfg, sink=out_path) + with open(out_path, "w") as fh: + json.dump(res, fh, indent=2) + print("->", out_path) diff --git a/scripts/dpfloor_fl_probe_hp.py b/scripts/dpfloor_fl_probe_hp.py new file mode 100644 index 000000000..fd6d59db4 --- /dev/null +++ b/scripts/dpfloor_fl_probe_hp.py @@ -0,0 +1,57 @@ +#!/usr/bin/env python3 +"""Certificate for PATH H: is the longdouble power iteration actually converged? + +At the PATH-H root of axis_n8 (and axis_n4) compare, at the SAME p: + (i) float64 LAPACK dense Perron root of R, + (ii) longdouble power iteration, cold start, forced 400 iterations (no early stop), + (iii) longdouble power iteration with the production stopping rule. +Also report Delta(root) reached by PATH H and the longdouble residual. +""" +from __future__ import annotations +import sys, os, json, math +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +sys.path.insert(0, "/workspace/sectorA/in/engine") +import numpy as np +import fl_common as C +import sector802_lib as L +import fl_hp as H + +LD = np.longdouble + + +def main(): + T = L.load_engine("/workspace/sectorA/in/engine") + hp = json.load(open("/workspace/dpfloor/out/pathH_longdouble.json")) + out = {"records": {}} + for tag in ("axis_n4", "axis_n6", "axis_n8"): + if tag not in hp["records"]: + continue + w = hp["records"][tag]["width"] + root = LD(hp["records"][tag]["p_root"]) + rec = {"width": w, "p_root_H": str(root)} + for name, matching in (("G4", False), ("G8", True)): + x = root if name == "G4" else (LD(1) - root) + agg = H.safe_counts(T, w, matching) + R, _ = H.assemble(agg, w, x) + rho_f = float(np.max(np.linalg.eigvals(np.asarray(R, dtype=float)).real)) + lam_c, _, it_c = H.perron_power(R, None, tol=LD(0), maxiter=400) + lam_p, _, it_p = H.perron_power(R, None, tol=LD(1e-30), maxiter=20000) + res_c = float(np.linalg.norm(R @ (R @ np.ones(R.shape[0], dtype=LD)) + / LD(R.shape[0]) - lam_c + * (R @ np.ones(R.shape[0], dtype=LD)) / LD(R.shape[0]))) + rec[name] = {"rho_lapack_float64": rho_f, + "rho_power_ld_forced400": str(lam_c), "iters_forced": it_c, + "rho_power_ld_production": str(lam_p), "iters_prod": it_p, + "rho_ld_minus_float64": str(LD(lam_c) - LD(rho_f))} + out["records"][tag] = rec + print(tag, json.dumps(rec, indent=1), flush=True) + # PATH H's own Delta at its root + print("PATH H Delta_at_root:", {t: hp["records"][t]["Delta_at_root"] + for t in out["records"]}) + with open("/workspace/dpfloor/out/step_hp_certificate.json", "w") as fh: + json.dump(out, fh, indent=2) + + +if __name__ == "__main__": + main() diff --git a/scripts/dpfloor_fl_shipped_vs_repro.py b/scripts/dpfloor_fl_shipped_vs_repro.py new file mode 100644 index 000000000..e29b6ab3e --- /dev/null +++ b/scripts/dpfloor_fl_shipped_vs_repro.py @@ -0,0 +1,87 @@ +#!/usr/bin/env python3 +"""Shipped settings vs re-run: how reproducible is the shipped oblique file? + +The shipped file's roots were produced with ARPACK tol=1e-10 and scipy brentq +xtol=3e-11. Re-running the SAME algorithm at (i) the shipped settings and +(ii) tight settings quantifies the configuration scatter of every geometry in +the file. (The root value itself does not depend on p_c, so this comparison is +free of any p_c convention question.) +""" +from __future__ import annotations +import sys, os, json + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +import numpy as np +import fl_common as C + +OUT = "/workspace/dpfloor/out" +LD = np.longdouble + + +def load(name): + p = os.path.join(OUT, name) + return json.load(open(p)) if os.path.exists(p) else None + + +def key_of(direction, n): + for tag, u, nn, _e in C.GEOMS: + if list(u) == list(direction) and nn == n: + return tag + return None + + +def main(): + shipped = json.load(open(sys.argv[1] if len(sys.argv) > 1 + else "/workspace/dpfloor/in/oblique-spin4-controls.json")) + loose = load("pathB_oblique_loose.json") + t1 = load("pathB_oblique_tight.json") + t2 = load("pathB_oblique_tight2.json") + tight = {} + for d in (t1, t2): + if d: + tight.update(d["records"]) + + rows = {} + for rec in shipped["records"]: + tag = key_of(rec["direction"], rec["n"]) + if tag is None: + continue + ps = LD(str(rec["charge_root_p"])) + pl = LD(str(loose["records"][tag]["p_root"])) if loose and tag in loose["records"] else None + pt = LD(str(tight[tag]["p_root"])) if tag in tight else None + r = {"direction": rec["direction"], "n": rec["n"], "ell": rec["physical_circumference"], + "p_shipped": str(ps), + "p_loose_rerun": str(pl) if pl is not None else None, + "p_tight": str(pt) if pt is not None else None} + if pl is not None: + r["shipped_minus_loose"] = str(ps - pl) + if pt is not None: + r["shipped_minus_tight"] = str(ps - pt) + r["loose_minus_tight"] = str(pl - pt) if pl is not None else None + r["Omega_shipped_minus_tight"] = str((ps - pt) * LD(r["ell"]) ** 4) + rows[tag] = r + def fmt(x): + return "-" if x is None else ("%.2e" % float(x)) + print("%-11s %-9s ell=%7.3f shipped-tight=%s loose-tight=%s" + % (tag, str(rec["direction"]), r["ell"], + fmt(r.get("shipped_minus_tight")), fmt(r.get("loose_minus_tight"))), + flush=True) + diffs = [abs(LD(r["shipped_minus_tight"])) for r in rows.values() if "shipped_minus_tight" in r] + dl = [abs(LD(r["loose_minus_tight"])) for r in rows.values() if r.get("loose_minus_tight") is not None] + summ = { + "n_geometries": len(rows), + "max_abs_shipped_minus_tight_p": float(max(diffs)) if diffs else None, + "max_abs_loose_minus_tight_p": float(max(dl)) if dl else None, + "statement": ("re-running the shipped algorithm with the shipped settings " + "reproduces the shipped roots only to ~1e-12 in p; with tight " + "settings the same geometries are reproducible to ~1e-16."), + } + print(json.dumps(summ, indent=2)) + with open(os.path.join(OUT, "step_shipped_vs_repro.json"), "w") as fh: + json.dump({"rows": rows, "summary": summ}, fh, indent=2) + print("->", os.path.join(OUT, "step_shipped_vs_repro.json")) + + +if __name__ == "__main__": + main() diff --git a/scripts/dpfloor_fl_solvers.py b/scripts/dpfloor_fl_solvers.py new file mode 100644 index 000000000..816d59a5b --- /dev/null +++ b/scripts/dpfloor_fl_solvers.py @@ -0,0 +1,153 @@ +#!/usr/bin/env python3 +"""PATH S ("solvers") -- SAME matrix (from sector802_lib / PATH A), different +linear algebra AND different root-finder. + +This path is explicitly NOT independent of PATH A for the automaton: it asks a +different question -- how much of the p_root value is set by the eigensolver and +the root finder rather than by the matrix. Solver variants: + + scipy_eig scipy.linalg.eig on dense R, argmax Re + numpy_eigvals numpy.linalg.eigvals on dense R, max Re + scipy_eigvals scipy.linalg.eigvals on dense R, max Re + arpack_1e10 scipy.sparse.linalg.eigs (ARPACK), tol=1e-10 + arpack_1e14 ARPACK, tol=1e-14 + power power iteration on dense R to a 1e-16 eigen-residual + lib2_sectorgap lib_sectorgap.perron (numpy.eig right + numpy.eig(R.T) left) + +Root finders: plain bisection to 1e-16, and scipy brentq at its default +xtol=2e-12 / rtol=8.9e-16. +""" +from __future__ import annotations +import sys, os, json, math, time + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +sys.path.insert(0, "/workspace/sectorA/in/engine") + +import numpy as np +import fl_common as C +import sector802_lib as L + +try: + import scipy.linalg as sla + from scipy.optimize import brentq + from scipy.sparse import csr_matrix + from scipy.sparse.linalg import eigs as sp_eigs + HAVE_SCIPY = True +except Exception: + HAVE_SCIPY = False + + +def power_perron(R, tol=1e-16, maxiter=200000): + n = R.shape[0] + v = np.ones(n) + v /= v.sum() + rho = 0.0 + lam_prev = 0.0 + for it in range(maxiter): + w = R @ v + s = w.sum() + if s <= 0: + break + lam = s / v.sum() + v = w / s + if it % 5 == 4: + if abs(lam - lam_prev) <= tol * abs(lam): + rho = lam + break + lam_prev = lam + else: + rho = lam_prev + # Rayleigh quotient with the converged (right) vector + rho = float(v @ (R @ v) / (v @ v)) + return rho, it + 1 + + +def build_R(agg, w, p): + R, _, _ = L.build_matrices(agg, w, p, want_dp=False, want_dg=False) + return R + + +SOLVERS = ["scipy_eig", "numpy_eigvals", "scipy_eigvals", "arpack_1e10", + "arpack_1e14", "power"] + + +def rho_of(R, name, sp=None): + if name == "scipy_eig": + ev = sla.eig(R, left=False, right=False) + return float(np.max(ev.real)) + if name == "numpy_eigvals": + return float(np.max(np.linalg.eigvals(R).real)) + if name == "scipy_eigvals": + return float(np.max(sla.eigvals(R).real)) + if name == "arpack_1e10": + val = sp_eigs(sp, k=1, which="LM", tol=1e-10, maxiter=500000, + return_eigenvectors=False)[0] + return float(val.real) + if name == "arpack_1e14": + val = sp_eigs(sp, k=1, which="LM", tol=1e-14, maxiter=500000, + return_eigenvectors=False)[0] + return float(val.real) + if name == "power": + return power_perron(R)[0] + raise ValueError(name) + + +def run(widths, pc, solvers=SOLVERS, sink=None): + T = L.load_engine("/workspace/sectorA/in/engine") + out = {"path": "S_solvers", "p_c": pc, "records": {}} + for w in widths: + t0 = time.time() + order4, index4, trans4, wind4 = L.enumerate_states(T, w, False) + aggN = L.aggregate(trans4, wind4, w, mark="none", mask_is_black=True) + order8, index8, trans8, wind8 = L.enumerate_states(T, w, True) + aggQ = L.aggregate(trans8, wind8, w, mark="none", mask_is_black=False) + tag = "axis_n%d" % w + rec = {"tag": tag, "width": w, "n_safe_G4": aggN["n"], "n_safe_G8": aggQ["n"], + "variants": {}} + for name in solvers: + def f(p, name=name): + R4 = build_R(aggN, w, p) + sp4 = csr_matrix(R4) if name.startswith("arpack") else None + l4 = rho_of(R4, name, sp4) + R8 = build_R(aggQ, w, 1.0 - p) + sp8 = csr_matrix(R8) if name.startswith("arpack") else None + l8 = rho_of(R8, name, sp8) + return -math.log(l4) + math.log(l8) + + root, width, nfev, fres = C.solve_root(f, 0.5, 0.8, xtol=1e-16) + vrec = {"p_root_bisect": root, "bracket_width": width, + "Delta_at_root": fres, + "Omega_bisect": C.omega(root, tag, pc)} + if HAVE_SCIPY: + rb = brentq(f, 0.5, 0.8) + vrec["p_root_brentq_default"] = rb + vrec["brentq_minus_bisect"] = rb - root + rec["variants"][name] = vrec + print("S %-9s %-13s p_root(bisect)=%.17g brentq-bisect=%+.2e" + % (tag, name, root, vrec.get("brentq_minus_bisect", float("nan"))), + flush=True) + roots = [v["p_root_bisect"] for v in rec["variants"].values()] + rec["spread_bisect"] = max(roots) - min(roots) + rec["seconds"] = round(time.time() - t0, 2) + out["records"][tag] = rec + print(" -> %s solver spread (p) = %.3e %.1fs" % (tag, rec["spread_bisect"], rec["seconds"]), + flush=True) + if sink: + with open(sink, "w") as fh: + json.dump(out, fh, indent=2) + out["note"] = ("same automaton and same matrix as PATH A; isolates the " + "eigensolver + root-finder contribution. NOT an independent " + "automaton path.") + return out + + +if __name__ == "__main__": + pc = float(sys.argv[1]) if len(sys.argv) > 1 else C.PC + ws = [int(x) for x in (sys.argv[2].split(",") if len(sys.argv) > 2 else ["4", "6", "8"])] + outp = sys.argv[3] if len(sys.argv) > 3 else "/workspace/dpfloor/out/pathS_solvers.json" + solvers = sys.argv[4].split(",") if len(sys.argv) > 4 else SOLVERS + res = run(ws, pc, solvers, sink=outp) + with open(outp, "w") as fh: + json.dump(res, fh, indent=2) + print("->", outp) diff --git a/scripts/dual_odd_scaling_collapse.py b/scripts/dual_odd_scaling_collapse.py new file mode 100644 index 000000000..d5213b0f5 --- /dev/null +++ b/scripts/dual_odd_scaling_collapse.py @@ -0,0 +1,129 @@ +#!/usr/bin/env python3 +"""Evaluate the fixed-width dual-odd charge scaling function. + +Uses the transparent safe transfer from fixed_width_charge_transfer.py and +reports + + F_w(X) = w * Theta_w(h_w + X*w^(-3/4)), + +where h is Bernoulli logit and h_w is the charge-coexistence root. It also +fits the antisymmetric part to a local odd polynomial and estimates the +quadratic analytic thermal-metric coefficient from the symmetric residual. + +This is a small-width deterministic transfer control, not a continuum proof. +""" +from __future__ import annotations + +import argparse +import json +import math +from pathlib import Path + +import numpy as np +from scipy.optimize import brentq + +from fixed_width_charge_transfer import SafeTransfer + + +def logistic(h: float) -> float: + return 1.0 / (1.0 + math.exp(-h)) + + +def logit(p: float) -> float: + return math.log(p / (1.0 - p)) + + +def run_width(width: int, x_grid: list[float]) -> dict[str, object]: + g4 = SafeTransfer(width, matching=False) + g8 = SafeTransfer(width, matching=True) + + def theta_p(p: float) -> float: + return g4.perron(p)["I0"] - g8.perron(1.0 - p)["I0"] + + root = brentq(theta_p, 0.5000000001, 0.999999999, xtol=2e-14) + h_root = logit(root) + + def scaled(x: float) -> float: + h = h_root + x * width ** (-0.75) + return width * theta_p(logistic(h)) + + curve = {format(x, ".12g"): scaled(x) for x in x_grid} + + positive = sorted({abs(x) for x in x_grid if x != 0.0}) + fit_x = [] + fit_y = [] + for x in positive: + if x <= 1.5: + odd = 0.5 * (scaled(x) - scaled(-x)) + fit_x.extend([x, -x]) + fit_y.extend([odd, -odd]) + fit_x = np.asarray(fit_x) + fit_y = np.asarray(fit_y) + design = np.column_stack([fit_x, fit_x**3, fit_x**5]) + a1, a3, a5 = np.linalg.lstsq(design, fit_y, rcond=None)[0] + + def odd_fit_prime(x: float) -> float: + return a1 + 3.0 * a3 * x * x + 5.0 * a5 * x**4 + + metric_points = [x for x in (0.5, 0.75, 1.0, 1.5) if x in positive] + numerator = 0.0 + denominator = 0.0 + point_estimates = {} + for x in metric_points: + symmetric_sum = scaled(x) + scaled(-x) + target = 2.0 * x * x * odd_fit_prime(x) * width ** (-0.75) + estimate = symmetric_sum / target + point_estimates[format(x, ".12g")] = estimate + numerator += symmetric_sum * target + denominator += target * target + c2 = numerator / denominator + + return { + "width": width, + "charge_root_p": root, + "charge_root_h": h_root, + "scaled_curve": curve, + "odd_polynomial": { + "a1_X": float(a1), + "a3_X3": float(a3), + "a5_X5": float(a5), + "fifth_derivative_120a5": float(120.0 * a5), + }, + "quadratic_metric_estimate": float(c2), + "pointwise_metric_estimates": point_estimates, + } + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--min-width", type=int, default=4) + parser.add_argument("--max-width", type=int, default=9) + parser.add_argument( + "--x-grid", + default="-1.5,-1,-0.75,-0.5,-0.25,0,0.25,0.5,0.75,1,1.5", + ) + parser.add_argument("--output", type=Path) + args = parser.parse_args() + + x_grid = [float(item) for item in args.x_grid.split(",")] + result = { + "schema": "dual-odd-charge-scaling-collapse-v1", + "definition": "F_w(X)=w*Theta_w(h_w+X*w^(-3/4))", + "claim_boundary": [ + "deterministic small-width transfer control", + "odd polynomial is only a local descriptive fit", + "quadratic metric interpretation is a scaling hypothesis", + ], + "records": [ + run_width(width, x_grid) + for width in range(args.min_width, args.max_width + 1) + ], + } + text = json.dumps(result, indent=2, sort_keys=True) + if args.output: + args.output.write_text(text + "\n", encoding="utf-8") + print(text) + + +if __name__ == "__main__": + main() diff --git a/scripts/essential_lineage_merger_control.py b/scripts/essential_lineage_merger_control.py new file mode 100644 index 000000000..d43232547 --- /dev/null +++ b/scripts/essential_lineage_merger_control.py @@ -0,0 +1,153 @@ +#!/usr/bin/env python3 +"""Small exact SITE interfaces for the no-merger proof, not a scaling experiment. + +The cylinder has periodic horizontal and FREE vertical boundaries. A pair of +nested masks is the exact three-colour common-label ensemble. No plane/cylinder +asymptotic, Markov approximation, or Poisson fit is inferred from this census. +""" +from __future__ import annotations +import argparse +from collections import Counter +from fractions import Fraction +from math import comb +from pathlib import Path +import json + + +def edges(w: int, h: int): + if w < 3 or h < 1: + raise ValueError('Use an honest width >= 3 and positive height') + out = [[] for _ in range(w*h)] + for y in range(h): + for x in range(w): + v=y*w+x + for dx,dy in ((1,0),(-1,0),(0,1),(0,-1)): + if 0 <= y+dy < h: + out[v].append(((y+dy)*w+(x+dx)%w, dx)) + return out + + +def components_bfs(mask: int, adj): + """Return (vertex mask, nonzero horizontal homology) for each component.""" + unseen=mask; result=[] + while unseen: + start=(unseen & -unseen).bit_length()-1 + stack=[start]; lift={start:0}; bits=0; winding=False + unseen &= ~(1<>v)&1: continue + proposed=lift[u]+dx + if v in lift: + winding |= lift[v] != proposed + else: + lift[v]=proposed; unseen &= ~(1<>u)&1: continue + for v,dx in adj[u]: + if u>=v or not (mask>>v)&1: continue + ru,du=find(u); rv,dv=find(v) + if ru==rv: wind[ru] |= dv-du != dx + else: + parent[rv]=ru; potential[rv]=dx+du-dv + wind[ru] |= wind[rv] + groups={} + for u in range(n): + if (mask>>u)&1: + r,_=find(u); groups[r]=groups.get(r,0)|(1<12: raise ValueError('This finite interface census is limited to 12 sites') + total=1<=w for b,wind in cc[m] if wind) + essential=[tuple(b for b,wind in c if wind) for c in cc] + # Inclusion-minimal increasing winding witnesses; they need not be entire clusters. + minimal=[] + for m in range(1,total): + if essential[m] and all(not essential[m^(1<>v)&1): + minimal.append(m) + double=[] + for m in range(total): + witnesses=[s for s in minimal if s&m==s] + double.append(any(not(a&b) for i,a in enumerate(witnesses) for b in witnesses[i+1:])) + hist=Counter(); configs=merging=0; witness=None + # Sum over A subset B: each site is early / added / still closed. + for late in range(total): + early=late + while True: + nums=[sum(a&c==a for a in essential[early]) for c in essential[late]] + mcount=sum(comb(k,2) for k in nums) + loss=len(essential[early])-sum(k>0 for k in nums) + assert sum(nums)==len(essential[early]) + assert 0<=loss<=mcount + assert (not mcount) or double[late] + assert mcount<=comb(h,2) + key=(early.bit_count(),(late^early).bit_count(),n-late.bit_count()) + hist[key,'normalizer']+=1 + hist[key,'pairs']+=mcount + hist[key,'loss']+=loss + hist[key,'event']+=bool(mcount) + if mcount: + merging+=1 + if witness is None or late.bit_count() float: + if h >= 0: + z = math.exp(-h) + return 1.0 / (1.0 + z) + z = math.exp(h) + return z / (1.0 + z) + + +def leading_moduli(transfer: SafeTransfer, p: float, count: int = 2) -> list[float]: + matrix = transfer.matrix(p) + n = matrix.shape[0] + if n <= count + 1: + vals = np.linalg.eigvals(matrix.toarray()) + else: + vals = eigs( + matrix, + k=count, + which="LM", + return_eigenvectors=False, + tol=1e-12, + maxiter=200000, + ) + moduli = sorted((abs(complex(v)) for v in vals), reverse=True) + return [float(x) for x in moduli[:count]] + + +def I_of_h(transfer: SafeTransfer, h: float) -> float: + return transfer.perron(logistic(h))["I0"] + + +def charge_of_h(g4: SafeTransfer, g8: SafeTransfer, h: float) -> float: + # Complement white logit is -h exactly. + return I_of_h(g4, h) - I_of_h(g8, -h) + + +def central_derivatives(g4: SafeTransfer, g8: SafeTransfer, h: float, eps: float): + f0 = charge_of_h(g4, g8, h) + fm1 = charge_of_h(g4, g8, h - eps) + fp1 = charge_of_h(g4, g8, h + eps) + fm2 = charge_of_h(g4, g8, h - 2 * eps) + fp2 = charge_of_h(g4, g8, h + 2 * eps) + d1 = (fp1 - fm1) / (2 * eps) + d2 = (fp1 - 2 * f0 + fm1) / eps**2 + d3 = (fp2 - 2 * fp1 + 2 * fm1 - fm2) / (2 * eps**3) + return d1, d2, d3 + + +def individual_second(transfer: SafeTransfer, h: float, eps: float) -> float: + return ( + I_of_h(transfer, h + eps) + - 2 * I_of_h(transfer, h) + + I_of_h(transfer, h - eps) + ) / eps**2 + + +def record(width: int, *, eps: float) -> dict[str, object]: + g4 = SafeTransfer(width, matching=False) + g8 = SafeTransfer(width, matching=True) + + def theta_p(p: float) -> float: + return g4.perron(p)["I0"] - g8.perron(1.0 - p)["I0"] + + root = brentq(theta_p, 0.5000000001, 0.999999999, xtol=2e-14) + hroot = math.log(root / (1.0 - root)) + d1, d2, d3 = central_derivatives(g4, g8, hroot, eps) + + # Repeat at half the step as a finite-difference stability control. + d1b, d2b, d3b = central_derivatives(g4, g8, hroot, eps / 2) + + c4 = individual_second(g4, hroot, eps) + c8 = individual_second(g8, -hroot, eps) + + spec4 = leading_moduli(g4, root, 2) + spec8 = leading_moduli(g8, 1.0 - root, 2) + gap4 = math.log(spec4[0] / spec4[1]) + gap8 = math.log(spec8[0] / spec8[1]) + + return { + "width": width, + "charge_root": root, + "logit_root": hroot, + "finite_difference_step": eps, + "charge_logit_derivatives": { + "d1": d1, + "d2": d2, + "d3": d3, + "half_step": {"d1": d1b, "d2": d2b, "d3": d3b}, + "scaled": { + "d1_times_w_quarter": d1 * width ** 0.25, + "d2_times_sqrt_w": d2 * math.sqrt(width), + "d3_over_w_5_over_4": d3 / width ** 1.25, + }, + }, + "individual_logit_second_derivatives": { + "G4": c4, + "G8_complement": c8, + "G4_over_sqrt_w": c4 / math.sqrt(width), + "G8_over_sqrt_w": c8 / math.sqrt(width), + }, + "safe_spectral_relaxation": { + "G4_lambda1": spec4[0], + "G4_lambda2_modulus": spec4[1], + "G4_gap": gap4, + "G4_gap_times_w": gap4 * width, + "G8_lambda1": spec8[0], + "G8_lambda2_modulus": spec8[1], + "G8_gap": gap8, + "G8_gap_times_w": gap8 * width, + }, + } + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--min-width", type=int, default=4) + parser.add_argument("--max-width", type=int, default=8) + parser.add_argument("--eps", type=float, default=7e-4) + parser.add_argument("--output", type=Path) + args = parser.parse_args() + + result = { + "schema": "fixed-width-charge-spectrum-derivatives-v1", + "claim_boundary": [ + "Finite differences are controls; exact first-derivative identities are in the accompanying notes.", + "Subleading safe-kernel eigenvalues are not yet formally identified with every periodic TL trace eigenvalue.", + "No continuum exponent is inferred solely from this file." + ], + "records": [record(w, eps=args.eps) for w in range(args.min_width, args.max_width + 1)], + } + text = json.dumps(result, indent=2, sort_keys=True) + if args.output: + args.output.write_text(text + "\n", encoding="utf-8") + print(text) + + +if __name__ == "__main__": + main() diff --git a/scripts/fixed_width_charge_transfer.py b/scripts/fixed_width_charge_transfer.py new file mode 100644 index 000000000..8998d7e67 --- /dev/null +++ b/scripts/fixed_width_charge_transfer.py @@ -0,0 +1,343 @@ +#!/usr/bin/env python3 +"""Safe-frontier transfer for fixed-width charge coexistence. + +For a cylinder of circumference w, construct the finite frontier automaton that +rejects a row transition as soon as horizontal homology is created. The Perron +root lambda^0_G,w(p) of the safe substochastic transfer gives + + I^0_G,w(p) = -log lambda^0_G,w(p). + +The charge-coexistence root solves + + lambda^0_4,w(p) = lambda^0_8,w(1-p). + +The implementation keeps lifted horizontal gains, so periodic seam winding is +detected rather than inferred from coarse connectivity flags. Width two +therefore retains the two physically distinct lifted horizontal bonds. + +This is a small-width research control, not a production pc estimator. +SciPy and NumPy are required. +""" +from __future__ import annotations + +import argparse +import json +import math +from collections import Counter, deque +from dataclasses import dataclass +from pathlib import Path + +import numpy as np +from scipy.optimize import brentq +from scipy.sparse import coo_matrix, csr_matrix +from scipy.sparse.linalg import eigs + + +@dataclass(frozen=True) +class State: + labels: tuple[int, ...] + gains: tuple[int, ...] + + +class DSU: + """Potential union-find with integer lifted-x differences.""" + + def __init__(self, n: int): + self.parent = list(range(n)) + self.delta = [0] * n + self.bad = False + + def find(self, a: int) -> tuple[int, int]: + if self.parent[a] != a: + root, gain = self.find(self.parent[a]) + self.delta[a] += gain + self.parent[a] = root + return self.parent[a], self.delta[a] + + def join(self, a: int, b: int, gain_b_minus_a: int) -> None: + ra, da = self.find(a) + rb, db = self.find(b) + if ra == rb: + if db - da != gain_b_minus_a: + self.bad = True + return + self.parent[rb] = ra + self.delta[rb] = gain_b_minus_a + da - db + + +def empty_state(width: int) -> State: + return State((-1,) * width, (0,) * width) + + +def step(state: State, mask: int, *, matching: bool) -> State | None: + """Advance one row; return None if horizontal winding is created.""" + width = len(state.labels) + dsu = DSU(2 * width) + representatives: dict[int, int] = {} + + for i, label in enumerate(state.labels): + if label < 0: + continue + if label in representatives: + dsu.join(representatives[label], i, state.gains[i]) + else: + representatives[label] = i + + occupied_new = [i for i in range(width) if (mask >> i) & 1] + + for i in occupied_new: + j = (i + 1) % width + if (mask >> j) & 1: + dsu.join(width + i, width + j, (i + 1) // width) + + displacements = (-1, 0, 1) if matching else (0,) + for i in occupied_new: + for dx in displacements: + j = (i + dx) % width + if state.labels[j] >= 0: + dsu.join(width + i, j, (i + dx) // width) + + if dsu.bad: + return None + + labels = [-1] * width + gains = [0] * width + canonical: dict[int, tuple[int, int]] = {} + next_label = 0 + for i in occupied_new: + root, gain = dsu.find(width + i) + if root not in canonical: + canonical[root] = (next_label, gain) + next_label += 1 + label, base_gain = canonical[root] + labels[i] = label + gains[i] = gain - base_gain + + return State(tuple(labels), tuple(gains)) + + +def build_automaton(width: int, *, matching: bool): + if width < 2: + raise ValueError("width must be >=2") + start = empty_state(width) + states = [start] + index = {start: 0} + queue = deque([start]) + transitions: list[list[int]] = [] + + while queue: + state = queue.popleft() + row: list[int] = [] + for mask in range(1 << width): + nxt = step(state, mask, matching=matching) + if nxt is None: + row.append(-1) + continue + if nxt not in index: + index[nxt] = len(states) + states.append(nxt) + queue.append(nxt) + row.append(index[nxt]) + transitions.append(row) + + if len(transitions) != len(states): + raise AssertionError("incomplete BFS") + return states, transitions + + +def central_trinomial(width: int) -> int: + return sum( + math.comb(width, 2 * r) * math.comb(2 * r, r) + for r in range(width // 2 + 1) + ) + + +def aggregate_transitions(transitions: list[list[int]], width: int): + counter: Counter[tuple[int, int, int]] = Counter() + for source, row in enumerate(transitions): + for mask, destination in enumerate(row): + if destination >= 0: + counter[(source, destination, mask.bit_count())] += 1 + keys = list(counter) + counts = np.array([counter[key] for key in keys], dtype=float) + src = np.array([key[0] for key in keys], dtype=np.int32) + dst = np.array([key[1] for key in keys], dtype=np.int32) + occ = np.array([key[2] for key in keys], dtype=np.int16) + return src, dst, occ, counts + + +class SafeTransfer: + def __init__(self, width: int, *, matching: bool): + self.width = width + self.matching = matching + self.states, transitions = build_automaton(width, matching=matching) + self.src, self.dst, self.occ, self.counts = aggregate_transitions( + transitions, width + ) + + def matrix(self, p: float) -> csr_matrix: + q = 1.0 - p + data = self.counts * (p ** self.occ) * (q ** (self.width - self.occ)) + return coo_matrix( + (data, (self.src, self.dst)), + shape=(len(self.states), len(self.states)), + ).tocsr() + + def derivative_matrix(self, p: float) -> csr_matrix: + q = 1.0 - p + weight = self.counts * (p ** self.occ) * (q ** (self.width - self.occ)) + factor = self.occ / p - (self.width - self.occ) / q + return coo_matrix( + (weight * factor, (self.src, self.dst)), + shape=(len(self.states), len(self.states)), + ).tocsr() + + @staticmethod + def _dominant_pair(matrix: csr_matrix): + n = matrix.shape[0] + if n <= 4: + dense = matrix.toarray() + vals, vecs = np.linalg.eig(dense) + k = int(np.argmax(vals.real)) + lam = float(vals[k].real) + right = np.asarray(vecs[:, k].real) + vals_l, vecs_l = np.linalg.eig(dense.T) + j = int(np.argmin(np.abs(vals_l - vals[k]))) + left = np.asarray(vecs_l[:, j].real) + else: + val, vec = eigs(matrix, k=1, which="LM", tol=1e-12, maxiter=200000) + lam = float(val[0].real) + right = np.asarray(vec[:, 0].real) + _, vec_l = eigs( + matrix.T, k=1, which="LM", tol=1e-12, maxiter=200000 + ) + left = np.asarray(vec_l[:, 0].real) + + if right.sum() < 0: + right = -right + if left.sum() < 0: + left = -left + inner = float(left @ right) + if inner <= 0: + raise ArithmeticError("Perron left/right normalization failed") + left /= inner + return lam, left, right + + def perron(self, p: float) -> dict[str, float]: + matrix = self.matrix(p) + derivative = self.derivative_matrix(p) + lam, left, right = self._dominant_pair(matrix) + dlam = float(left @ (derivative @ right)) + I0 = -math.log(lam) + I0_prime = -dlam / lam + mean_occupied = self.width * p - p * (1 - p) * I0_prime + residual = float( + np.linalg.norm(matrix @ right - lam * right, ord=np.inf) + / max(1.0, np.linalg.norm(right, ord=np.inf)) + ) + return { + "lambda": lam, + "I0": I0, + "I0_prime": I0_prime, + "mean_occupied_safe_row": mean_occupied, + "right_residual_inf": residual, + } + + +def coexistence_record( + width: int, + *, + reference_pc: float, + root_xtol: float = 2e-14, +) -> dict[str, object]: + g4 = SafeTransfer(width, matching=False) + g8 = SafeTransfer(width, matching=True) + + expected = central_trinomial(width) + if len(g4.states) != expected or len(g8.states) != expected: + raise AssertionError( + f"safe-state count mismatch at w={width}: " + f"{len(g4.states)}, {len(g8.states)}, expected {expected}" + ) + same_state_set = set(g4.states) == set(g8.states) + + def log_ratio(p: float) -> float: + a = g4.perron(p)["lambda"] + b = g8.perron(1.0 - p)["lambda"] + return math.log(a) - math.log(b) + + root = brentq(log_ratio, 0.5000000001, 0.999999999, xtol=root_xtol) + p4 = g4.perron(root) + p8 = g8.perron(1.0 - root) + theta_slope = p4["I0_prime"] + p8["I0_prime"] + + pc4 = g4.perron(reference_pc) + pc8 = g8.perron(1.0 - reference_pc) + theta_pc = pc4["I0"] - pc8["I0"] + + return { + "width": width, + "safe_state_count_G4": len(g4.states), + "safe_state_count_G8": len(g8.states), + "central_trinomial_count": expected, + "G4_G8_safe_state_sets_identical": same_state_set, + "charge_coexistence_root": root, + "log_perron_ratio_at_root": log_ratio(root), + "lambda_root_G4": p4["lambda"], + "lambda_root_G8_complement": p8["lambda"], + "theta_slope_at_root": theta_slope, + "theta_slope_times_w_quarter": theta_slope * width ** 0.25, + "mean_occupied_G4_safe_row_at_root": p4["mean_occupied_safe_row"], + "mean_occupied_G8_safe_row_at_complement_root": p8[ + "mean_occupied_safe_row" + ], + "theta_at_reference_pc": theta_pc, + "theta_at_reference_pc_times_w_17_over_4": theta_pc + * width ** (17.0 / 4.0), + "reference_pc_minus_root": reference_pc - root, + "root_shift_times_w4": (reference_pc - root) * width**4, + "perron_residuals": { + "G4_root": p4["right_residual_inf"], + "G8_root": p8["right_residual_inf"], + "G4_reference": pc4["right_residual_inf"], + "G8_reference": pc8["right_residual_inf"], + }, + } + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--min-width", type=int, default=2) + parser.add_argument("--max-width", type=int, default=8) + parser.add_argument( + "--reference-pc", + type=float, + default=0.59274605079, + help="diagnostic reference only; not used to locate charge roots", + ) + parser.add_argument("--output", type=Path) + args = parser.parse_args() + + records = [ + coexistence_record(width, reference_pc=args.reference_pc) + for width in range(args.min_width, args.max_width + 1) + ] + result = { + "schema": "fixed-width-charge-transfer-v1", + "model": "independent square-site NN / complementary matching site", + "claim_boundary": [ + "Charge roots are located only from equality of safe Perron roots.", + "reference_pc is used only for finite-size scaling diagnostics.", + "This small-width implementation is not a new production pc estimator.", + ], + "reference_pc": args.reference_pc, + "records": records, + } + text = json.dumps(result, indent=2, sort_keys=True) + if args.output: + args.output.write_text(text + "\n", encoding="utf-8") + print(text) + + +if __name__ == "__main__": + main() diff --git a/scripts/giant_cluster_random_information.py b/scripts/giant_cluster_random_information.py new file mode 100644 index 000000000..4b28ea556 --- /dev/null +++ b/scripts/giant_cluster_random_information.py @@ -0,0 +1,337 @@ +#!/usr/bin/env python3 +"""Complete-cluster column scores: exact all-height controls, no sampling. + +Uses the unchanged lifted frontier engine, but reconstructs a NEW direct +activity with even/odd COLUMN occupation and distinct-boundary marks. The +lumping preserves these marks and is not the homogeneous unmarked quotient. +""" +from __future__ import annotations +import argparse +from collections import Counter, deque +from fractions import Fraction as F +import hashlib +import importlib.util +from itertools import product +import json +from math import comb, factorial, sqrt +from pathlib import Path +import sys + +ENGINE_BLOB = '52f3611990ce2b1331d9e5296e0262f5e402e0d7' + +def load_engine(): + here = Path(__file__).resolve() + paths = (here.with_name('tagged_winding_span.py'), + here.parents[2]/'source_inputs/tagged_winding_span.py') + for path in paths: + if not path.is_file(): + continue + raw = path.read_bytes() + got = hashlib.sha1(b'blob '+str(len(raw)).encode()+b'\0'+raw).hexdigest() + if got != ENGINE_BLOB: + raise ValueError(f'Unexpected engine blob {got}: {path}') + spec = importlib.util.spec_from_file_location('random_info_pinned', path) + mod = importlib.util.module_from_spec(spec) + sys.modules[spec.name] = mod + spec.loader.exec_module(mod) + return mod + raise FileNotFoundError('Pinned tagged_winding_span.py is required.') + +def parity_counts(mask, width): + return tuple(sum((mask >> x) & 1 for x in range(k, width, 2)) for k in (0, 1)) + +def colored_activity(engine, width, matching=False, cap=10000): + if width not in (2, 3, 4): + raise ValueError('Bounded reference controls support widths 2, 3, 4.') + states, index, source = [], {}, Counter() + def add(state): + if state not in index: + if len(states) >= cap: + raise RuntimeError('State cap reached; nothing truncated.') + index[state] = len(states); states.append(state) + return index[state] + for mask in range(1, 1 << width): + state, _ = engine.step(engine.empty(width), mask, matching, False) + bd = parity_counts(engine.expand_mask(mask, width, matching), width) + source[(add((0, state)), (0, 0, *bd))] += 1 + full, transitions = (1 << width)-1, [] + for old, state in states: + current = sum(1 << x for x, label in enumerate(state.labels) if label >= 0) + k = parity_counts(current, width) + horizontal = engine.expand_mask(current, width, True) + row = [] + for mask in range(1 << width): + nxt, outcome = engine.step(state, mask, matching, False, True) + bd = (horizontal | engine.expand_mask(old, width, matching) + | engine.expand_mask(mask, width, matching)) & (~current) & full + b = parity_counts(bd, width) + if outcome == 1: + upper = parity_counts(engine.expand_mask(current, width, matching), width) + row.append((-1, (*k, b[0]+upper[0], b[1]+upper[1]))) + elif outcome == 0: + row.append((add((current, nxt)), (*k, *b))) + transitions.append(row) + blocks = [0]*len(states) + while True: + labels, nxt = {}, [] + for row in transitions: + sig = tuple(sorted(Counter((blocks[j] if j >= 0 else j, mark) + for j, mark in row).items())) + if sig not in labels: labels[sig] = len(labels) + nxt.append(labels[sig]) + if nxt == blocks: break + blocks = nxt + n = max(blocks)+1 + reduced = [[(blocks[j] if j >= 0 else j, mark) + for j, mark in transitions[blocks.index(i)]] for i in range(n)] + src = Counter() + for (i, mark), count in source.items(): src[(blocks[i], mark)] += count + return reduced, src, len(states) + +def activity_weight(mark, qe, qo): + ke, ko, be, bo = mark + return qe**ke * qo**ko * (1-qe)**be * (1-qo)**bo + +def dot(a, b): return sum((x*y for x, y in zip(a,b)), F(0)) +def mv(A, x): return [dot(row, x) for row in A] + +def inverse_fraction(A): + """Exact Gauss-Jordan; build once and reuse for moment right-hand sides.""" + n = len(A) + M = [[F(v) for v in row]+[F(i == j) for j in range(n)] for i,row in enumerate(A)] + for j in range(n): + k = next((i for i in range(j,n) if M[i][j]), None) + if k is None: raise ValueError('Singular exact activity resolvent.') + M[j], M[k] = M[k], M[j] + d = M[j][j]; M[j] = [v/d for v in M[j]] + for i in range(n): + if i != j and M[i][j]: + d = M[i][j]; M[i] = [a-d*b for a,b in zip(M[i],M[j])] + return [row[n:] for row in M] + +def indices(d, order): + return sorted((a for a in product(range(order+1),repeat=d) if sum(a)<=order), + key=lambda a:(sum(a),a)) + +def all_height_moments(tr, src, qe, qo, forms, order=4): + """Raw joint moments of additive marks using differentiated resolvents. + + forms is a list of length-4 rational vectors on (Ke,Ko,Be,Bo). + Source boundary marks and BOTH terminal boundary rows are included. + """ + qe,qo = F(qe),F(qo) + if not (0 < qe < 1 and 0 < qo < 1): raise ValueError('Invalid site probabilities.') + keys = indices(len(forms),order); zero = (0,)*len(forms); n = len(tr) + R = {}; b = {}; alpha = {} + for a in keys: + R[a] = [[F(0)]*n for _ in range(n)]; b[a] = [F(0)]*n; alpha[a] = [F(0)]*n + def weighted(mark, a): + z = activity_weight(mark,qe,qo) + for power,form in zip(a,forms): z *= dot(form,mark)**power + return z + for i,row in enumerate(tr): + for j,mark in row: + for a in keys: + value = weighted(mark,a) + if j<0: b[a][i] += value + else: R[a][i][j] += value + for (i,mark),count in src.items(): + for a in keys: alpha[a][i] += count*weighted(mark,a) + G = inverse_fraction([[F(i==j)-R[zero][i][j] for j in range(n)] for i in range(n)]) + y, moments = {}, {} + for a in keys: + rhs = b[a][:] + for h in keys: + if h==zero or any(x>z for x,z in zip(h,a)): continue + prev = tuple(z-x for x,z in zip(h,a)) + factor = 1 + for z,x in zip(a,h): factor *= comb(z,x) + term = mv(R[h],y[prev]); rhs = [v+factor*u for v,u in zip(rhs,term)] + y[a] = mv(G,rhs) + total = F(0) + for h in keys: + if any(x>z for x,z in zip(h,a)): continue + prev = tuple(z-x for x,z in zip(h,a)); factor = 1 + for z,x in zip(a,h): factor *= comb(z,x) + total += factor*dot(alpha[h],y[prev]) + moments[a] = total + nu = moments[zero] + return nu,{a:v/nu for a,v in moments.items()} + +def physical_shapes(width,height,matching): + """Independent lifted BFS, not the frontier engine; boundary masks exact.""" + steps=[(-1,0),(1,0),(0,-1),(0,1)] + if matching: steps += [(a,b) for a in (-1,1) for b in (-1,1)] + records=[] + for mask in range(1,1<<(width*height)): + if not mask & ((1<> (width*(height-1)): continue + C={(x,y) for y in range(height) for x in range(width) if mask>>(y*width+x)&1} + root=min(C); lift={root:0}; Q=deque([root]); wind=False + while Q: + u=Q.popleft() + for dx,dy in steps: + v=((u[0]+dx)%width,u[1]+dy) + if v not in C: continue + expected=lift[u]+dx + if v in lift: wind |= lift[v] != expected + else: lift[v]=expected;Q.append(v) + if len(lift)!=len(C) or not wind: continue + B={((x+dx)%width,y+dy) for x,y in C for dx,dy in steps}-C + mark=tuple(sum(x%2==j for x,y in D) for D in (C,B) for j in (0,1)) + records.append(mark) + return records + +def finite_coefficients(tr,src,height): + states=Counter(src) + exits=Counter() + for level in range(1,height+1): + nxt=Counter() + for (i,mark),count in states.items(): + for j,m in tr[i]: + total=tuple(a+b for a,b in zip(mark,m)) + if j<0: + if level==height: exits[total]+=count + elif level>v&1):C.add(v);todo.append(v) + Q=(C|{v for u in C for v in adj[u]})-{n} + wt=q**mask.bit_count()*p**(n-mask.bit_count()) + fullscore=[(F(mask>>i&1)-q)/(p*q) for i in range(n)] + xi=[fullscore[i] if i in Q else F(0) for i in range(n)] + prodv=[F(1)]*(1< tuple[tuple[int, bool], ...]: + """Physical NN components, with horizontal winding from lifted BFS.""" + unseen = {i for i in range(N) if mask >> i & 1} + ans = [] + while unseen: + root = min(unseen) + unseen.remove(root) + todo = [root] + lift = {root: 0} + bits = 0 + wraps = False + for v in todo: + bits |= 1 << v + x, y = v % W, v // W + for dx, dy in ((1, 0), (-1, 0), (0, 1), (0, -1)): + if not 0 <= y + dy < H: + continue + u = (y + dy) * W + (x + dx) % W + if not mask >> u & 1: + continue + z = lift[v] + dx + if u not in lift: + lift[u] = z + unseen.remove(u) + todo.append(u) + elif lift[u] != z: + assert (lift[u] - z) % W == 0 + wraps = True + ans.append((bits, wraps)) + return tuple(ans) + +@lru_cache(None) +def winding_count(mask: int) -> int: + return sum(w for _, w in components(mask)) + +def occupancy(mask: int, p: F) -> F: + k = mask.bit_count() + return p**k * (1-p)**(N-k) + +@lru_cache(None) +def nu(p: F) -> F: + """Finite-strip TOTAL expected component count, not infinite intensity.""" + return sum((occupancy(m,p)*winding_count(m) for m in range(1 << N)), F(0)) + +@lru_cache(None) +def clock_bernstein(r: F) -> tuple[F, ...]: + """Normalized degree-N Bernstein coefficients of beta(p;r), t=p/r. + +Each final essential component contributes its increasing first-winding +reliability polynomial. Lower-degree polynomials are elevated exactly. +""" + if not 0 < r < 1: + raise ValueError('terminal parameter must lie in (0,1)') + raw = [F(0) for _ in range(N+1)] + for final in range(1 << N): + weight = occupancy(final,r) + for cmask, wraps in components(final): + if not wraps: + continue + n = cmask.bit_count() + count = [0]*(n+1) + sub = cmask + while True: + if winding_count(sub): + count[sub.bit_count()] += 1 + if sub == 0: + break + sub = (sub-1)&cmask + for j in range(N+1): + a = sum(count[k]*comb(N-n,j-k) + for k in range(n+1) if 0 <= j-k <= N-n) + raw[j] += weight*a + return tuple(raw[j]/comb(N,j) for j in range(N+1)) + +def bernstein_eval(coeffs: tuple[F,...], t: F) -> F: + n = len(coeffs)-1 + return sum((a*comb(n,j)*t**j*(1-t)**(n-j) + for j,a in enumerate(coeffs)),F(0)) + +def beta(p: F, r: F) -> F: + if not 0 <= p <= r < 1: + raise ValueError('require 0 <= p <= r < 1') + return bernstein_eval(clock_bernstein(r),p/r) + +def beta_prime(p: F, r: F) -> F: + c = clock_bernstein(r) + return N/r*bernstein_eval(tuple(c[j+1]-c[j] for j in range(N)),p/r) + +def direct_pair_values(p: F, r: F) -> tuple[F,F,F,int]: + """Integrate early/final common labels over all 3^N pairs.""" + hits = loss = pairs = F(0) + checked = 0 + for final in range(1 << N): + fc = [b for b,w in components(final) if w] + early = final + while True: + ec = [b for b,w in components(early) if w] + ns = [sum((e & c)==e for e in ec) for c in fc] + assert sum(ns)==len(ec) + hit = sum(k>0 for k in ns) + lost = len(ec)-hit + pair = sum(k*(k-1)//2 for k in ns) + assert 0 <= lost <= pair + wt = p**early.bit_count() * (r-p)**(final.bit_count()-early.bit_count()) * (1-r)**(N-final.bit_count()) + hits += wt*hit + loss += wt*lost + pairs += wt*pair + checked += 1 + if early==0: + break + early=(early-1)&final + return hits,loss,pairs,checked + +def invert_beta(target: F, r: F, steps: int=48) -> tuple[F,F]: + if not 0 <= target <= beta(r,r): + raise ValueError('target outside clock range') + lo,hi=F(0),r + for _ in range(steps): + mid=(lo+hi)/2 + if beta(mid,r) dict: + h=F(-527,625) + h8=2*h*h-1 + leak0=(h8-h)/(1-h) + leak4=(1-h8)/(1-h) + assert leak0==F(429,625) and leak4==F(196,625) + return {'H4_oblique':h,'H8_oblique':h8, + 'hidden_H8_into_scalar':leak0,'hidden_H8_into_spin4':leak4, + 'pair_projection_spin4_coefficient':1/(1-h)} + +def alignment_controls() -> dict: + # b=x+g*x^2, e=b^2: all pure reparametrization, but normalized + # single-charge curve changes by g*x^2 (root=0, root slope=1). + checks=0 + for g in (F(-1,4), F(0), F(1,4)): + for x in (F(-1,4),F(-1,8),F(0),F(1,8),F(1,4)): + b=x+g*x*x + bp=1+2*g*x + bg=x*x + ep=2*b*bp + eg=2*b*bg + assert bp>0 + assert eg-ep*bg/bp==0 + assert b-x==g*x*x + checks+=1 + # b=x, e=x^2+g*x: scalar charge curve unchanged, true N=x. + for x in (F(-1,4),F(-1,8),F(0),F(1,8),F(1,4)): + assert (x - (2*x+F(1,4))*F(0))==x + checks+=1 + return {'exact_controls':checks, + 'pure_tangent_false_shape_signal_at_x_1_4_g_1_4':F(1,64), + 'pure_normal_invisible_to_single_charge_at_x_1_4':F(1,4)} + +@lru_cache(None) +def torus_rank(mask: int) -> int: + """Independent 2D lifted BFS on the 3x3 NN torus (not the free strip).""" + unseen = {i for i in range(N) if mask >> i & 1} + periods = [] + while unseen: + root=min(unseen); unseen.remove(root) + lift={root:(0,0)}; queue=[root] + for v in queue: + x,y=v%W,v//W + for dx,dy in ((1,0),(-1,0),(0,1),(0,-1)): + u=((y+dy)%H)*W+(x+dx)%W + if not mask>>u&1: + continue + z=(lift[v][0]+dx,lift[v][1]+dy) + if u not in lift: + lift[u]=z;unseen.remove(u);queue.append(u) + else: + a,b=z[0]-lift[u][0],z[1]-lift[u][1] + assert a%W==0 and b%H==0 + if a or b: + periods.append((a//W,b//H)) + if not periods: + return 0 + a,b=periods[0] + return 2 if any(a*y-b*x for x,y in periods) else 1 + +def pair_marks(mask: int) -> tuple[int,int,int]: + """Horizontal NN pair sources; each horizontal edge counted once.""" + black=white=0 + for y in range(H): + for x in range(W): + a=(mask>>(y*W+x))&1 + b=(mask>>(y*W+(x+1)%W))&1 + black+=a*b;white+=(1-a)*(1-b) + k=mask.bit_count() + assert white==N-2*k+black + return k,black,white + +def rank_response_controls() -> dict: + data=[(m,torus_rank(m),pair_marks(m)) for m in range(1< dict: + """Postprocess PRINTED existing floats as exact decimal rationals. + +These are not new eigenvalue calculations/certificates. A decimal input's +last-digit error remains present in the resulting displayed values. +""" + p=F('0.5927460507921');q=1-p + bs=['-1.350049753766846','-1.7138942463383808','-2.0732497837477912', + '-2.4300006319440906','-2.7852763705546106'] + ws=['-0.6080813474300464','-0.7864337384173812','-0.9602971742425906', + '-1.1315559208546921','-1.3013395578810114'] + tb=['-0.3116848217980418','-0.3102994990346319','-0.3098063945349776', + '-0.30962331020127565','-0.30954105615205907'] + tw=['0.17111151832689642','0.17249684109030636','0.17298994558996025', + '0.17317302992366124','0.17325528397288223'] + result=[] + for w,b,a,t1,t2 in zip(range(4,9),bs,ws,tb,tw): + nb=F(b)+w*p*p;nw=F(a)+w*q*q + diffT=F(t2)-F(t1)-2*p*q + assert abs(nb-nw) dict: + c=F(0);h=F(5,96);hbar=h + eigen=h*h-(c+2)*h/12+c*(5*c+22)/2880 + assert h-hbar==0 and eigen==F(-55,9216) + return {'c':c,'h':h,'hbar':hbar,'spatial_momentum_weight':h-hbar, + 'chiral_spin4_I3_eigenvalue':eigen, + 'meaning':'A Virasoro zero-mode counterexample to blanket spin exclusion, not a site field identification.', + 'source':'https://arxiv.org/html/1810.11053v2 equation (2)'} + +def encode(obj): + if isinstance(obj,F): + return {'fraction':str(obj),'decimal':float(obj)} + if isinstance(obj,dict): + return {str(k):encode(v) for k,v in obj.items()} + if isinstance(obj,(list,tuple)): + return [encode(v) for v in obj] + return obj + +def report() -> dict: + terminals=(F(1,3),F(1,2),F(3,4)) + cases=[] + total_pairs=0 + for r in terminals: + coeff=clock_bernstein(r) + assert coeff[0]==0 and coeff[-1]==nu(r) + assert all(coeff[j+1]>=coeff[j] for j in range(N)) + for p in (r/3,2*r/3): + a,l,m,n=direct_pair_values(p,r) + total_pairs+=n + assert a==beta(p,r) and nu(p)==a+l and 0<=l<=m + assert 0=3 same-ell D4 classes, so " + "there is no cheaper modulus. Extrapolating the measured " + "x2.85-per-row state growth to N=1105 (memory_G8 = |a|+|b| = " + "37,41,43,47) gives 1e8..1e21 states: the F-based GO is not " + "actionable."), + }, + "reference_md5": { + "/workspace/dpfloor/scripts/fl_path_oblique.py": + "c402cdf6a0d92d317864d793b6eece44", + "rev769-repo/scripts/oblique_charge_transfer.py (local copy)": + "d5c0e5a4896bc6a244be8460e6929dd3", + "rev769-repo/scripts/diagonal_charge_transfer.py (local copy)": + "0ea1018e4d43a010035e142ce85864b1", + "rev769-repo/scripts/oblique_winding_necklace.py (local copy)": + "ba82c6076017f7aa27297778153caefe", + "rev769-repo/scripts/oblique_winding_corridor.py (local copy)": + "a94d2bcbf6bb5db1499d00beaee55644", + }, + "scripts_md5": {s: md5(os.path.join("/workspace/n325rec/scripts", s)) + for s in SCRIPTS + if os.path.exists(os.path.join("/workspace/n325rec/scripts", s))}, + } + for name in FILES: + p = os.path.join(OUT, name + ".json") + if os.path.exists(p): + rec[name] = json.load(open(p)) + else: + rec[name] = None + p = "/workspace/n325rec/out/rec.json" + with open(p, "w") as fh: + json.dump(rec, fh, indent=1) + print("wrote", p, os.path.getsize(p), "bytes") + for s, m in rec["scripts_md5"].items(): + print(" %-20s %s" % (s, m)) + + +if __name__ == "__main__": + main() diff --git a/scripts/n325rec_oblique_indep.py b/scripts/n325rec_oblique_indep.py new file mode 100644 index 000000000..dd4e5673e --- /dev/null +++ b/scripts/n325rec_oblique_indep.py @@ -0,0 +1,440 @@ +#!/usr/bin/env python3 +"""n325rec -- SECOND, independent oblique safe-charge-transfer implementation. + +Object (identical to PATH B by construction; see the SHARED layer below): + primitive direction u=(a,b), repeat count n (circumference ell = n*|u|), + frontier = `memory` rows of `width`=n cells in the SL(2,Z) frame + x = s*u + t*v. G4 = black NN graph at density p, G8 = white NN+matching graph + at density 1-p. The SAFE transfer keeps only frontier states with no + cylinder-wrapping cluster; Delta(p) = log lambda0_G4(p) - log lambda0_G8(1-p) + and p_root is its zero. + +INDEPENDENCE -- where this file differs from B_oblique + (B = /workspace/dpfloor/scripts/fl_path_oblique.py, md5 c402cdf6a0d92d317864d793b6eece44, + a verbatim copy of rev769 scripts/oblique_charge_transfer.py, + md5 d5c0e5a4896bc6a244be8460e6929dd3): + + LAYER 1 -- state encoding. B stores (labels, gains) per frontier cell with + labels=-1 meaning "empty": the occupancy is only implicit and the gains live + inside a DSU's parent/delta arrays. Here a state is an explicit 3-tuple + (occupancy bitmask, label tuple, lift tuple): a real bitmask, cells ordered + NEWEST ROW FIRST (B orders oldest row first), and an explicit integer lift per + cell. Canonical form: labels by first occurrence, lifts relative to each + component's first cell. + + LAYER 2 -- winding test. B rejects a transition the instant a DSU union + closes a cycle with a non-zero deck gain (eager, incremental, inside the DSU). + Here the constraint graph of the candidate transition (new-row cells + old + components as super-nodes, weighted edges carrying the deck displacement) is + materialised FIRST and every edge is checked while the spanning forest is + grown; reject iff any edge closes a non-zero fundamental cycle. + Same MATHEMATICAL predicate, different algorithm and a different failure + moment (see SHARED layer S2). + + LAYER 3 -- eigensolver / root finder / arithmetic. B uses dense LAPACK + eigvals (n<2500) or ARPACK tol<=1e-15 (n>=2500) and float64 brentq/bisection. + Here lambda0 comes from an explicit power iteration on the sparse count-list + matrix, warm-started between successive p, evaluated in np.longdouble (IEEE + binary128 on this aarch64 host, ~34 decimal digits), and the root is found by + longdouble bisection with xtol=1e-25. No LAPACK, no ARPACK, no scipy. + + SHARED LAYER -- NOT covered by this independence (stated explicitly because a + second path cannot certify it): + (S1) the choice of Bezout complement v=(c,d), a*d-b*c=1. This is a + CONVENTION, not a theorem: v -> v+k*u gives the same cylinder only when + n|k. Both paths take the extended-gcd complement, so a wrong + convention would be wrong in both. + (S2) the predicate "wrapping <=> the frontier constraint graph admits no + consistent integer potential". + (S3) row memory = max dt and the frontier bookkeeping itself. +""" +from __future__ import annotations + +import json +import math +import sys +import time +from collections import deque + +import numpy as np + +# ------------------------------------------------------------------ the frame +def egcd_iter(a: int, b: int): + """Iterative extended gcd (no recursion). a*x + b*y = g.""" + old_r, r = a, b + old_s, s = 1, 0 + old_t, t = 0, 1 + while r != 0: + q = old_r // r + old_r, r = r, old_r - q * r + old_s, s = s, old_s - q * s + old_t, t = t, old_t - q * t + if old_r < 0: + old_r, old_s, old_t = -old_r, -old_s, -old_t + return old_r, old_s, old_t + + +def bezout_complement_iter(a: int, b: int): + """(c,d) with a*d - b*c = 1, same normalisation as the reference code.""" + g, x, y = egcd_iter(a, b) + if g != 1: + raise ValueError("direction (%d,%d) is not primitive" % (a, b)) + d, c = x, -y + if a * d - b * c != 1: + raise AssertionError("bezout failure for (%d,%d)" % (a, b)) + return c, d + + +def frame(direction, matching, complement=None): + """Return (comp, edges, memory) for the SL(2,Z) frame of `direction`.""" + a, b = direction + c, d = complement if complement is not None else bezout_complement_iter(a, b) + if a * d - b * c != 1: + raise ValueError("complement must satisfy det(u,v)=1") + gens = [(1, 0), (0, 1)] + if matching: + gens = gens + [(1, 1), (1, -1)] + edges = set() + for dx, dy in gens: + ds = d * dx - c * dy + dt = -b * dx + a * dy + if dt < 0 or (dt == 0 and ds < 0): + ds, dt = -ds, -dt + edges.add((ds, dt)) + ordered = sorted(edges, key=lambda e: (e[1], e[0])) + return (c, d), ordered, max(dt for _, dt in ordered) + + +# ------------------------------------------------------------- the automaton +class IndependentOblique: + """Safe transfer automaton for (width=n, direction=u, matching).""" + + def __init__(self, width, direction, matching, state_cap=300000, + complement=None): + self.width = width + self.direction = tuple(direction) + self.matching = bool(matching) + self.comp, self.edges, self.memory = frame(direction, matching, + complement=complement) + t0 = time.time() + self._build(state_cap) + self.build_seconds = time.time() - t0 + self._warm = None + + # ---- one transition ----------------------------------------------------- + def _step(self, state, mask): + W = self.width + M = self.memory + occ, lab, lif = state + parent = list(range(W)) + delta = [0] * W + comp_of_node = {} + + def find(x): + root, acc = x, 0 + while parent[root] != root: + acc += delta[root] + root = parent[root] + cur, g = x, 0 + while parent[cur] != cur: + nxt = parent[cur] + dcur = delta[cur] + parent[cur] = root + delta[cur] = acc - g + g += dcur + cur = nxt + return root, acc + + def join(u, v, g): + """enforce lift[v] - lift[u] = g; False on contradiction.""" + ru, gu = find(u) + rv, gv = find(v) + if ru == rv: + return (gv - gu) == g + parent[rv] = ru + delta[rv] = g + gu - gv + return True + + def oldnode(lbl): + node = comp_of_node.get(lbl) + if node is None: + node = len(parent) + comp_of_node[lbl] = node + parent.append(node) + delta.append(0) + return node + + # --- constraints from the frame edges + for ds, dt in self.edges: + if dt == 0: + for i in range(W): + if not (mask >> i) & 1: + continue + j = (i + ds) % W + if not (mask >> j) & 1: + continue + if not join(i, j, ds): + return None + else: + row = dt - 1 # 0 == newest retained row + for i in range(W): + idx = row * W + i + if not (occ >> idx) & 1: + continue + j = (i + ds) % W + if not (mask >> j) & 1: + continue + node = oldnode(lab[idx]) + if not join(node, j, lif[idx] + ds): + return None + + # --- new frontier occupancy: new row on top, old rows shifted down + new_occ = mask & ((1 << W) - 1) + for r in range(M - 1): + base = r * W + if (occ >> base) & ((1 << W) - 1): + new_occ |= (((occ >> base) & ((1 << W) - 1)) << ((r + 1) * W)) + + newlab = [-1] * (M * W) + newlif = [0] * (M * W) + root2lab = {} + for r in range(M): + for c in range(W): + nidx = r * W + c + if not (new_occ >> nidx) & 1: + continue + if r == 0: + node, base = c, 0 + else: + oidx = (r - 1) * W + c + node, base = oldnode(lab[oidx]), lif[oidx] + root, g = find(node) + if root not in root2lab: + root2lab[root] = (len(root2lab), g + base) + nl, b0 = root2lab[root] + newlab[nidx] = nl + newlif[nidx] = (g + base) - b0 + return (new_occ, tuple(newlab), tuple(newlif)) + + def _build(self, state_cap): + W = self.width + M = self.memory + start = (0, tuple([-1] * (M * W)), tuple([0] * (M * W))) + states = [start] + index = {start: 0} + queue = deque([0]) + trans = [] + while queue: + si = queue.popleft() + st = states[si] + row = [] + for mask in range(1 << W): + nxt = self._step(st, mask) + if nxt is None: + row.append(-1) + continue + j = index.get(nxt) + if j is None: + if len(states) >= state_cap: + raise RuntimeError( + "state cap %d reached (%s n=%d matching=%s)" + % (state_cap, self.direction, W, self.matching)) + j = len(states) + index[nxt] = j + states.append(nxt) + queue.append(j) + row.append(j) + trans.append(row) + if len(trans) != len(states): + raise AssertionError("BFS bookkeeping") + self.states = states + self.trans = trans + self.n = len(states) + agg = {} + for src, row in enumerate(trans): + for mask, dst in enumerate(row): + if dst < 0: + continue + k = (src, dst, bin(mask).count("1")) + agg[k] = agg.get(k, 0) + 1 + keys = sorted(agg) + self.src = np.array([k[0] for k in keys], dtype=np.int64) + self.dst = np.array([k[1] for k in keys], dtype=np.int64) + self.occ = np.array([k[2] for k in keys], dtype=np.int64) + self.cnt = np.array([agg[k] for k in keys], dtype=float) + self.nnz = len(keys) + + # ---- lambda0 ------------------------------------------------------------ + def matrix_f64(self, p): + w = self.width + pw = p ** self.occ + qw = (1.0 - p) ** (w - self.occ) + R = np.zeros((self.n, self.n)) + np.add.at(R, (self.src, self.dst), self.cnt * pw * qw) + return R + + def matvec(self, p, v, dtype): + d = dtype(p) + q = dtype(1) - d + w = self.width + pw = np.empty(w + 1, dtype=dtype) + qw = np.empty(w + 1, dtype=dtype) + pw[0] = dtype(1) + qw[0] = dtype(1) + for i in range(1, w + 1): + pw[i] = pw[i - 1] * d + qw[i] = qw[i - 1] * q + coeff = self.cnt.astype(dtype) * pw[self.occ] * qw[w - self.occ] + out = np.zeros(self.n, dtype=dtype) + np.add.at(out, self.dst, coeff * v[self.src]) + return out + + def lambda0(self, p, dtype=np.longdouble, tol=1e-25, maxit=1000, + warm=True): + """Perron root by power iteration (1-norm growth ratio). + + Returns (lambda, iters, rel_residual). The ratio ||R v||_1/||v||_1 + converges to rho(R) as v -> the Perron vector. + """ + dty = dtype + if warm and self._warm is not None: + v = self._warm + else: + v = np.ones(self.n, dtype=dty) + v = v / v.max() + prev = None + resid = float("nan") + it = 0 + for it in range(1, maxit + 1): + w = self.matvec(p, v, dty) + s = w.sum() + vs = v.sum() + if not (s > 0): + raise RuntimeError("zero transfer matrix") + lam = s / vs + v = w / w.max() + if prev is not None: + resid = float(abs(lam - prev) / abs(lam)) + if resid <= tol: + self._warm = v + return float(lam), it, resid + prev = lam + self._warm = v + return float(prev), it, resid + + +# ------------------------------------------------------------------ geometry +def solve_p_root(t4, t8, pc, dtype=np.longdouble, xtol=None, lo=0.50, hi=0.80, + maxiter=400, tol=1e-25): + """Delta(p) = log l4(p) - log l8(1-p) ; zero by longdouble bisection.""" + if xtol is None: + xtol = 1e-25 if dtype is np.longdouble else 1e-16 + dt = dtype + l4 = lambda p: t4.lambda0(p, dtype=dt, tol=tol)[0] + l8 = lambda p: t8.lambda0(p, dtype=dt, tol=tol)[0] + + def f(p): + return math.log(l4(p)) - math.log(l8(dt(1) - dt(p))) + + a, b = dt(lo), dt(hi) + fa, fb = f(a), f(b) + if (fa > 0) == (fb > 0): + raise ValueError("no sign change on [%s,%s]: f=%r,%r" % (lo, hi, fa, fb)) + for _ in range(maxiter): + if b - a < dt(xtol): + break + m = (a + b) / 2 + fm = f(m) + if fm == 0: + a = b = m + break + if (fm > 0) == (fa > 0): + a, fa = m, fm + else: + b, fb = m, fm + root = (a + b) / 2 + return root, f(root), abs(b - a) + + +GEOMS_REF = { + "diag_n4": ((1, 1), 4), + "diag_n5": ((1, 1), 5), + "slope21_n3": ((2, 1), 3), + "slope21_n4": ((2, 1), 4), + "slope31_n3": ((3, 1), 3), + "slope32_n2": ((3, 2), 2), + "slope52_n2": ((5, 2), 2), + "axis_n4": ((1, 0), 4), + "axis_n6": ((1, 0), 6), + "axis_n8": ((1, 0), 8), + "n325_1_18": ((1, 18), 1), + "n325_6_17": ((6, 17), 1), + "n325_2_3_n5": ((2, 3), 5), + "n25_3_4": ((3, 4), 1), + "n25_0_1_n5": ((0, 1), 5), +} + + +def run(spec, pc, state_cap=300000, dtype=np.longdouble, xtol=None, + complement=None, sink=None, tol=1e-25): + out = {"path": "n325rec_independent_oblique", "p_c": pc, + "dtype": str(np.dtype(dtype)), "tol": tol, "records": {}} + for tag, u, n in spec: + t0 = time.time() + try: + t4 = IndependentOblique(n, u, False, state_cap=state_cap, + complement=complement) + t8 = IndependentOblique(n, u, True, state_cap=state_cap, + complement=complement) + build_t = time.time() - t0 + root, resid, width = solve_p_root(t4, t8, pc, dtype=dtype, + xtol=xtol, tol=tol) + ell = n * math.hypot(u[0], u[1]) + a, b = u + cos4 = (a ** 4 - 6 * a * a * b * b + b ** 4) / (a * a + b * b) ** 2 + rec = {"tag": tag, "direction": list(u), "n": n, "ell": ell, + "cos4": cos4, + "comp": list(t4.comp), + "edges_G4": [list(e) for e in t4.edges], + "edges_G8": [list(e) for e in t8.edges], + "memory_G4": t4.memory, "memory_G8": t8.memory, + "nnz_G4": t4.nnz, "nnz_G8": t8.nnz, + "n_safe_G4": t4.n, "n_safe_G8": t8.n, + "build_seconds": round(build_t, 2), + "p_root_ld": repr(root), + "p_root_float": float(root), + "root_minus_pc": float(root - pc), + "Delta_at_root": float(resid), + "bracket_width": float(width), + "Omega": float(-(root - pc) * ell ** 4), + "seconds": round(time.time() - t0, 2)} + except RuntimeError as exc: + rec = {"tag": tag, "direction": list(u), "n": n, + "error": str(exc), "seconds": round(time.time() - t0, 2)} + out["records"][tag] = rec + if "error" in rec: + print("N %-13s ABORT %s" % (tag, rec["error"]), flush=True) + else: + print("N %-13s n_safe=%d/%d p_root=%s Delta=%.3e %.1fs" + % (tag, rec["n_safe_G4"], rec["n_safe_G8"], + rec["p_root_ld"], rec["Delta_at_root"], rec["seconds"]), + flush=True) + if sink: + with open(sink, "w") as fh: + json.dump(out, fh, indent=1) + return out + + +if __name__ == "__main__": + pc = float(sys.argv[1]) if len(sys.argv) > 1 else 0.5927460507921 + outp = sys.argv[2] if len(sys.argv) > 2 else \ + "/workspace/n325rec/out/s3_indep.json" + only = sys.argv[3].split(",") if len(sys.argv) > 3 else None + spec = [(t, u, n) for t, (u, n) in GEOMS_REF.items()] + if only: + spec = [s for s in spec if s[0] in only] + print("restricted to", [s[0] for s in spec], flush=True) + res = run(spec, pc, sink=outp) + with open(outp, "w") as fh: + json.dump(res, fh, indent=1) + print("->", outp) diff --git a/scripts/n325rec_s1_orient.py b/scripts/n325rec_s1_orient.py new file mode 100644 index 000000000..69e7c4b52 --- /dev/null +++ b/scripts/n325rec_s1_orient.py @@ -0,0 +1,251 @@ +#!/usr/bin/env python3 +"""n325rec step 1 -- EXACT orientation table for a^2+b^2=N. + +Independent computation (no numbers are copied from n1105mix): + * all integer representations a^2+b^2=N, classified into D4 orbits + (D4 = the 8 symmetries (a,b)->(+-a,+-b),(+-b,+-a)); + * primitivity gcd(a,b)=1 and, crucially, the "same-ell realization": + a non-primitive representative (k*a0,k*b0) is the SAME cylinder as the + primitive u=(a0,b0) with index n=k (both give period vector k*u0), so it + is usable exactly when written that way; + * exact cos(4m theta) as Fraction, from cos(4m th) = Re((a+ib)^{4m})/N^{2m}; + * design matrix A[i][m] = cos(4m theta_i) (m=0,1,2 -> H0,H4,H8), its EXACT + rank, the exact H0/H4/H8 projector weights w (solve A^T w = e_0), and + |w|_1, |w|_2 (the L2 norm is the orientation-noise amplification); + * floating point is used only for the final sqrt / L2 display. + +Pure stdlib (fractions). No numpy. +""" +from __future__ import annotations +import json +import math +import sys +from fractions import Fraction + + +# ---------------------------------------------------------------- integer reps +def reps(N: int): + """All (a,b), 0<=a<=b, a^2+b^2=N, as D4-class representatives.""" + out = [] + a = 0 + while a * a <= N: + b2 = N - a * a + b = math.isqrt(b2) + if b * b == b2 and b >= a: + out.append((a, b)) + a += 1 + return out + + +def d4_orbit(p): + a, b = p + return sorted({(a, b), (b, a), (-a, b), (a, -b), (-b, a), (b, -a), + (-a, -b), (-b, -a)}) + + +# ------------------------------------------------------------ exact harmonics +def ipow(a: int, b: int, k: int): + """(a+ib)^k exactly.""" + re, im = 1, 0 + for _ in range(k): + re, im = re * a - im * b, re * b + im * a + return re, im + + +def cos4m(a: int, b: int, N: int, m: int) -> Fraction: + """cos(4*m*theta) exactly, theta = angle of (a,b).""" + re, _im = ipow(a, b, 4 * m) + return Fraction(re, N ** (2 * m)) + + +# --------------------------------------------------------- exact linear algebra +def solve_exact(M, rhs): + """Solve M x = rhs exactly (Fraction Gauss-Jordan). Returns (x, rank, ok).""" + n = len(M) + m = len(M[0]) + aug = [[Fraction(M[i][j]) for j in range(m)] + [Fraction(rhs[i])] + for i in range(n)] + if n != m: + raise ValueError("solve_exact wants square") + rank = 0 + piv = [] + r = 0 + for c in range(m): + pr = None + for i in range(r, n): + if aug[i][c] != 0: + pr = i + break + if pr is None: + continue + aug[r], aug[pr] = aug[pr], aug[r] + pv = aug[r][c] + aug[r] = [x / pv for x in aug[r]] + for i in range(n): + if i != r and aug[i][c] != 0: + f = aug[i][c] + aug[i] = [aug[i][j] - f * aug[r][j] for j in range(m + 1)] + piv.append(c) + r += 1 + if r == n: + break + rank = r + for i in range(r, n): + if all(aug[i][j] == 0 for j in range(m)) and aug[i][m] != 0: + return None, rank, False + if rank < m: + return None, rank, False + x = [aug[i][m] for i in range(m)] + return x, rank, True + + +def rank_exact(M): + """Exact rank of a rectangular Fraction matrix.""" + if not M: + return 0 + A = [[Fraction(v) for v in row] for row in M] + rows, cols = len(A), len(A[0]) + r = 0 + for c in range(cols): + pr = None + for i in range(r, rows): + if A[i][c] != 0: + pr = i + break + if pr is None: + continue + A[r], A[pr] = A[pr], A[r] + pv = A[r][c] + for i in range(r + 1, rows): + if A[i][c] != 0: + f = A[i][c] / pv + A[i] = [A[i][j] - f * A[r][j] for j in range(cols)] + r += 1 + if r == rows: + break + return r + + +# ---------------------------------------------------------------------- driver +def analyse(N: int, n_harm: int = 3): + rp = reps(N) + orbits = [] + seen = set() + for p in rp: + if p in seen: + continue + orb = [q for q in d4_orbit(p)] + for q in rp: + if q in orb: + seen.add(q) + orbits.append(orb) + + classes = [] + for orb in orbits: + rep = None + for q in orb: + if q in rp: + rep = q + break + a, b = rep + g = math.gcd(a, b) + c4 = cos4m(a, b, N, 1) + c8 = cos4m(a, b, N, 2) + c12 = cos4m(a, b, N, 3) + harm = [Fraction(1), c4, c8, c12][:n_harm] + classes.append({ + "rep": [a, b], + "orbit_size": len({q for q in orb}), + "gcd": g, + "primitive": g == 1, + "same_ell_realization": ({"u": [a // g, b // g], "n": g} + if g > 1 else {"u": [a, b], "n": 1}), + "theta_deg": math.degrees(math.atan2(b, a)), + "cos4": str(c4), "cos4_float": float(c4), + "cos8": str(c8), "cos8_float": float(c8), + "cos12": str(c12), "cos12_float": float(c12), + "_harm": harm, + }) + + A = [c["_harm"] for c in classes] + A = [list(row) for row in A] + rk = rank_exact(A) + out = { + "N": N, + "sqrt_N": math.sqrt(N), + "ell4_exact": N ** 2, + "all_representations": [list(p) for p in rp], + "n_d4_classes": len(classes), + "n_primitive_classes": sum(1 for c in classes if c["primitive"]), + "design_matrix_rank": rk, + "design_matrix_shape": [len(A), n_harm], + "classes": classes, + } + # H0/H4/... projector weights when the square system is invertible + if len(classes) == n_harm and rk == n_harm: + M = [[A[i][j] for i in range(n_harm)] for j in range(n_harm)] # A^T + for target in range(n_harm): + e = [Fraction(1 if k == target else 0) for k in range(n_harm)] + w, _r, ok = solve_exact([row[:] for row in M], e) + if ok: + l1 = sum(abs(x) for x in w) + l2 = math.sqrt(float(sum(x * x for x in w))) + out.setdefault("projector_weights", {})["H%d" % (4 * target)] = { + "w_num": [str(x) for x in w], + "w_float": [float(x) for x in w], + "L1": float(l1), + "L2": l2, + } + # error propagation of a per-point Omega error sigma + if len(classes) == n_harm and rk == n_harm: + B = [[A[i][j] for i in range(n_harm)] for j in range(n_harm)] + Binv = [] + # invert A exactly (A^T w = e solved above gives rows of A^{-T}) + for target in range(n_harm): + e = [Fraction(1 if k == target else 0) for k in range(n_harm)] + w, _r, ok = solve_exact([row[:] for row in B], e) + Binv.append(w) + # B^{-1} = A^{-T}; P = (A^T A)^{-1}A^T Omega = A^{-1} Omega for square A + # A^{-1} = (A^{-T})^T + Ainv = [[Binv[j][i] for j in range(n_harm)] for i in range(n_harm)] + amps = [] + for m in range(n_harm): + amps.append(math.sqrt(float(sum(Ainv[m][i] ** 2 + for i in range(n_harm))))) + out["coefficient_noise_amplification"] = { + "H%d" % (4 * m): amps[m] for m in range(n_harm)} + for c in classes: + c.pop("_harm", None) + return out + + +def main(): + res = {"schema": "n325rec.orientation-table.v1", + "p_c": 0.5927460507921, + "note": ("exact cos(4m theta); L2 of the H0 projector weight = " + "orientation-noise amplification; same_ell_realization " + "shows the primitive (u,n) pair with n*|u| = sqrt(N)")} + for N in (25, 325, 1105): + res["N=%d" % N] = analyse(N, n_harm=(2 if N == 25 else (3 if N == 325 else 4))) + out = sys.argv[1] if len(sys.argv) > 1 else "/workspace/n325rec/out/s1_orient.json" + with open(out, "w") as fh: + json.dump(res, fh, indent=1) + for key in ("N=25", "N=325", "N=1105"): + d = res[key] + print("%s: classes=%d (primitive %d) rank=%d sqrtN=%.6f" + % (key, d["n_d4_classes"], d["n_primitive_classes"], + d["design_matrix_rank"], d["sqrt_N"])) + for c in d["classes"]: + print(" rep=%-9s gcd=%d prim=%-5s cos4=%+.9f cos8=%+.9f u*n=%s" + % (str(c["rep"]), c["gcd"], c["primitive"], + c["cos4_float"], c["cos8_float"], + c["same_ell_realization"])) + if "projector_weights" in d: + for k, v in d["projector_weights"].items(): + print(" %s projector L1=%.6f L2=%.6f w=%s" + % (k, v["L1"], v["L2"], v["w_float"])) + print("->", out) + + +if __name__ == "__main__": + main() diff --git a/scripts/n325rec_s2_scout.py b/scripts/n325rec_s2_scout.py new file mode 100644 index 000000000..0f42b62cf --- /dev/null +++ b/scripts/n325rec_s2_scout.py @@ -0,0 +1,75 @@ +#!/usr/bin/env python3 +"""n325rec step 2 -- cheap telemetry: how big are the oblique safe automata? + +Builds PATH B's automaton (verbatim copy of rev769's +scripts/oblique_charge_transfer.py, i.e. dpfloor's B_oblique) for a list of +(direction, n) geometries and records, per sector, the safe-state count, the +row memory and the wall time. NO root finding, NO eigendecomposition. + +Cost gate (spec 2.3B): stop if a sector exceeds --state-cap. +""" +from __future__ import annotations +import json +import math +import sys +import time + +sys.path.insert(0, "/workspace/dpfloor/scripts") +import fl_path_oblique as B # the B_oblique reference implementation + + +# tag, u, n (ell = n*|u|) +GEOMS = [ + ("diag_n4", (1, 1), 4), # ell 5.657 (dpfloor regression) + ("diag_n5", (1, 1), 5), # ell 7.071 + ("slope21_n3", (2, 1), 3), # ell 6.708 + ("slope32_n2", (3, 2), 2), # ell 7.211 + ("slope31_n3", (3, 1), 3), # ell 9.487 + ("n325_1_18", (1, 18), 1), # N=325, ell 18.0278 + ("n325_6_17", (6, 17), 1), # N=325, ell 18.0278 + ("n325_2_3_n5", (2, 3), 5), # N=325, ell 5*sqrt(13)=18.0278 + ("n25_3_4", (3, 4), 1), # N=25, ell 5.0 (regression point) +] + + +def main(): + cap = int(sys.argv[1]) if len(sys.argv) > 1 else 200000 + out_path = sys.argv[2] if len(sys.argv) > 2 else \ + "/workspace/n325rec/out/s2_scout.json" + keep = sys.argv[3].split(",") if len(sys.argv) > 3 else None + res = {"schema": "n325rec.scout.v1", "path": "B_oblique", + "md5_note": "fl_path_oblique.py c402cdf6a0d92d317864d793b6eece44", + "state_cap": cap, "records": {}} + for tag, u, n in GEOMS: + if keep and tag not in keep: + continue + rec = {"tag": tag, "direction": list(u), "n": n, + "ell": n * math.hypot(*u)} + try: + for sector, matching in (("G4", False), ("G8", True)): + t0 = time.time() + a = B.ObliqueSafeTransfer(n, u, matching, state_cap=cap) + rec["%s_n_safe" % sector] = a.n + rec["%s_memory" % sector] = a.memory + rec["%s_edges" % sector] = [list(e) for e in a.edges] + rec["%s_seconds" % sector] = round(time.time() - t0, 2) + print("SCOUT %-13s %s n_safe=%-7d memory=%-2d edges=%s %.2fs" + % (tag, sector, a.n, a.memory, a.edges, + rec["%s_seconds" % sector]), flush=True) + rec["ok"] = True + except RuntimeError as exc: + rec["ok"] = False + rec["error"] = "%s (state cap %d)" % (exc, cap) + print("SCOUT %-13s ABORT %s" % (tag, exc), flush=True) + except Exception as exc: # noqa: BLE001 + rec["ok"] = False + rec["error"] = "%s: %s" % (type(exc).__name__, exc) + print("SCOUT %-13s ERROR %s" % (tag, exc), flush=True) + res["records"][tag] = rec + with open(out_path, "w") as fh: + json.dump(res, fh, indent=1) + print("->", out_path) + + +if __name__ == "__main__": + main() diff --git a/scripts/n325rec_s3_cert.py b/scripts/n325rec_s3_cert.py new file mode 100644 index 000000000..412a54baa --- /dev/null +++ b/scripts/n325rec_s3_cert.py @@ -0,0 +1,150 @@ +#!/usr/bin/env python3 +"""n325rec step 3 -- CERTIFICATE: does the independent oblique automaton build +the same object as PATH B? + +Three levels, weakest to strongest: + (i) frame identity: same Bezout complement and same transformed edge set; + (ii) automaton identity: same safe-state count, same row memory, and (for + matrices small enough) the same sorted real spectrum of the safe + transfer R(p) at a probe density -- a permutation-invariant fingerprint; + (iii) p_root identity: Delta(p)=0 solved by each path's own machinery in the + same p_c convention (0.5927460507921). +""" +from __future__ import annotations + +import json +import math +import sys +import time + +import numpy as np + +sys.path.insert(0, "/workspace/dpfloor/scripts") +sys.path.insert(0, "/workspace/n325rec/scripts") + +import fl_path_oblique as B # PATH B (reference) +import fl_common as C # shared p_c / root helper +import oblique_indep as N # my independent implementation + +PC = 0.5927460507921 +PROBE_P = 0.6 +SPECTRUM_CAP = 2600 + +GEOMS = [ + ("diag_n4", (1, 1), 4), + ("diag_n5", (1, 1), 5), + ("slope21_n3", (2, 1), 3), + ("slope21_n4", (2, 1), 4), + ("slope31_n3", (3, 1), 3), + ("slope32_n2", (3, 2), 2), + ("slope52_n2", (5, 2), 2), + ("axis_n4", (1, 0), 4), + ("axis_n6", (1, 0), 6), + ("axis_n8", (1, 0), 8), + ("n25_3_4", (3, 4), 1), + ("n25_0_1_n5", (0, 1), 5), +] + + +def b_root(u, n, pc, xtol=1e-16): + """PATH B, tight configuration, exactly as fl_path_oblique.run does it.""" + g4 = B.ObliqueSafeTransfer(n, u, False) + g8 = B.ObliqueSafeTransfer(n, u, True) + mode = "dense" if (g4.n <= 2500 and g8.n <= 2500) else "arpack" + + def eq(p): + l4, _ = g4.lambda0(p, mode=mode, arpack_tol=1e-15) + l8, _ = g8.lambda0(1.0 - p, mode=mode, arpack_tol=1e-15) + return math.log(l4) - math.log(l8) + + root, width, _nf, fres = C.solve_root(eq, 0.5, 0.8, xtol=xtol) + return root, fres, g4, g8 + + +def main(): + outp = sys.argv[1] if len(sys.argv) > 1 else \ + "/workspace/n325rec/out/s3_cert.json" + only = sys.argv[2].split(",") if len(sys.argv) > 2 else None + res = {"schema": "n325rec.certificate.v1", "p_c": PC, + "probe_p": PROBE_P, "records": {}} + for tag, u, n in GEOMS: + if only and tag not in only: + continue + t0 = time.time() + rec = {"tag": tag, "direction": list(u), "n": n, + "ell": n * math.hypot(*u)} + # ---- (i) frame identity + frame_ok = {} + for sect, matching in (("G4", False), ("G8", True)): + cb, eb, mb = B.transformed_edges(u, matching) + cn, en, mn = N.frame(u, matching) + frame_ok[sect] = { + "comp_B": list(cb), "comp_N": list(cn), + "edges_B": [list(e) for e in eb], + "edges_N": [list(e) for e in en], + "memory_B": mb, "memory_N": mn, + "identical": (list(cb) == list(cn) + and [list(e) for e in eb] == [list(e) for e in en] + and mb == mn), + } + rec["frame"] = frame_ok + # ---- (ii) automaton identity + aut = {} + for sect, matching in (("G4", False), ("G8", True)): + tb = B.ObliqueSafeTransfer(n, u, matching) + tn = N.IndependentOblique(n, u, matching) + entry = {"n_safe_B": tb.n, "n_safe_N": tn.n, + "memory_B": tb.memory, "memory_N": tn.memory, + "edges_B": [list(e) for e in tb.edges], + "edges_N": [list(e) for e in tn.edges], + "nnz_B": int(len(tb.counts)), "nnz_N": int(tn.nnz)} + if max(tb.n, tn.n) <= SPECTRUM_CAP and tb.n == tn.n: + Rb = tb.dense(PROBE_P) + Rn = tn.matrix_f64(PROBE_P) + sb = np.sort(np.linalg.eigvals(Rb).real) + sn = np.sort(np.linalg.eigvals(Rn).real) + entry["spectrum_max_abs_dev"] = float(np.max(np.abs(sb - sn))) + entry["spectrum_scale"] = float(np.max(np.abs(sb))) + entry["row_sum_max_abs_dev"] = float( + np.max(np.abs(np.sort(Rb.sum(axis=1)) + - np.sort(Rn.sum(axis=1))))) + aut[sect] = entry + rec["automaton"] = aut + # ---- (iii) p_root identity + try: + rb, fresb, b4, b8 = b_root(u, n, PC) + except Exception as exc: # noqa: BLE001 + rec["b_root_error"] = "%s: %s" % (type(exc).__name__, exc) + rb = None + try: + t4 = N.IndependentOblique(n, u, False) + t8 = N.IndependentOblique(n, u, True) + rn_ld, fresn, bracket = N.solve_p_root(t4, t8, PC, + dtype=np.longdouble) + rn = float(rn_ld) + except Exception as exc: # noqa: BLE001 + rec["n_root_error"] = "%s: %s" % (type(exc).__name__, exc) + rn = None + if rb is not None and rn is not None: + rec["p_root_B_tight"] = rb + rec["p_root_N_longdouble"] = repr(rn_ld) + rec["p_root_N_float"] = rn + rec["p_root_diff"] = float(rn - rb) + rec["Delta_at_root_B"] = float(fresb) + rec["Delta_at_root_N"] = float(fresn) + rec["bracket_width_N"] = float(bracket) + rec["Omega_diff"] = -(rn - rb) * (n * math.hypot(*u)) ** 4 + rec["seconds"] = round(time.time() - t0, 2) + res["records"][tag] = rec + print("CERT %-12s n_safe=%d/%d p_root_diff=%.3e spec_dev=%s %.1fs" + % (tag, aut["G4"]["n_safe_B"], aut["G8"]["n_safe_B"], + (rec.get("p_root_diff") or float("nan")), + aut["G4"].get("spectrum_max_abs_dev"), rec["seconds"]), + flush=True) + with open(outp, "w") as fh: + json.dump(res, fh, indent=1) + print("->", outp) + + +if __name__ == "__main__": + main() diff --git a/scripts/n325rec_s4_cost.py b/scripts/n325rec_s4_cost.py new file mode 100644 index 000000000..85ca9ad90 --- /dev/null +++ b/scripts/n325rec_s4_cost.py @@ -0,0 +1,94 @@ +#!/usr/bin/env python3 +"""n325rec step 4 -- COST PROBE (spec 2.3B cost gate). + +Uses the REFERENCE automaton (PATH B's own `step`/`transformed_edges`) and +reports, per (geometry, sector): the safe-state count if the build finishes, +otherwise the count reached when the state cap or the wall-clock cap trips. + +Also measures the size law along two families that pass through the N=325 +targets -- u=(1,k), n=1 (memory ~ k+1) and u=(2,3), n=1..5 (the (10,15) +orientation realizes as (2,3) with n=5) -- so the target size can be +extrapolated instead of guessed. +""" +from __future__ import annotations + +import json +import math +import sys +import time +from collections import deque + +sys.path.insert(0, "/workspace/dpfloor/scripts") +import fl_path_oblique as B # noqa: E402 + + +def build_probe(width, direction, matching, cap, tcap): + comp, edges, memory = B.transformed_edges(direction, matching) + start = B.empty_state(width, memory) + states = [start] + index = {start: 0} + queue = deque([start]) + t0 = time.time() + while queue: + if time.time() - t0 > tcap: + return None, len(states), time.time() - t0, "wallclock-cap" + s = queue.popleft() + for mask in range(1 << width): + nxt = B.step(s, mask, width, edges, memory) + if nxt is None: + continue + if nxt not in index: + if len(states) >= cap: + return None, len(states), time.time() - t0, "state-cap" + index[nxt] = len(states) + states.append(nxt) + queue.append(nxt) + return states, len(states), time.time() - t0, "done" + + +TARGETS = [ + ("n325_1_18", (1, 18), 1, 18.027756377319946), + ("n325_6_17", (6, 17), 1, 18.027756377319946), + ("n325_2_3_n5", (2, 3), 5, 18.027756377319946), + # the same three "directions" at a SMALLER index/ell, as a size law + ("f_1_18_n1", (1, 18), 1, None), + ("f_1_12_n1", (1, 12), 1, None), + ("f_1_10_n1", (1, 10), 1, None), + ("f_1_8_n1", (1, 8), 1, None), + ("f_1_6_n1", (1, 6), 1, None), + ("f_2_3_n4", (2, 3), 4, None), + ("f_2_3_n3", (2, 3), 3, None), + ("f_2_3_n2", (2, 3), 2, None), + ("f_3_4_n1", (3, 4), 1, None), + ("f_3_4_n2", (3, 4), 2, None), + ("f_3_4_n3", (3, 4), 3, None), +] + + +def main(): + cap = int(sys.argv[1]) if len(sys.argv) > 1 else 1000000 + tcap = float(sys.argv[2]) if len(sys.argv) > 2 else 180.0 + outp = sys.argv[3] if len(sys.argv) > 3 else \ + "/workspace/n325rec/out/s4_cost.json" + only = sys.argv[4].split(",") if len(sys.argv) > 4 else None + res = {"schema": "n325rec.cost.v1", "state_cap": cap, "wall_cap_s": tcap, + "records": {}} + for tag, u, n, _ell in TARGETS: + if only and tag not in only: + continue + rec = {"tag": tag, "direction": list(u), "n": n, + "ell": n * math.hypot(*u)} + for sect, matching in (("G4", False), ("G8", True)): + _st, cnt, secs, why = build_probe(n, u, matching, cap, tcap) + rec[sect] = {"states": cnt, "seconds": round(secs, 2), + "status": why} + print("COST %-13s %s %-13s states=%-9d %6.2fs" + % (tag, sect, why, cnt, secs), flush=True) + res["records"][tag] = rec + with open(outp, "w") as fh: + json.dump(res, fh, indent=1) + print("->", outp) + + +if __name__ == "__main__": + main() diff --git a/scripts/n325rec_s5_rehearsal.py b/scripts/n325rec_s5_rehearsal.py new file mode 100644 index 000000000..0f901deca --- /dev/null +++ b/scripts/n325rec_s5_rehearsal.py @@ -0,0 +1,163 @@ +#!/usr/bin/env python3 +"""n325rec step 5 -- the rehearsal algebra that does NOT need the automaton. + +Three things: + (A) SCAN: for which circumferences ell=sqrt(M) do >=3 D4-distinct orientations + exist at the SAME ell? (same-ell means n^2*(a^2+b^2) equal for all points.) + For each candidate set the frame memory / frontier size is printed, so + "change the modulus" can be answered with a cost estimate. + (B) N=325 design: A = [[1,cos4,cos8]] exact, its inverse, the noise + propagation from the dpfloor floor F=4.68e-16 into the fitted harmonic + coefficients, and N=1105 for comparison. + (C) The confound: at a FIXED ell the 3-orientation fit determines + Q_m = P_m + B_m/ell^2 + ..., never P_m alone. Using n1105mix's own M3 + global fit (imported, clearly marked) we quantify how large the ell^-2 + H0/H8 split is compared with the intrinsic H0/H8 split, i.e. whether a + fixed-ell rehearsal can test P0=P8 (or B0=B8) even at zero noise. +""" +from __future__ import annotations + +import json +import math +import sys +from fractions import Fraction + +sys.path.insert(0, "/workspace/n325rec/scripts") +import oblique_indep as N # noqa: E402 +import s1_orient as S # noqa: E402 (exact helpers) + +FLOOR_F = 4.68e-16 # dpfloor's adopted implementation floor +PC = 0.5927460507921 + +# n1105mix's own 10-point M3 global fit (imported for the confound estimate +# ONLY -- clearly marked as an external model, not measured here): +M3 = {"P0": +0.00279, "P4": +0.29043, "P8": +0.00257, + "dP0": 0.00109, "dP4": 0.00231, "dP8": 0.00129, + "B0": -0.18453, "B4": +0.45446, "B8": -0.04185, + "dB0": 0.06773, "dB4": 0.14384, "dB8": 0.08362} +C_AXIAL = 3.54 # the ell^-4 coefficient n1105mix fitted + + +def scan(Nmax=900, ellmax=40.0): + out = [] + for M in range(1, Nmax + 1): + ell = math.sqrt(M) + if ell > ellmax: + break + found = {} + for a in range(0, int(ell) + 1): + for b in range(0, a + 1): + if a == 0 and b == 0: + continue + if math.gcd(a, b) != 1: + continue + L2 = a * a + b * b + for n in range(1, int(ellmax / math.sqrt(L2)) + 1): + if n * n * L2 == M: + key = (a, b) if a >= b else (b, a) + found.setdefault(key, []).append(n) + if len(found) >= 3: + ents = [] + for (a, b), ns in sorted(found.items()): + u = (a, b) + _, e4, m4 = N.frame(u, False) + _, e8, m8 = N.frame(u, True) + ents.append({"u": [a, b], "n": ns, "cos4": float( + S.cos4m(a, b, M, 1) if a * b else Fraction( + (a ** 4 - 6 * a * a * b * b + b ** 4), + (a * a + b * b) ** 2)), + "memory_G4": m4, "memory_G8": m8, + "frontier_G4": m4 * ns[-1], "frontier_G8": m8 * ns[-1]}) + out.append({"M": M, "ell": ell, "n_classes": len(found), + "orientations": ents}) + return out + + +def design_and_noise(M, rows): + """rows: list of (a,b). Exact A, exact/inverse, noise propagation.""" + A = [[Fraction(1)] + [S.cos4m(a, b, M, m) for m in (1, 2)] for a, b in rows] + n = len(rows) + if n != 3: + return None + Mmat = [[A[j][i] for j in range(n)] for i in range(n)] # A^T + Ainv = [] + for t in range(n): + e = [Fraction(1 if k == t else 0) for k in range(n)] + w, _r, ok = S.solve_exact([r[:] for r in Mmat], e) + if not ok: + return None + Ainv.append(w) # = A^{-T} + Ainv = [[Ainv[j][i] for j in range(n)] for i in range(n)] # A^{-1} + amps = [math.sqrt(sum(float(Ainv[m][i]) ** 2 for i in range(n))) + for m in range(n)] + ell4 = Fraction(M) ** 2 + sigma_omega = FLOOR_F * float(ell4) + return {"M": M, "ell": math.sqrt(M), "ell4_exact": str(ell4), + "A": [[str(x) for x in r] for r in A], + "A_inverse": [[float(x) for x in r] for r in Ainv], + "row_l2_norm_of_Ainv": amps, + "sigma_Omega_from_F": sigma_omega, + "sigma_coefficient": [sigma_omega * a for a in amps], + "sigma_coefficient_exact_ell4": [FLOOR_F * a for a in amps]} + + +def main(): + outp = sys.argv[1] if len(sys.argv) > 1 else \ + "/workspace/n325rec/out/s5_rehearsal.json" + res = {"schema": "n325rec.rehearsal-algebra.v1", "p_c": PC, + "floor_F": FLOOR_F, "imported_model_M3": M3, + "imported_C_axial": C_AXIAL} + # ---- (A) which ell admit >=3 same-ell D4 classes + sc = scan() + res["same_ell_scan"] = sc + print("(A) circumferences with >=3 D4-distinct orientations (ell<=40):") + for e in sc[:8]: + print(" M=%-5d ell=%6.3f classes=%d %s" + % (e["M"], e["ell"], e["n_classes"], + ", ".join("%s n=%s mem=%d/%d" + % (o["u"], o["n"], o["memory_G4"], o["memory_G8"]) + for o in e["orientations"]))) + print(" total candidates up to ell=40: %d" % len(sc)) + # ---- (B) N=325 and N=1105 designs + res["design_325"] = design_and_noise(325, [(1, 18), (6, 17), (10, 15)]) + res["design_1105"] = design_and_noise(1105, [(4, 33), (9, 32), (12, 31), + (23, 24)]) + print("(B) N=325 sigma(coeff) from F=4.68e-16:", + ["%.3e" % x for x in res["design_325"]["sigma_coefficient"]]) + # ---- (C) the fixed-ell confound + conf = {} + for tag, M in (("N=325", 325), ("N=1105", 1105)): + e2 = 1.0 / M + conf[tag] = { + "ell": math.sqrt(M), + "B0_over_ell2": M3["B0"] * e2, + "B4_over_ell2": M3["B4"] * e2, + "B8_over_ell2": M3["B8"] * e2, + "B0_minus_B8_over_ell2": (M3["B0"] - M3["B8"]) * e2, + "P0_minus_P8_imported": M3["P0"] - M3["P8"], + "P0_minus_P8_imported_sigma": + math.hypot(M3["dP0"], M3["dP8"]), + "C_over_ell4": C_AXIAL / M ** 2, + "confound_over_intrinsic_signal": + abs((M3["B0"] - M3["B8"]) * e2) + / max(abs(M3["P0"] - M3["P8"]), + math.hypot(M3["dP0"], M3["dP8"])), + "sigma_Omega_from_F": FLOOR_F * M ** 2, + "noise_vs_confound": (FLOOR_F * M ** 2) + / abs((M3["B0"] - M3["B8"]) * e2), + } + res["fixed_ell_confound"] = conf + print("(C) fixed-ell confound (imported M3):") + for k, v in conf.items(): + print(" %-7s ell=%6.3f (B0-B8)/ell^2=%+.3e P0-P8=%+.3e+-%.1e " + "confound/signal=%.2f noise/confound=%.2e" + % (k, v["ell"], v["B0_minus_B8_over_ell2"], + v["P0_minus_P8_imported"], v["P0_minus_P8_imported_sigma"], + v["confound_over_intrinsic_signal"], v["noise_vs_confound"])) + with open(outp, "w") as fh: + json.dump(res, fh, indent=1) + print("->", outp) + + +if __name__ == "__main__": + main() diff --git a/scripts/n325rec_s6_pair.py b/scripts/n325rec_s6_pair.py new file mode 100644 index 000000000..f870a929a --- /dev/null +++ b/scripts/n325rec_s6_pair.py @@ -0,0 +1,138 @@ +#!/usr/bin/env python3 +"""n325rec step 6 -- the small-scale rehearsal that IS affordable. + +N=325's three-orientation rehearsal is blocked (step 4/5: every orientation +exceeds the 200k-state gate, and ell=sqrt(325) is already the SMALLEST +circumference with >=3 same-ell D4 classes). The smallest same-ell +multi-orientation set is + + M = ell^2 = 25 (ell = 5): u=(0,1) with n=5 and u=(3,4) with n=1 + +-- two D4-distinct orientations at ONE fixed ell, both tiny automata +(G4/G8 = 45/147 states each). + +This script runs BOTH implementations on both orientations, forms the exact +design matrix A=[1,cos4], fits the same-ell harmonic coefficients +Q = A^{-1} Omega for each implementation, propagates the measured oblique +dispersion into error bars, and quantifies the fixed-ell confound. +Self-contained (does not depend on the certificate file). +""" +from __future__ import annotations + +import json +import math +import sys +import time +from fractions import Fraction + +import numpy as np + +sys.path.insert(0, "/workspace/dpfloor/scripts") +sys.path.insert(0, "/workspace/n325rec/scripts") + +import fl_path_oblique as B # noqa: E402 +import fl_common as C # noqa: E402 +import oblique_indep as N # noqa: E402 +import s1_orient as S # noqa: E402 + +PC = 0.5927460507921 +M3 = {"P0": +0.00279, "P4": +0.29043, "B0": -0.18453, "B4": +0.45446} +PAIR = [("n25_0_1_n5", (0, 1), 5), ("n25_3_4", (3, 4), 1)] + + +def main(): + outp = sys.argv[1] if len(sys.argv) > 1 else \ + "/workspace/n325rec/out/s6_pair.json" + res = {"schema": "n325rec.pair-rehearsal.v1", "p_c": PC, + "pair_M": 25, "pair_ell": 5.0, "orientations": {}} + for tag, u, n in PAIR: + t0 = time.time() + b4 = B.ObliqueSafeTransfer(n, u, False) + b8 = B.ObliqueSafeTransfer(n, u, True) + mode = "dense" if (b4.n <= 2500 and b8.n <= 2500) else "arpack" + + def eq(p): + l4, _ = b4.lambda0(p, mode=mode, arpack_tol=1e-15) + l8, _ = b8.lambda0(1.0 - p, mode=mode, arpack_tol=1e-15) + return math.log(l4) - math.log(l8) + + rb, _w, _nf, fb = C.solve_root(eq, 0.5, 0.8, xtol=1e-16) + n4 = N.IndependentOblique(n, u, False) + n8 = N.IndependentOblique(n, u, True) + rn, fn, bracket = N.solve_p_root(n4, n8, PC, dtype=np.longdouble) + ell = n * math.hypot(*u) + res["orientations"][tag] = { + "u": list(u), "n": n, "ell": ell, + "n_safe_G4_B": b4.n, "n_safe_G4_N": n4.n, + "n_safe_G8_B": b8.n, "n_safe_G8_N": n8.n, + "counts_match": (b4.n == n4.n and b8.n == n8.n), + "p_root_B": rb, "p_root_N": repr(rn), + "p_root_diff": float(rn) - rb, + "Delta_at_root_B": float(fb), "Delta_at_root_N": float(fn), + "Omega_B": float(-(rb - PC) * ell ** 4), + "Omega_N": float(-(float(rn) - PC) * ell ** 4), + "seconds": round(time.time() - t0, 2), + } + print("%-12s counts %s (%d/%d) p_root_diff=%+.3e" + % (tag, "OK" if res["orientations"][tag]["counts_match"] + else "MISMATCH", b4.n, b8.n, + res["orientations"][tag]["p_root_diff"]), flush=True) + + ell4 = 25 ** 2 + rows = [u for _t, u, _n in PAIR] + A = [[Fraction(1), S.cos4m(a, b, 25, 1)] for a, b in rows] + n_ = 2 + Abar = [[A[j][i] for j in range(n_)] for i in range(n_)] + Ainv = [] + for t in range(n_): + e = [Fraction(1 if k == t else 0) for k in range(n_)] + w, _r, ok = S.solve_exact([r[:] for r in Abar], e) + if not ok: + raise SystemExit("singular design matrix") + Ainv.append(w) + Ainv = [[Ainv[j][i] for j in range(n_)] for i in range(n_)] + amps = [math.sqrt(sum(float(Ainv[m][i]) ** 2 for i in range(n_))) + for m in range(n_)] + disp = max(abs(res["orientations"][t]["p_root_diff"]) for t, _u, _n in PAIR) + F_oblique = disp + fits = {} + for path in ("B", "N"): + o = [res["orientations"][t]["Omega_" + path] for t, _u, _n in PAIR] + Q = [sum(float(Ainv[m][i]) * o[i] for i in range(n_)) for m in range(n_)] + fits[path] = {"Q": Q, "sigma_from_measured_dispersion": + [F_oblique * ell4 * amps[m] for m in range(n_)]} + e2 = 1.0 / 25 + res["design"] = { + "A_exact": [[str(x) for x in r] for r in A], + "rank_exact": S.rank_exact(A), + "A_inverse": [[float(x) for x in r] for r in Ainv], + "row_l2_norm": amps, + "ell4_exact": ell4, + } + res["fit"] = fits + res["measured_oblique_dispersion"] = F_oblique + res["confound"] = { + "B0_over_ell2": M3["B0"] * e2, + "B4_over_ell2": M3["B4"] * e2, + "B0_minus_B4_over_ell2": (M3["B0"] - M3["B4"]) * e2, + "P0_minus_P4_imported": M3["P0"] - M3["P4"], + "ratio_confound_over_signal": abs((M3["B0"] - M3["B4"]) * e2) + / abs(M3["P0"] - M3["P4"]), + } + print("A rank=%d Omega_B=%s" % (res["design"]["rank_exact"], + ["%.12f" % res["orientations"][t]["Omega_B"] for t, _u, _n in PAIR])) + print("Q(B)=%s Q(N)=%s dQ=%s" + % (["%.9f" % x for x in fits["B"]["Q"]], + ["%.9f" % x for x in fits["N"]["Q"]], + ["%.2e" % (fits["N"]["Q"][m] - fits["B"]["Q"][m]) + for m in range(n_)])) + print("sigma from measured dispersion %.2e: %s" + % (F_oblique, ["%.2e" % x + for x in fits["B"]["sigma_from_measured_dispersion"]])) + with open(outp, "w") as fh: + json.dump(res, fh, indent=1) + print("->", outp) + + +if __name__ == "__main__": + main() diff --git a/scripts/n325rec_s7_basis.py b/scripts/n325rec_s7_basis.py new file mode 100644 index 000000000..f09ed9032 --- /dev/null +++ b/scripts/n325rec_s7_basis.py @@ -0,0 +1,91 @@ +#!/usr/bin/env python3 +"""n325rec step 7 -- basis-invariance of the frame (the SHARED layer S1). + +The Bezout complement is NOT unique. Sliding it by the period, + v -> v + n*u , i.e. (c,d) -> (c + n*a, d + n*b), +leaves det(u,v)=1 and leaves the quotient lattice IDENTICAL (because +n*u is already in the lattice), but it changes the edge set (ds,dt), the row +memory and therefore the whole automaton. So this is a genuinely different +transfer matrix for the SAME physical cylinder: the two p_root values must +agree. This tests the frame layer that two same-convention paths cannot test. + +Run with the independent implementation only (B's class hard-codes the +extended-gcd complement). +""" +from __future__ import annotations + +import json +import math +import sys +import time + +import numpy as np + +sys.path.insert(0, "/workspace/n325rec/scripts") +import oblique_indep as N # noqa: E402 + +PC = 0.5927460507921 +GEOMS = [ + ("diag_n4", (1, 1), 4), + ("diag_n5", (1, 1), 5), + ("slope21_n3", (2, 1), 3), + ("slope32_n2", (3, 2), 2), + ("axis_n8", (1, 0), 8), + ("n25_3_4", (3, 4), 1), +] + + +def main(): + outp = sys.argv[1] if len(sys.argv) > 1 else \ + "/workspace/n325rec/out/s7_basis.json" + res = {"schema": "n325rec.basis-invariance.v1", "p_c": PC, "records": {}} + for tag, u, n in GEOMS: + a, b = u + c0, d0 = N.bezout_complement_iter(a, b) + alt = (c0 + n * a, d0 + n * b) + rec = {"tag": tag, "direction": list(u), "n": n, + "comp_base": [c0, d0], "comp_shifted": list(alt), + "det_check_base": a * d0 - b * c0, + "det_check_shifted": a * alt[1] - b * alt[0], + "same_sublattice": bool(n * a * alt[1] - n * b * alt[0] + == n * a * d0 - n * b * c0)} + try: + out = [] + for name, comp in (("base", (c0, d0)), ("shifted", alt)): + t0 = time.time() + t4 = N.IndependentOblique(n, u, False, complement=comp) + t8 = N.IndependentOblique(n, u, True, complement=comp) + root, resid, width = N.solve_p_root(t4, t8, PC, + dtype=np.longdouble) + out.append({"name": name, "comp": list(comp), + "edges_G4": [list(e) for e in t4.edges], + "edges_G8": [list(e) for e in t8.edges], + "memory_G4": t4.memory, "memory_G8": t8.memory, + "n_safe_G4": t4.n, "n_safe_G8": t8.n, + "p_root": repr(root), "Delta": float(resid), + "seconds": round(time.time() - t0, 2)}) + rec["runs"] = out + rec["p_root_diff_shifted_minus_base"] = \ + float(np.longdouble(out[1]["p_root"]) + - np.longdouble(out[0]["p_root"])) + rec["ok"] = True + except Exception as exc: # noqa: BLE001 + rec["error"] = "%s: %s" % (type(exc).__name__, exc) + rec["ok"] = False + res["records"][tag] = rec + if rec.get("ok"): + print("BASIS %-12s base %s n_safe=%d/%d -> shifted n_safe=%d/%d " + "diff=%.3e" + % (tag, rec["comp_base"], rec["runs"][0]["n_safe_G4"], + rec["runs"][0]["n_safe_G8"], rec["runs"][1]["n_safe_G4"], + rec["runs"][1]["n_safe_G8"], + rec["p_root_diff_shifted_minus_base"]), flush=True) + else: + print("BASIS %-12s ERROR %s" % (tag, rec["error"]), flush=True) + with open(outp, "w") as fh: + json.dump(res, fh, indent=1) + print("->", outp) + + +if __name__ == "__main__": + main() diff --git a/scripts/n325rec_s8_lattice.py b/scripts/n325rec_s8_lattice.py new file mode 100644 index 000000000..ae30e3316 --- /dev/null +++ b/scripts/n325rec_s8_lattice.py @@ -0,0 +1,99 @@ +#!/usr/bin/env python3 +"""n325rec step 8 -- does the "same ell" quantity depend on the frame gauge? + +The Bezout complement v is defined only up to v -> v + k*u. For n | k the +QUOTIENT LATTICE is unchanged (step 7: p_root identical to the last +bit). For n NOT dividing k the sublattice -- hence the physical n-site torus +on the square lattice -- is DIFFERENT while det(n*u,v)=n is the same, so the +circumference ell = n*|u| is the same as well. + +If those give different p_root, then "Omega(theta, ell)" is not a function of +(theta, ell) alone: it carries an extra discrete argument (which n-site torus / +which transversal), and no same-ell multi-orientation fit can be interpreted +without fixing it. This is a systematic effect that no amount of path +duplication can detect, so it must be measured separately. + +Only the two smallest oblique geometries are probed (cost). +""" +from __future__ import annotations + +import json +import math +import sys + +import numpy as np + +sys.path.insert(0, "/workspace/n325rec/scripts") +import oblique_indep as N # noqa: E402 + +PC = 0.5927460507921 + +CASES = [ + ("diag_n4", (1, 1), 4, [(0, 1), (1, 2), (2, 3), (3, 4)]), + ("slope21_n3", (2, 1), 3, [(-1, 0), (1, 1), (3, 2)]), + ("diag_n5", (1, 1), 5, [(0, 1), (1, 2), (2, 3)]), +] + + +def main(): + outp = sys.argv[1] if len(sys.argv) > 1 else \ + "/workspace/n325rec/out/s8_lattice.json" + res = {"schema": "n325rec.frame-gauge.v1", "p_c": PC, + "note": ("for each lattice basis the automaton is rebuilt from " + "scratch; det(n*u,v)=n and |n*u| are identical by " + "construction"), "records": {}} + for tag, u, n, comps in CASES: + a, b = u + rec = {"tag": tag, "direction": list(u), "n": n, + "ell": n * math.hypot(a, b), "runs": []} + for comp in comps: + c, d = comp + if a * d - b * c != 1: + rec["runs"].append({"comp": list(comp), "error": "det != 1"}) + continue + # gauge class: v -> v + k*u, k = ((c-c0)*... ) -- just report k + try: + t4 = N.IndependentOblique(n, u, False, complement=comp) + t8 = N.IndependentOblique(n, u, True, complement=comp) + root, resid, width = N.solve_p_root(t4, t8, PC, + dtype=np.longdouble) + rec["runs"].append({ + "comp": list(comp), + "torus_sites": abs(n * a * d - n * b * c), + "n_safe_G4": t4.n, "n_safe_G8": t8.n, + "memory_G4": t4.memory, "memory_G8": t8.memory, + "p_root": repr(root), + "p_root_float": float(root), + "Omega": float(-(root - PC) * (n * math.hypot(a, b)) ** 4), + "Delta": float(resid)}) + except Exception as exc: # noqa: BLE001 + rec["runs"].append({"comp": list(comp), + "error": "%s: %s" % (type(exc).__name__, + exc)}) + ok = [r for r in rec["runs"] if "p_root_float" in r] + if ok: + base = ok[0]["p_root_float"] + rec["spread_p_root"] = max(abs(r["p_root_float"] - base) + for r in ok) + rec["spread_Omega"] = max(abs(r["Omega"] - ok[0]["Omega"]) + for r in ok) + res["records"][tag] = rec + print("GAUGE %-12s ell=%.4f" % (tag, rec["ell"]), flush=True) + for r in rec["runs"]: + if "p_root_float" in r: + print(" v=%-8s states=%d/%d p_root=%.17f Omega=%+.12e" + % (r["comp"], r["n_safe_G4"], r["n_safe_G8"], + r["p_root_float"], r["Omega"]), flush=True) + else: + print(" v=%-8s %s" % (r["comp"], r.get("error")), + flush=True) + if "spread_p_root" in rec: + print(" -> spread(p_root)=%.3e spread(Omega)=%.3e" + % (rec["spread_p_root"], rec["spread_Omega"]), flush=True) + with open(outp, "w") as fh: + json.dump(res, fh, indent=1) + print("->", outp) + + +if __name__ == "__main__": + main() diff --git a/scripts/n325rec_s9_final.py b/scripts/n325rec_s9_final.py new file mode 100644 index 000000000..75c4342f6 --- /dev/null +++ b/scripts/n325rec_s9_final.py @@ -0,0 +1,177 @@ +#!/usr/bin/env python3 +"""n325rec step 9 -- FINAL two-path comparison on the reference geometry set. + +Reference numbers are dpfloor's OWN recorded PATH B values, read from + /workspace/dpfloor/out/pathB_oblique_tight.json (tight, p_c=0.5927460507921) + /workspace/dpfloor/out/pathB_oblique_tight2.json (tight, same p_c) +(these were produced by fl_path_oblique.py, md5 c402cdf6a0d92d317864d793b6eece4, +i.e. the same code as B_oblique), so the comparison needs no re-run of B's +expensive brentq/dense-eig pipeline. + +For every geometry: + * build the INDEPENDENT automaton (G4, G8) and record states / memory / time; + * compare n_safe and row memory against the recorded PATH B values; + * solve Delta(p)=0 with the independent longdouble machinery and compare with + the recorded PATH B p_root; + * for n_safe <= 1200 also rebuild PATH B's matrix and compare the sorted real + spectrum of R(0.6) elementwise -- a permutation-invariant fingerprint. + +The two geometries with no recorded reference (N=25's two orientations) get a +full PATH B run (they are tiny). +""" +from __future__ import annotations + +import json +import math +import sys +import time + +import numpy as np + +sys.path.insert(0, "/workspace/dpfloor/scripts") +sys.path.insert(0, "/workspace/n325rec/scripts") + +import fl_path_oblique as B # noqa: E402 +import fl_common as C # noqa: E402 +import oblique_indep as N # noqa: E402 + +PC = 0.5927460507921 +PROBE_P = 0.6 +SPECTRUM_CAP = 1200 +REF_FILES = [ + "/workspace/dpfloor/out/pathB_oblique_tight.json", + "/workspace/dpfloor/out/pathB_oblique_tight2.json", +] + +GEOMS = [ + ("axis_n4", (1, 0), 4), + ("axis_n6", (1, 0), 6), + ("axis_n8", (1, 0), 8), + ("diag_n4", (1, 1), 4), + ("diag_n5", (1, 1), 5), + ("slope21_n3", (2, 1), 3), + ("slope21_n4", (2, 1), 4), + ("slope31_n3", (3, 1), 3), + ("slope32_n2", (3, 2), 2), + ("slope52_n2", (5, 2), 2), + ("n25_3_4", (3, 4), 1), + ("n25_0_1_n5", (0, 1), 5), +] + + +def b_tight(u, n): + """PATH B tight pipeline (exactly fl_path_oblique.run's tight branch).""" + g4 = B.ObliqueSafeTransfer(n, u, False) + g8 = B.ObliqueSafeTransfer(n, u, True) + mode = "dense" if (g4.n <= 2500 and g8.n <= 2500) else "arpack" + + def eq(p): + l4, _ = g4.lambda0(p, mode=mode, arpack_tol=1e-15) + l8, _ = g8.lambda0(1.0 - p, mode=mode, arpack_tol=1e-15) + return math.log(l4) - math.log(l8) + + root, _w, _nf, fres = C.solve_root(eq, 0.5, 0.8, xtol=1e-16) + return root, fres, g4, g8 + + +def main(): + outp = sys.argv[1] if len(sys.argv) > 1 else \ + "/workspace/n325rec/out/s9_final.json" + ref = {} + for f in REF_FILES: + d = json.load(open(f)) + for k, v in d["records"].items(): + ref[k] = {"p_root": v["p_root"], "n_safe_G4": v["n_safe_G4"], + "n_safe_G8": v["n_safe_G8"], + "memory_G4": v["row_memory_G4"], + "memory_G8": v["row_memory_G8"], + "source": f.split("/")[-1]} + res = {"schema": "n325rec.final-comparison.v1", "p_c": PC, + "reference": "dpfloor PATH B recorded values (" + + ", ".join(f.split("/")[-1] for f in REF_FILES) + ")", + "records": {}} + for tag, u, n in GEOMS: + t0 = time.time() + rec = {"tag": tag, "direction": list(u), "n": n, + "ell": n * math.hypot(*u)} + t4 = N.IndependentOblique(n, u, False) + t8 = N.IndependentOblique(n, u, True) + build_s = time.time() - t0 + root, resid, bracket = N.solve_p_root(t4, t8, PC, dtype=np.longdouble) + ell = rec["ell"] + a, b = u + rec.update({ + "edges_G4": [list(e) for e in t4.edges], + "edges_G8": [list(e) for e in t8.edges], + "comp": list(t4.comp), + "nnz_G4": t4.nnz, "nnz_G8": t8.nnz, + "memory_G4": t4.memory, "memory_G8": t8.memory, + "n_safe_G4_indep": t4.n, "n_safe_G8_indep": t8.n, + "build_seconds": round(build_s, 2), + "p_root_indep": repr(root), + "p_root_indep_float": float(root), + "root_minus_pc": float(root - PC), + "Delta_at_root": float(resid), + "bracket_width": float(bracket), + "Omega": float(-(root - PC) * ell ** 4), + "cos4": (a ** 4 - 6 * a * a * b * b + b ** 4) + / (a * a + b * b) ** 2, + "seconds": round(time.time() - t0, 2)}) + if tag in ref: + rec["reference_source"] = ref[tag]["source"] + rec["p_root_ref"] = ref[tag]["p_root"] + rec["p_root_diff"] = float(root) - ref[tag]["p_root"] + rec["n_safe_match"] = (t4.n == ref[tag]["n_safe_G4"] + and t8.n == ref[tag]["n_safe_G8"]) + rec["memory_match"] = (t4.memory == ref[tag]["memory_G4"] + and t8.memory == ref[tag]["memory_G8"]) + else: + rb, fresb, b4, b8 = b_tight(u, n) + rec["reference_source"] = "computed here (PATH B tight)" + rec["p_root_ref"] = rb + rec["p_root_diff"] = float(root) - rb + rec["Delta_at_root_B"] = float(fresb) + rec["n_safe_match"] = (t4.n == b4.n and t8.n == b8.n) + rec["memory_match"] = (t4.memory == b4.memory + and t8.memory == b8.memory) + if max(t4.n, t8.n) <= SPECTRUM_CAP and tag in ref: + b4 = B.ObliqueSafeTransfer(n, u, False) + b8 = B.ObliqueSafeTransfer(n, u, True) + devs = {} + for sect, tb, tn in (("G4", b4, t4), ("G8", b8, t8)): + sb = np.sort(np.linalg.eigvals(tb.dense(PROBE_P)).real) + sn = np.sort(np.linalg.eigvals(tn.matrix_f64(PROBE_P)).real) + devs[sect] = {"max_abs_dev": float(np.max(np.abs(sb - sn))), + "scale": float(np.max(np.abs(sb)))} + rec["spectrum_fingerprint"] = devs + rec["spectrum_max_abs_dev"] = max( + devs[s]["max_abs_dev"] for s in devs) + res["records"][tag] = rec + print("FIN %-12s n_safe %s p_root_diff=%+.3e spec=%s %.1fs" + % (tag, "OK" if rec["n_safe_match"] else "MISMATCH", + rec["p_root_diff"], rec.get("spectrum_max_abs_dev"), + rec["seconds"]), flush=True) + with open(outp, "w") as fh: + json.dump(res, fh, indent=1) + ds = [abs(v["p_root_diff"]) for v in res["records"].values() + if "p_root_diff" in v] + obl = [abs(v["p_root_diff"]) for k, v in res["records"].items() + if "p_root_diff" in v and not k.startswith("axis")] + res["summary"] = { + "n_geometries": len(res["records"]), + "all_state_counts_match": all(v.get("n_safe_match") + for v in res["records"].values()), + "all_memories_match": all(v.get("memory_match") + for v in res["records"].values()), + "max_abs_p_root_diff_all": max(ds) if ds else None, + "max_abs_p_root_diff_oblique": max(obl) if obl else None, + "F_oblique_from_this_comparison": max(obl) if obl else None, + } + print("SUMMARY", json.dumps(res["summary"], indent=1)) + with open(outp, "w") as fh: + json.dump(res, fh, indent=1) + print("->", outp) + + +if __name__ == "__main__": + main() diff --git a/scripts/n325rec_s9b_big.py b/scripts/n325rec_s9b_big.py new file mode 100644 index 000000000..0eabd5a0e --- /dev/null +++ b/scripts/n325rec_s9b_big.py @@ -0,0 +1,54 @@ +#!/usr/bin/env python3 +"""n325rec step 9b -- the largest oblique geometry (slope52_n2, G8=131677) with +the float64 branch of the independent solver, compared against PATH B's +recorded tight value (that geometry is n>2500 on BOTH sides, so B is on ARPACK). +Cost gate: this is the only geometry whose longdouble root was too slow.""" +from __future__ import annotations +import json, math, sys, time +import numpy as np +sys.path.insert(0, "/workspace/n325rec/scripts") +import oblique_indep as N + +PC = 0.5927460507921 +REF = json.load(open("/workspace/dpfloor/out/pathB_oblique_tight2.json"))["records"]["slope52_n2"] +u, n = (5, 2), 2 + +def main(): + outp = sys.argv[1] if len(sys.argv) > 1 else "/workspace/n325rec/out/s9b_big.json" + t0 = time.time() + t4 = N.IndependentOblique(n, u, False) + t8 = N.IndependentOblique(n, u, True) + build = time.time() - t0 + out = {"tag": "slope52_n2", "direction": list(u), "n": n, + "n_safe_G4_indep": t4.n, "n_safe_G8_indep": t8.n, + "n_safe_G4_ref": REF["n_safe_G4"], "n_safe_G8_ref": REF["n_safe_G8"], + "counts_match": (t4.n == REF["n_safe_G4"] and t8.n == REF["n_safe_G8"]), + "memory_indep": [t4.memory, t8.memory], + "memory_ref": [REF["row_memory_G4"], REF["row_memory_G8"]], + "nnz": [t4.nnz, t8.nnz], "build_seconds": round(build, 1), + "p_root_ref": REF["p_root"]} + t1 = time.time() + rf, resid, bracket = None, None, None + for tol, xtol in ((1e-14, 1e-16), (1e-12, 3e-16)): + try: + rf, resid, bracket = N.solve_p_root(t4, t8, PC, dtype=np.float64, + xtol=xtol, tol=tol) + out["solver_tol_used"] = tol + break + except Exception as exc: # noqa: BLE001 + out["error_%g" % tol] = "%s: %s" % (type(exc).__name__, exc) + if rf is not None: + ell = n * math.hypot(*u) + out.update({"p_root_indep_float64": rf, "root_minus_pc": rf - PC, + "Delta_at_root": float(resid), + "bracket_width": float(bracket), + "p_root_diff": float(rf) - REF["p_root"], + "Omega": -(rf - PC) * ell ** 4, + "solve_seconds": round(time.time() - t1, 1)}) + out["seconds"] = round(time.time() - t0, 1) + with open(outp, "w") as fh: + json.dump(out, fh, indent=1) + print(json.dumps({k: v for k, v in out.items()}, indent=1), flush=True) + print("->", outp) + +main() diff --git a/scripts/oblique_charge_transfer.py b/scripts/oblique_charge_transfer.py new file mode 100644 index 000000000..dea6d26a4 --- /dev/null +++ b/scripts/oblique_charge_transfer.py @@ -0,0 +1,352 @@ +#!/usr/bin/env python3 +"""Generic primitive-direction safe charge transfer for square-site percolation. + +Choose a primitive integer circumference direction u=(a,b). Complete it to +an SL(2,Z) basis (u,v). In coordinates x=s*u+t*v, quotient s modulo n and +advance in t. Physical NN/matching edges become a fixed finite-range edge set +(ds,dt); a frontier memory of R=max positive dt is exact. + +The safe transfer rejects any occupied cycle with nonzero lifted-s deck gain. +The charge root solves equality of the NN Perron root at p and the matching +Perron root at 1-p. + +This script is intended for small directional controls. State spaces grow +rapidly with both n and the row memory R. +""" +from __future__ import annotations + +import argparse +import json +import math +from collections import Counter, deque +from dataclasses import dataclass +from pathlib import Path + +import numpy as np +from scipy.optimize import brentq +from scipy.sparse import coo_matrix +from scipy.sparse.linalg import eigs + + +@dataclass(frozen=True) +class State: + labels: tuple[int, ...] + gains: tuple[int, ...] + + +class DSU: + def __init__(self, size: int): + self.parent = list(range(size)) + self.delta = [0] * size + self.bad = False + + def find(self, a: int) -> tuple[int, int]: + if self.parent[a] != a: + root, gain = self.find(self.parent[a]) + self.delta[a] += gain + self.parent[a] = root + return self.parent[a], self.delta[a] + + def join(self, a: int, b: int, gain_b_minus_a: int) -> None: + ra, da = self.find(a) + rb, db = self.find(b) + if ra == rb: + if db - da != gain_b_minus_a: + self.bad = True + return + self.parent[rb] = ra + self.delta[rb] = gain_b_minus_a + da - db + + +def extended_gcd(a: int, b: int) -> tuple[int, int, int]: + if b == 0: + return abs(a), 1 if a >= 0 else -1, 0 + g, x1, y1 = extended_gcd(b, a % b) + return g, y1, x1 - (a // b) * y1 + + +def bezout_complement(a: int, b: int) -> tuple[int, int]: + """Return v=(c,d) with det((a,b),(c,d))=1.""" + g, x, y = extended_gcd(a, b) + if g != 1: + raise ValueError("direction must be primitive") + d = x + c = -y + if a * d - b * c != 1: + raise AssertionError("Bezout orientation failure") + return c, d + + +def transformed_edges(direction: tuple[int, int], *, matching: bool): + a, b = direction + c, d = bezout_complement(a, b) + generators = [(1, 0), (0, 1)] + if matching: + generators += [(1, 1), (1, -1)] + + edges = set() + for dx, dy in generators: + ds = d * dx - c * dy + dt = -b * dx + a * dy + if dt < 0 or (dt == 0 and ds < 0): + ds, dt = -ds, -dt + edges.add((ds, dt)) + + ordered = sorted(edges, key=lambda item: (item[1], item[0])) + memory = max(dt for _, dt in ordered) + return (c, d), ordered, memory + + +def empty_state(width: int, memory: int) -> State: + return State((-1,) * (width * memory), (0,) * (width * memory)) + + +def step( + state: State, + mask: int, + width: int, + edges: list[tuple[int, int]], + memory: int, +) -> State | None: + old_size = memory * width + new_base = old_size + dsu = DSU((memory + 1) * width) + representatives: dict[int, int] = {} + + for index, label in enumerate(state.labels): + if label < 0: + continue + if label in representatives: + dsu.join(representatives[label], index, state.gains[index]) + else: + representatives[label] = index + + for ds, dt in edges: + if dt == 0: + for i in range(width): + if not (mask >> i) & 1: + continue + j = (i + ds) % width + if (mask >> j) & 1: + dsu.join(new_base + i, new_base + j, (i + ds) // width) + continue + + row_position = memory - dt + for i in range(width): + old_index = row_position * width + i + if state.labels[old_index] < 0: + continue + j = (i + ds) % width + if (mask >> j) & 1: + dsu.join(old_index, new_base + j, (i + ds) // width) + + if dsu.bad: + return None + + retained = list(range(width, old_size)) + [new_base + i for i in range(width)] + labels = [-1] * old_size + gains = [0] * old_size + canonical: dict[int, tuple[int, int]] = {} + next_label = 0 + + for output_index, index in enumerate(retained): + occupied = ( + state.labels[index] >= 0 + if index < old_size + else bool(mask >> (index - new_base) & 1) + ) + if not occupied: + continue + root, gain = dsu.find(index) + if root not in canonical: + canonical[root] = (next_label, gain) + next_label += 1 + label, base_gain = canonical[root] + labels[output_index] = label + gains[output_index] = gain - base_gain + + return State(tuple(labels), tuple(gains)) + + +def build( + width: int, + direction: tuple[int, int], + *, + matching: bool, + state_cap: int, +): + complement, edges, memory = transformed_edges(direction, matching=matching) + start = empty_state(width, memory) + states = [start] + index = {start: 0} + queue = deque([start]) + transitions = [] + + while queue: + state = queue.popleft() + row = [] + for mask in range(1 << width): + nxt = step(state, mask, width, edges, memory) + if nxt is None: + row.append(-1) + continue + if nxt not in index: + if len(states) >= state_cap: + raise RuntimeError("state cap reached") + index[nxt] = len(states) + states.append(nxt) + queue.append(nxt) + row.append(index[nxt]) + transitions.append(row) + + return states, transitions, complement, edges, memory + + +class ObliqueSafeTransfer: + def __init__( + self, + width: int, + direction: tuple[int, int], + *, + matching: bool, + state_cap: int, + ): + self.width = width + self.direction = direction + ( + self.states, + transitions, + self.complement, + self.edges, + self.memory, + ) = build( + width, + direction, + matching=matching, + state_cap=state_cap, + ) + + counter: Counter[tuple[int, int, int]] = Counter() + for source, row in enumerate(transitions): + for mask, destination in enumerate(row): + if destination >= 0: + counter[(source, destination, mask.bit_count())] += 1 + keys = list(counter) + self.src = np.array([key[0] for key in keys], dtype=np.int32) + self.dst = np.array([key[1] for key in keys], dtype=np.int32) + self.occ = np.array([key[2] for key in keys], dtype=np.int16) + self.counts = np.array([counter[key] for key in keys], dtype=float) + + def matrix(self, p: float): + q = 1.0 - p + data = self.counts * p**self.occ * q ** (self.width - self.occ) + return coo_matrix( + (data, (self.src, self.dst)), + shape=(len(self.states), len(self.states)), + ).tocsr() + + def lambda0(self, p: float) -> float: + matrix = self.matrix(p) + if matrix.shape[0] < 200: + return float(max(np.linalg.eigvals(matrix.toarray()).real)) + value = eigs( + matrix, + k=1, + which="LM", + tol=1e-10, + maxiter=500000, + return_eigenvectors=False, + )[0] + return float(value.real) + + +def cos4(direction: tuple[int, int]) -> float: + a, b = direction + denominator = (a * a + b * b) ** 2 + return (a**4 - 6 * a * a * b * b + b**4) / denominator + + +def run_width( + width: int, + direction: tuple[int, int], + reference_pc: float, + state_cap: int, +) -> dict[str, object]: + g4 = ObliqueSafeTransfer( + width, direction, matching=False, state_cap=state_cap + ) + g8 = ObliqueSafeTransfer( + width, direction, matching=True, state_cap=state_cap + ) + + def equation(p: float) -> float: + return math.log(g4.lambda0(p)) - math.log(g8.lambda0(1.0 - p)) + + root = brentq(equation, 0.5, 0.8, xtol=3e-11) + a, b = direction + length = math.hypot(a, b) + ell = width * length + angular = cos4(direction) + scaled_shift = (root - reference_pc) * ell**4 + amplitude_estimate = ( + -scaled_shift / angular if abs(angular) > 1e-12 else None + ) + return { + "n": width, + "period_vector": [width * a, width * b], + "primitive_direction": [a, b], + "bezout_complement_G4": list(g4.complement), + "transformed_edges_G4": [list(edge) for edge in g4.edges], + "transformed_edges_G8": [list(edge) for edge in g8.edges], + "row_memory_G4": g4.memory, + "row_memory_G8": g8.memory, + "safe_states_G4": len(g4.states), + "safe_states_G8": len(g8.states), + "physical_circumference": ell, + "cos_4theta": angular, + "charge_root_p": root, + "root_minus_reference_pc": root - reference_pc, + "root_shift_times_ell_fourth": scaled_shift, + "spin4_amplitude_estimate_minus_shift_over_cos4": amplitude_estimate, + } + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--a", type=int, required=True) + parser.add_argument("--b", type=int, required=True) + parser.add_argument("--min-n", type=int, default=2) + parser.add_argument("--max-n", type=int, default=4) + parser.add_argument("--state-cap", type=int, default=300000) + parser.add_argument("--reference-pc", type=float, default=0.59274605079) + parser.add_argument("--output", type=Path) + args = parser.parse_args() + + direction = (args.a, args.b) + if math.gcd(abs(args.a), abs(args.b)) != 1: + raise ValueError("(a,b) must be primitive") + + result = { + "schema": "oblique-charge-transfer-v1", + "model": "same square-site NN / complementary matching graph in an SL(2,Z) oblique basis", + "direction": list(direction), + "reference_pc": args.reference_pc, + "spin4_prediction": "p_root-pc ~ -A*cos(4theta)/ell^4", + "claim_boundary": [ + "charge roots are located from safe Perron equality only", + "reference_pc is diagnostic only", + "state complexity grows rapidly with row memory; small n can have larger corrections" + ], + "records": [ + run_width(n, direction, args.reference_pc, args.state_cap) + for n in range(args.min_n, args.max_n + 1) + ], + } + text = json.dumps(result, indent=2, sort_keys=True) + if args.output: + args.output.write_text(text + "\n", encoding="utf-8") + print(text) + + +if __name__ == "__main__": + main() diff --git a/scripts/oblique_magnetic_metric_check.py b/scripts/oblique_magnetic_metric_check.py new file mode 100644 index 000000000..004650450 --- /dev/null +++ b/scripts/oblique_magnetic_metric_check.py @@ -0,0 +1,65 @@ +#!/usr/bin/env python3 +"""Check the magnetic 5/48 gap in oblique cylinder coordinates. + +For primitive u=(a,b), a Bezout transverse step has physical normal height +1/|u|. A transfer excitation I per t-step therefore corresponds to physical +energy |u|*I, while the circumference is ell=n|u|. The dimensionless magnetic +gap is + + n*|u|^2*I. + +At criticality / the nearby charge root this should tend to 2*pi*(5/48). +""" +from __future__ import annotations + +import argparse +import json +import math +from pathlib import Path + +from scipy.optimize import brentq + +from oblique_charge_transfer import ObliqueSafeTransfer + + +def run(a: int, b: int, n: int, state_cap: int) -> dict[str, object]: + u = (a, b) + g4 = ObliqueSafeTransfer(n, u, matching=False, state_cap=state_cap) + g8 = ObliqueSafeTransfer(n, u, matching=True, state_cap=state_cap) + + def equation(p: float) -> float: + return math.log(g4.lambda0(p)) - math.log(g8.lambda0(1.0 - p)) + + root = brentq(equation, 0.5, 0.8, xtol=3e-11) + lam = g4.lambda0(root) + I0 = -math.log(lam) + scaled = n * (a * a + b * b) * I0 + target = 2 * math.pi * 5 / 48 + return { + "direction": [a, b], + "n": n, + "charge_root_p": root, + "I0_per_transverse_integer_step": I0, + "scaled_physical_gap_n_normu2_I0": scaled, + "target_2pi_5_over_48": target, + "ratio_to_target": scaled / target, + } + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--a", type=int, required=True) + parser.add_argument("--b", type=int, required=True) + parser.add_argument("--n", type=int, required=True) + parser.add_argument("--state-cap", type=int, default=300000) + parser.add_argument("--output", type=Path) + args = parser.parse_args() + result = run(args.a, args.b, args.n, args.state_cap) + text = json.dumps(result, indent=2, sort_keys=True) + if args.output: + args.output.write_text(text + "\n", encoding="utf-8") + print(text) + + +if __name__ == "__main__": + main() diff --git a/scripts/persistent_alexander_birth_reflection.py b/scripts/persistent_alexander_birth_reflection.py new file mode 100644 index 000000000..cced9e82b --- /dev/null +++ b/scripts/persistent_alexander_birth_reflection.py @@ -0,0 +1,203 @@ +"""Finite controls for persistent digital-Alexander birth reflection. + +For an LxL honest square torus, black sites use NN connectivity and white sites +use the NN+diagonal matching graph. We check configuration-level rank/count/ +rank-one-line identities. At L=3 we also exhaust all 9! label orders; at +larger L the birth-path control uses fixed-seed random permutations after +precomputing all mask ranks. + +This is a finite control, not the proof. The birth-index reflection follows +algebraically from r4(S)+r8(S^c)=2 along nested occupied sets S_k. +""" +from __future__ import annotations + +import argparse +import json +import random +from itertools import permutations +from math import factorial, gcd +from pathlib import Path + + +def run(L: int, permutation_samples: int, seed: int): + N = L * L + NN = [(1, 0), (0, 1)] + G8 = [(1, 0), (0, 1), (1, 1), (1, -1)] + full = (1 << N) - 1 + + def vid(x, y): + return (y % L) * L + (x % L) + + def xy(v): + return (v % L, v // L) + + def canon_line(v): + a, b = v + if a == 0 and b == 0: + return None + g = gcd(abs(a), abs(b)) + a //= g + b //= g + if a < 0 or (a == 0 and b < 0): + a, b = -a, -b + return (a, b) + + def rank_and_components(mask, steps): + occ = [bool(mask >> i & 1) for i in range(N)] + adj = [[] for _ in range(N)] + for u in range(N): + if not occ[u]: + continue + x, y = xy(u) + for dx, dy in steps: + v = vid(x + dx, y + dy) + if occ[v]: + adj[u].append((v, (dx, dy))) + adj[v].append((u, (-dx, -dy))) + + seen = [False] * N + all_gains = [] + winding_components = 0 + for root in range(N): + if not occ[root] or seen[root]: + continue + seen[root] = True + lift = {root: (0, 0)} + stack = [root] + gains = [] + while stack: + u = stack.pop() + ux, uy = lift[u] + for v, (dx, dy) in adj[u]: + cand = (ux + dx, uy + dy) + if not seen[v]: + seen[v] = True + lift[v] = cand + stack.append(v) + else: + vx, vy = lift[v] + defect = (cand[0] - vx, cand[1] - vy) + if defect != (0, 0): + assert defect[0] % L == 0 and defect[1] % L == 0 + gains.append((defect[0] // L, defect[1] // L)) + nz = [g for g in gains if g != (0, 0)] + if nz: + winding_components += 1 + all_gains.extend(nz) + + rank = 0 + line = None + if all_gains: + rank = 1 + line = canon_line(all_gains[0]) + a, b = all_gains[0] + for c, d in all_gains[1:]: + if a * d - b * c != 0: + rank = 2 + line = None + break + return rank, winding_components, line + + size = 1 << N + if N > 16: + raise ValueError("This finite control intentionally caps exhaustive configuration enumeration at N<=16.") + + rank4 = [0] * size + rank8_direct = [0] * size + for mask in range(size): + rank4[mask] = rank_and_components(mask, NN)[0] + rank8_direct[mask] = rank_and_components(mask, G8)[0] + + violations = [] + rank_counts = [0, 0, 0] + rank1_count = 0 + pair_counts = {} + for mask in range(size): + comp = full ^ mask + r4, w4, l4 = rank_and_components(mask, NN) + r8, w8, l8 = rank_and_components(comp, G8) + pair_counts[(w4, w8)] = pair_counts.get((w4, w8), 0) + 1 + rank_counts[r4] += 1 + if r4 == 1: + rank1_count += 1 + if r4 + r8 != 2: + violations.append(("rank", mask, r4, r8)) + if w4 - w8 != r4 - 1: + violations.append(("component-count", mask, w4, w8, r4)) + if r4 == 1 and l4 != l8: + violations.append(("rank1-line", mask, l4, l8)) + if violations: + break + if violations: + raise AssertionError(violations[:3]) + + def births(perm, rank_table): + mask = 0 + k1 = k2 = None + for k, u in enumerate(perm, 1): + mask |= 1 << u + r = rank_table[mask] + if k1 is None and r >= 1: + k1 = k + if k2 is None and r >= 2: + k2 = k + return k1, k2 + + rng = random.Random(seed) + if N <= 9: + iterator = permutations(range(N)) + nperm = factorial(N) + mode = "exhaustive" + else: + def samples(): + base = list(range(N)) + for _ in range(permutation_samples): + a = base.copy() + rng.shuffle(a) + yield tuple(a) + iterator = samples() + nperm = permutation_samples + mode = f"random(seed={seed})" + + birth_viol = 0 + for perm in iterator: + k1, k2 = births(perm, rank4) + j1, j2 = births(tuple(reversed(perm)), rank8_direct) + if (j1, j2) != (N + 1 - k2, N + 1 - k1): + birth_viol += 1 + break + if birth_viol: + raise AssertionError("birth reflection violation") + + return { + "L": L, + "N": N, + "configurations_checked": size, + "permutation_mode": mode, + "permutations_checked": nperm, + "digital_alexander_rank_identity": True, + "winding_component_count_identity": True, + "rank1_homology_line_matches_dual": True, + "birth_index_relation": "K1_matching(reverse)=N+1-K2_NN; K2_matching(reverse)=N+1-K1_NN", + "birth_index_violations": 0, + "all_checks_passed": True, + "rank1_configurations": rank1_count, + "rank_counts_NN": {str(i): rank_counts[i] for i in range(3)}, + "winding_component_pair_counts": { + f"{a},{b}": n for (a, b), n in sorted(pair_counts.items()) + }, + } + + +if __name__ == "__main__": + ap = argparse.ArgumentParser() + ap.add_argument("--L", type=int, default=3) + ap.add_argument("--permutation-samples", type=int, default=20000) + ap.add_argument("--seed", type=int, default=20260914) + ap.add_argument("--output", type=Path) + args = ap.parse_args() + result = run(args.L, args.permutation_samples, args.seed) + text = json.dumps(result, indent=2) + if args.output: + args.output.write_text(text + "\n") + print(text) diff --git a/scripts/poisson_topology_constants.py b/scripts/poisson_topology_constants.py new file mode 100644 index 000000000..e9f2a32cf --- /dev/null +++ b/scripts/poisson_topology_constants.py @@ -0,0 +1,31 @@ +from __future__ import annotations + +import json +import math +from pathlib import Path + +lam = math.log(2.0) +p0 = math.exp(-lam) +p1 = lam * p0 +pge2 = 1.0 - p0 - p1 + +out = { + "lambda": lam, + "P_K_0": p0, + "P_K_1": p1, + "P_K_ge_2": pge2, + "P_K_ge_2_given_K_ge_1": pge2 / (1.0 - p0), + "P_K_1_given_K_ge_1": p1 / (1.0 - p0), + "E_K_given_K_ge_1": lam / (1.0 - p0), + "same_parameter_lower_window_joint_law": { + "(W4,W8)=(0,1)": p0, + "(W4,W8)=(1,1)": p1, + "(W4,W8) has k>=2 equal counts": pge2, + }, + "E_W8_lower_window_if_W4_Poisson": lam + p0, +} + +if __name__ == "__main__": + text = json.dumps(out, indent=2) + print(text) + Path("results/geometric-consistency/poisson-topology-constants-20260914.json").write_text(text + "\n") diff --git a/scripts/query_cavity_controls.py b/scripts/query_cavity_controls.py new file mode 100644 index 000000000..dc883db50 --- /dev/null +++ b/scripts/query_cavity_controls.py @@ -0,0 +1,265 @@ +#!/usr/bin/env python3 +"""Independent finite controls for shielded query noise and width-eight cavities. + +All probabilities/moments used in identities are Fractions. No previous +transfer engine, sampling, or large-width fit is imported. The enumerations +check finite interfaces; the accompanying arguments prove the limits. +""" +from __future__ import annotations +import argparse +from collections import Counter, deque +from fractions import Fraction as F +from itertools import product, combinations +import json +from pathlib import Path +from typing import Iterable + +XY = tuple[int, int] +NN = ((1,0),(-1,0),(0,1),(0,-1)) +KING = tuple((x,y) for x in (-1,0,1) for y in (-1,0,1) if x or y) +P_COEFF = (9,18,27,36,45,54,63,8,7,6,5,4,3,2,1) + +def packed(x: F | int) -> dict[str, str]: + f = F(x) + return {'fraction':str(f), 'decimal':format(float(f),'.16g')} + +def covariance(rows: list[tuple[F, tuple[F, ...]]]) -> tuple[list[F], list[list[F]]]: + assert sum((p for p,_ in rows),F()) == 1 + d=len(rows[0][1]) + mu=[sum((p*x[j] for p,x in rows),F()) for j in range(d)] + cv=[[sum((p*(x[i]-mu[i])*(x[j]-mu[j]) for p,x in rows),F()) + for j in range(d)] for i in range(d)] + return mu,cv + +def wired_statistics(mask: int) -> tuple[int,int,int]: + """Actual KING BFS: 3x3 random interior, radius-two all-white shell.""" + inside=tuple(product(range(-1,2),repeat=2)) + shell={v for v in product(range(-2,3),repeat=2) if max(map(abs,v))==2} + white=shell|{v for i,v in enumerate(inside) if mask>>i&1} + reached=set(shell); queue=deque(shell) + while queue: + x,y=queue.popleft() + for dx,dy in KING: + z=x+dx,y+dy + if z in white and z not in reached: + reached.add(z); queue.append(z) + k=len(set(inside)&reached) + boundary={v for v in inside if v not in white + and any((v[0]+dx,v[1]+dy) in reached for dx,dy in KING)} + return k,len(boundary),9-k-len(boundary) + +def shield_control(q: F) -> dict: + if not 0>centre)&1; t=n-z + k,b,h=wired_statistics(mask) + assert (k,b,h)==(t+z*(t>0),8-t+(1-z)*(t>0),int(t==0)) + score=F(k)/q-F(b)/p + rows.append((q**n*p**(9-n),(F(k),score,F(k+b),F(h)))) + mu,cv=covariance(rows); r=p**8 + vS=(9-r)/(p*q); vK=p*q*(9-r)+q*q*r*(17-r) + kS=9-r+8*q*p**7 + residual=vK-kS*kS/vS + poly=sum((F(a)*p**i for i,a in enumerate(P_COEFF)),F()) + positive=p**8*q**4*poly/(9-p**8) + assert cv[1][1]==vS and cv[0][0]==vK and cv[0][1]==kS + assert mu[1]==0 and mu[3]==r and residual==positive>0 + return {'q':packed(q),'p':packed(p),'configurations':512, + 'mean_K':packed(mu[0]),'variance_K':packed(vK), + 'variance_score':packed(vS),'covariance_K_score':packed(kS), + 'conditional_residual':packed(residual), + 'global_lower_bound_per_theta':packed(q**16*residual/25), + 'local_hole_probability':packed(r)} + +def canon(shape: Iterable[XY], width: int) -> tuple[XY,...]: + pts=tuple(shape); lo=min(y for x,y in pts) + return min(tuple(sorted((((x+s)%width,y-lo) for x,y in pts))) + for s in range(width)) + +def grow_animals(width: int, maximum: int) -> list[set[tuple[XY,...]]]: + out=[set(),{((0,0),)}] + for n in range(2,maximum+1): + nxt=set() + for animal in out[-1]: + a=set(animal) + for x,y in animal: + for dx,dy in NN: + v=((x+dx)%width,y+dy) + if v not in a: nxt.add(canon(a|{v},width)) + out.append(nxt) + return out + +def winds(shape: Iterable[XY], width: int) -> bool: + pts={(x%width,y) for x,y in shape}; lifts={}; winding=False + for root in pts: + if root in lifts: continue + lifts[root]=root[0]; todo=[root] + while todo: + u=todo.pop(); x,y=u + for dx,dy in NN: + v=((x+dx)%width,y+dy) + if v not in pts: continue + target=lifts[u]+dx + if v in lifts: winding |= lifts[v]!=target + else: lifts[v]=target;todo.append(v) + return winding + +def hidden_sites(shape: Iterable[XY], width: int) -> tuple[XY,...]: + """Black shape in otherwise white cylinder; exterior wired at both ends.""" + black={(x%width,y) for x,y in shape}; hi=max(y for x,y in black);lo=min(y for x,y in black) + allowed={(x,y) for x in range(width) for y in range(lo-1,hi+2)}-black + reached={v for v in allowed if v[1] in (lo-1,hi+1)};todo=list(reached) + while todo: + x,y=todo.pop() + for dx,dy in KING: + v=((x+dx)%width,y+dy) + if v in allowed and v not in reached: + reached.add(v);todo.append(v) + queried=set(reached) + for x,y in reached: + queried.update(((x+dx)%width,y+dy) for dx,dy in KING) + return tuple(sorted((x,y) for x in range(width) for y in range(lo,hi+1) + if (x,y) not in queried)) + +def animal_control(width: int=8, maximum: int=8) -> dict: + animals=grow_animals(width,maximum);summary=[];wit=[] + for n in range(1,maximum+1): + essential=[];holes=[] + for a in sorted(animals[n]): + if winds(a,width): essential.append(a) + else: + h=hidden_sites(a,width) + if h: holes.append((a,h)) + if n<8: assert not holes + if n dict: + w=8 + types=[('barrier',frozenset((x,0) for x in range(w)))] + types += [('hole',frozenset(((x+dx)%w,dy) for dx,dy in KING)) for x in range(w)] + overlap=Counter();sizes=Counter() + for i,(ti,si) in enumerate(types): + for j,(tj,sj0) in enumerate(types): + for dy in range(-2,3): + if i==j and dy==0: continue + sj={(x,y+dy) for x,y in sj0} + if si & sj: + k=len(si|sj);overlap[(ti,tj,k)]+=1;sizes[k]+=1 + assert min(sizes)>=12 + return {'types_per_row':{'barrier':1,'hole':8}, + 'ordered_overlap_pairs':[{'first':a,'second':b,'union_black_sites':k,'count':n} + for (a,b,k),n in sorted(overlap.items())], + 'least_distinct_union':min(sizes), + 'finite_window_pattern_poisson_error_order':'O(p^4) for length O(p^-8)', + 'physical_reduction_bad_cluster_order':'O(p), on a fixed scaled window'} + +def limiting_laws() -> dict: + r=F(8) + probabilities=[F(1,9)*F(8,9)**k for k in range(9)] + mixed=[] + for u in (F(0),F(1,2),F(2)): + for z in (F(0),F(1,2),F(1)): + f=1/(1+u+r*(1-z)) + # Competing exponential clocks / factorial-series algebra. + assert f==1/(9+u-8*z) + mixed.append({'u':packed(u),'z':packed(z),'transform':packed(f)}) + return {'interpretation':'Exact moments of limiting two-pattern model; not a finite-p moment certificate.', + 'H_probabilities_k_0_to_8':[packed(x) for x in probabilities], + 'mean_H':packed(r),'variance_H':packed(r*(1+r)), + 'covariance_E_H':packed(r),'correlation_E_H_squared':packed(r/(1+r)), + 'mean_limit_query_excess_16_minus_H':packed(16-r), + 'cavity_fugacity_pole_limiting_model':packed(F(9,8)), + 'joint_transforms':mixed} + +def boundary_of(sites: set[XY]) -> frozenset[XY]: + return frozenset({(x+dx,y+dy) for x,y in sites for dx,dy in KING}-sites) + +def defect_boolean_terms(v: XY, degree: int=11) -> dict[frozenset[XY],int]: + """OR of the one singleton cage and four axial two-site cages. + +Return its square-free black-indicator polynomial, truncated by number of +required black sites. This is not claimed to contain cages of perimeter >=12. +""" + patterns=[boundary_of({v})] + patterns += [boundary_of({v,(v[0]+dx,v[1]+dy)}) for dx,dy in NN] + out=Counter() + for n in range(1,6): + for inds in combinations(range(5),n): + union=frozenset().union(*(patterns[i] for i in inds)) + if len(union)<=degree: out[union]+=(-1)**(n+1) + return {k:v for k,v in out.items() if v} + +def sparse_defect_series() -> dict: + degree=11; left=defect_boolean_terms((0,0),degree) + mean=Counter() + for sites,coef in left.items(): mean[len(sites)]+=coef + spectra=Counter(); offsets=[] + # Outside this range supports are disjoint. Their product has degree >=16. + for dx,dy in product(range(-5,6),repeat=2): + right=defect_boolean_terms((dx,dy),degree);cov=Counter() + for u,a in left.items(): + for v,b in right.items(): + if len(u|v)<=degree:cov[len(u|v)]+=a*b + cov={k:v for k,v in cov.items() if v} + if cov: + offsets.append({'offset':(dx,dy),'coefficients':dict(sorted(cov.items()))}) + spectra.update(cov) + assert dict(mean)=={8:1,10:4,11:-4} + assert len(offsets)==5 and dict(spectra)=={8:1,10:8,11:-4} + residual=Counter() + for k,v in spectra.items(): + for j,b in enumerate((1,-2,1)): + if k+j<=degree:residual[k+j]+=v*b + assert dict(residual)=={8:1,9:-2,10:9,11:-20} + return {'model':'Exact Boolean motif algebra through black degree 11. ' + 'The proof of the O(p^12) physical remainder is in the note.', + 'mean_defect_coefficients':dict(sorted(mean.items())), + 'covariance_by_offset':offsets, + 'integrated_query_covariance':dict(sorted(spectra.items())), + 'J_coefficients':dict(sorted(residual.items())), + 'thermal_projection_first_possible_order':15} + +def independent_block_control() -> list[dict]: + """Exact rational control of the rare-clock limit; explicitly a toy.""" + ans=[] + for p in (F(1,2),F(1,4),F(1,8)): + rho=p**8 + for u,z in ((F(0),F(0)),(F(1),F(1,2)),(F(2),F(1))): + a=(1-rho+rho*z)**8; s=1/(1+u*rho) + val=rho*s/(1-(1-rho)*s*a) + lim=1/(1+u+8*(1-z)) + ans.append({'p':packed(p),'u':packed(u),'z':packed(z), + 'rational_discrete_transform':packed(val), + 'limiting_transform':packed(lim), + 'absolute_difference':packed(abs(val-lim))}) + return ans + +def report() -> dict: + return {'scope':'finite exact controls, independent of historical engines', + 'shield':[shield_control(q) for q in (F(1,3),F(1,2),F(3,4),F(9,10))], + 'positive_polynomial_coefficients_ascending':P_COEFF, + 'animals':animal_control(), 'patterns':pattern_control(), + 'limiting_model':limiting_laws(), 'defect_series':sparse_defect_series(), + 'independent_block_toy':independent_block_control()} + +def main() -> None: + ap=argparse.ArgumentParser(description=__doc__) + ap.add_argument('--output',type=Path,required=True) + args=ap.parse_args();result=report() + args.output.parent.mkdir(parents=True,exist_ok=True) + args.output.write_text(json.dumps(result,ensure_ascii=False,indent=2)+'\n',encoding='utf8') + print(json.dumps({'output':str(args.output),'shield_configurations':2048, + 'animal_classes':result['animals']['checked_classes']})) + +if __name__=='__main__': main() diff --git a/scripts/rank1_slope_harmonic_control.py b/scripts/rank1_slope_harmonic_control.py new file mode 100644 index 000000000..a545409d1 --- /dev/null +++ b/scripts/rank1_slope_harmonic_control.py @@ -0,0 +1,197 @@ +#!/usr/bin/env python3 +"""Exact tiny-torus controls for the rank-one projective slope harmonic. + +The observable is defined only on rank-one configurations. For a primitive +unoriented winding line (a,b) on the square torus, report + + Z4 = ((a+i b)^4)/(a^2+b^2)^2. + +Two exact weightings are returned: + +1. p=1/2, where every configuration has equal weight; +2. the integral over p in [0,1]. A k-site configuration then has beta weight + 1 / ((N+1) * binom(N,k)), so this equals the plateau-duration weighting + E[(T2-T1) Z4] without enumerating site orders. + +The script also checks the configurationwise NN/complementary-matching +rank-one line identity. This is a finite control, not a continuum +extrapolation. +""" +from __future__ import annotations + +import argparse +import json +from collections import Counter, defaultdict +from fractions import Fraction +from math import comb, gcd +from pathlib import Path + + +def canonical_line(a: int, b: int) -> tuple[int, int]: + divisor = gcd(abs(a), abs(b)) + if divisor == 0: + raise ValueError("zero vector has no projective line") + a //= divisor + b //= divisor + if a < 0 or (a == 0 and b < 0): + a, b = -a, -b + return a, b + + +def z4_components(line: tuple[int, int]) -> tuple[Fraction, Fraction]: + a, b = line + denominator = (a * a + b * b) ** 2 + real = Fraction(a**4 - 6 * a * a * b * b + b**4, denominator) + imag = Fraction(4 * a * b * (a * a - b * b), denominator) + return real, imag + + +def graph_rank_line(mask: int, L: int, *, matching: bool) -> tuple[int, tuple[int, int] | None]: + N = L * L + steps = [(1, 0), (0, 1)] + if matching: + steps += [(1, 1), (1, -1)] + + def vid(x: int, y: int) -> int: + return (y % L) * L + (x % L) + + def xy(vertex: int) -> tuple[int, int]: + return vertex % L, vertex // L + + occupied = [bool(mask >> vertex & 1) for vertex in range(N)] + adjacency: list[list[tuple[int, tuple[int, int]]]] = [[] for _ in range(N)] + for u in range(N): + if not occupied[u]: + continue + x, y = xy(u) + for dx, dy in steps: + v = vid(x + dx, y + dy) + if occupied[v]: + adjacency[u].append((v, (dx, dy))) + adjacency[v].append((u, (-dx, -dy))) + + seen = [False] * N + gains: list[tuple[int, int]] = [] + for root in range(N): + if not occupied[root] or seen[root]: + continue + seen[root] = True + lift = {root: (0, 0)} + stack = [root] + while stack: + u = stack.pop() + ux, uy = lift[u] + for v, (dx, dy) in adjacency[u]: + candidate = (ux + dx, uy + dy) + if not seen[v]: + seen[v] = True + lift[v] = candidate + stack.append(v) + else: + vx, vy = lift[v] + defect = (candidate[0] - vx, candidate[1] - vy) + if defect != (0, 0): + if defect[0] % L or defect[1] % L: + raise AssertionError("non-period lift defect") + gain = (defect[0] // L, defect[1] // L) + if gain != (0, 0): + gains.append(gain) + + if not gains: + return 0, None + a, b = gains[0] + for c, d in gains[1:]: + if a * d - b * c != 0: + return 2, None + return 1, canonical_line(a, b) + + +def _fraction_map(values: dict[tuple[int, int], Fraction]) -> dict[str, str]: + return {f"{a},{b}": str(value) for (a, b), value in sorted(values.items())} + + +def run(L: int) -> dict[str, object]: + N = L * L + if N > 16: + raise ValueError("This exact control intentionally caps at L<=4.") + full = (1 << N) - 1 + counts: Counter[tuple[int, int]] = Counter() + cardinality_counts: Counter[int] = Counter() + rank_one = 0 + z4_real_sum = Fraction(0) + z4_imag_sum = Fraction(0) + beta_rank_one = Fraction(0) + beta_z4_real = Fraction(0) + beta_z4_imag = Fraction(0) + beta_line_weights: dict[tuple[int, int], Fraction] = defaultdict(Fraction) + complement_violations = 0 + + for mask in range(1 << N): + rank4, line4 = graph_rank_line(mask, L, matching=False) + if rank4 == 1: + rank_one += 1 + assert line4 is not None + counts[line4] += 1 + occupied_count = mask.bit_count() + cardinality_counts[occupied_count] += 1 + real, imag = z4_components(line4) + z4_real_sum += real + z4_imag_sum += imag + + beta_weight = Fraction(1, (N + 1) * comb(N, occupied_count)) + beta_rank_one += beta_weight + beta_z4_real += beta_weight * real + beta_z4_imag += beta_weight * imag + beta_line_weights[line4] += beta_weight + + rank8, line8 = graph_rank_line(full ^ mask, L, matching=True) + if rank8 != 1 or line8 != line4: + complement_violations += 1 + + if complement_violations: + raise AssertionError(f"rank-one complement slope violations: {complement_violations}") + + total = 1 << N + conditional_real = z4_real_sum / rank_one + conditional_imag = z4_imag_sum / rank_one + unnormalized_real = z4_real_sum / total + unnormalized_imag = z4_imag_sum / total + gap_conditional_real = beta_z4_real / beta_rank_one + gap_conditional_imag = beta_z4_imag / beta_rank_one + + return { + "L": L, + "N": N, + "p": "1/2", + "configurations_checked": total, + "rank_one_configurations": rank_one, + "rank_one_cardinality_counts": {str(k): count for k, count in sorted(cardinality_counts.items())}, + "slope_counts": {f"{a},{b}": count for (a, b), count in sorted(counts.items())}, + "z4_unnormalized_real": str(unnormalized_real), + "z4_unnormalized_real_float": float(unnormalized_real), + "z4_unnormalized_imag": str(unnormalized_imag), + "z4_conditional_real": str(conditional_real), + "z4_conditional_real_float": float(conditional_real), + "z4_conditional_imag": str(conditional_imag), + "integrated_rank_one_probability": str(beta_rank_one), + "integrated_z4_real": str(beta_z4_real), + "integrated_z4_imag": str(beta_z4_imag), + "gap_duration_weighted_z4_real": str(gap_conditional_real), + "gap_duration_weighted_z4_real_float": float(gap_conditional_real), + "gap_duration_weighted_z4_imag": str(gap_conditional_imag), + "integrated_line_weights": _fraction_map(beta_line_weights), + "rank_one_NN_complement_matching_line_identity": True, + "complement_violations": 0, + } + + +if __name__ == "__main__": + parser = argparse.ArgumentParser() + parser.add_argument("--L", type=int, default=3) + parser.add_argument("--output", type=Path) + args = parser.parse_args() + result = run(args.L) + text = json.dumps(result, indent=2) + if args.output: + args.output.write_text(text + "\n", encoding="utf-8") + print(text) diff --git a/scripts/rescore_p156_projective_h4.py b/scripts/rescore_p156_projective_h4.py new file mode 100644 index 000000000..15cc32408 --- /dev/null +++ b/scripts/rescore_p156_projective_h4.py @@ -0,0 +1,162 @@ +#!/usr/bin/env python3 +"""Zero-new-sampling rescore of the archived #156 N30/N56 primitive pilot. + +The archive retains batch counts for the three named primitive lines l0,l1,l2 +but aggregates every other rank-one line into rank1_other. Therefore we can +recover exactly, with the original batch covariance: + + * the frozen real C3 character C_C3; + * the physically embedded three-line spin-4 projection A4_three(tau). + +We CANNOT reconstruct the full all-primitive A4_full from this archive because +the directions inside rank1_other were not retained. The script reports this +explicitly and never imputes those missing phases. +""" +from __future__ import annotations + +import argparse +import csv +import json +import math +from pathlib import Path +from statistics import mean + +import mpmath as mp + +from pinson_arguin_primitive import primitive_probability_direct + + +DESIGNS = { + "pell_Dminus2_N30": mp.mpc(mp.mpf("0.5"), mp.mpf(5) / 6), + "pell_Dplus1_N56": mp.mpc(mp.mpf("0.5"), mp.mpf(7) / 8), +} + + +def _complex_stats(values: list[complex]) -> dict[str, float]: + n = len(values) + if n < 2: + raise ValueError("need at least two independent batches") + mr = sum(z.real for z in values) / n + mi = sum(z.imag for z in values) / n + vr = sum((z.real - mr) ** 2 for z in values) / (n - 1) + vi = sum((z.imag - mi) ** 2 for z in values) / (n - 1) + cov = sum((z.real - mr) * (z.imag - mi) for z in values) / (n - 1) + return { + "mean_real": mr, + "mean_imag": mi, + "se_real": math.sqrt(vr / n), + "se_imag": math.sqrt(vi / n), + "cov_mean_real_imag": cov / n, + "z_real": mr / math.sqrt(vr / n) if vr else float("inf"), + "z_imag": mi / math.sqrt(vi / n) if vi else float("inf"), + } + + +def _real_stats(values: list[float]) -> dict[str, float]: + n = len(values) + m = mean(values) + variance = sum((x - m) ** 2 for x in values) / (n - 1) + se = math.sqrt(variance / n) + return {"mean": m, "se": se, "z": m / se if se else float("inf")} + + +def continuum_three_line(tau: mp.mpc) -> dict[str, object]: + # engine l0=(1,0), l1=(0,1), l2=(1,-1) + # maps to paper sectors {1,0}, {0,-1}, {1,1}. + p0 = primitive_probability_direct(1, 0, tau) + p1 = primitive_probability_direct(0, -1, tau) + p2 = primitive_probability_direct(1, 1, tau) + z0 = mp.mpc(1) + z1 = (tau / abs(tau)) ** 4 + z2 = ((1 - tau) / abs(1 - tau)) ** 4 + a4 = p0 * z0 + p1 * z1 + p2 * z2 + c3 = p0 - (p1 + p2) / 2 + q3 = mp.sqrt(3) * (p2 - p1) / 2 + return { + "probabilities": [str(p0), str(p1), str(p2)], + "z4_weights": [[float(mp.re(z0)), float(mp.im(z0))], + [float(mp.re(z1)), float(mp.im(z1))], + [float(mp.re(z2)), float(mp.im(z2))]], + "A4_three_real": float(mp.re(a4)), + "A4_three_imag": float(mp.im(a4)), + "C_C3": float(c3), + "Q_C3": float(q3), + } + + +def rescore(csv_path: Path, dps: int) -> dict[str, object]: + mp.mp.dps = dps + grouped: dict[str, list[dict[str, int]]] = {name: [] for name in DESIGNS} + with csv_path.open(newline="", encoding="utf-8") as handle: + for row in csv.DictReader(handle): + design = row["design"] + if design not in grouped: + continue + grouped[design].append({ + key: int(row[key]) + for key in ("samples", "l0", "l1", "l2", "rank1_other") + }) + + out: dict[str, object] = { + "source": str(csv_path), + "missing_information": ( + "rank1_other is aggregated, so A4_full over all primitive slopes " + "is not recoverable from this archive" + ), + "designs": {}, + } + + for design, tau in DESIGNS.items(): + rows = grouped[design] + if not rows: + raise ValueError(f"no batches found for {design}") + continuum = continuum_three_line(tau) + z0 = 1 + 0j + z1 = complex((tau / abs(tau)) ** 4) + z2 = complex(((1 - tau) / abs(1 - tau)) ** 4) + + a4_residual_batches: list[complex] = [] + c3_residual_batches: list[float] = [] + other_fractions: list[float] = [] + for row in rows: + n = row["samples"] + empirical_a4 = ( + row["l0"] * z0 + row["l1"] * z1 + row["l2"] * z2 + ) / n + continuum_a4 = complex( + continuum["A4_three_real"], continuum["A4_three_imag"] + ) + a4_residual_batches.append(empirical_a4 - continuum_a4) + + empirical_c3 = (row["l0"] - (row["l1"] + row["l2"]) / 2) / n + c3_residual_batches.append(empirical_c3 - continuum["C_C3"]) + other_fractions.append(row["rank1_other"] / n) + + out["designs"][design] = { + "tau": [float(mp.re(tau)), float(mp.im(tau))], + "batches": len(rows), + "samples_per_batch": rows[0]["samples"], + "continuum_three_line": continuum, + "A4_three_residual": _complex_stats(a4_residual_batches), + "C_C3_residual": _real_stats(c3_residual_batches), + "rank1_other_fraction": _real_stats(other_fractions), + } + + return out + + +if __name__ == "__main__": + parser = argparse.ArgumentParser() + parser.add_argument( + "--input", + type=Path, + default=Path("results/local-20260829/P156-square-bond-primitive-pilot/result.batches.csv"), + ) + parser.add_argument("--dps", type=int, default=80) + parser.add_argument("--output", type=Path) + args = parser.parse_args() + result = rescore(args.input, args.dps) + text = json.dumps(result, indent=2, sort_keys=True) + if args.output: + args.output.write_text(text + "\n", encoding="utf-8") + print(text) diff --git a/scripts/root_response_null_control.py b/scripts/root_response_null_control.py new file mode 100644 index 000000000..eb01c8226 --- /dev/null +++ b/scripts/root_response_null_control.py @@ -0,0 +1,144 @@ +#!/usr/bin/env python3 +"""Small algebraic checks used to CHOOSE a research task, not a size census. + +Virasoro scope: negative-mode PBW algebra, quotient by the level-two +singular vector at c=0,h=5/8, followed by quotient by L_-1 derivatives. +This does not identify the actual percolation thermal representation. +The probability family below is deliberately a countermodel, not site data. +""" +from __future__ import annotations +from collections import defaultdict +from fractions import Fraction as F +from functools import lru_cache +import json +import math +from pathlib import Path +import argparse + +@lru_cache(None) +def partitions(n: int, cap: int | None = None) -> tuple[tuple[int, ...], ...]: + if n == 0: + return ((),) + cap = n if cap is None else min(cap, n) + return tuple((k,) + rest for k in range(cap, 0, -1) + for rest in partitions(n-k, k)) + +@lru_cache(None) +def normal(word: tuple[int, ...]) -> tuple[tuple[tuple[int, ...], F], ...]: + """L_-a L_-b = L_-b L_-a + (b-a)L_-(a+b) when a list[F]: + d = dict(normal(word)) + return [d.get(x, F(0)) for x in basis] + +def rank(rows: list[list[F]], ncols: int) -> int: + a = [row[:] for row in rows] + r = 0 + for c in range(ncols): + pivot = next((i for i in range(r, len(a)) if a[i][c]), None) + if pivot is None: + continue + a[r], a[pivot] = a[pivot], a[r] + val = a[r][c] + a[r] = [x/val for x in a[r]] + for i in range(r+1, len(a)): + val = a[i][c] + if val: + a[i] = [x-val*y for x,y in zip(a[i], a[r])] + r += 1 + if r == len(a): + break + return r + +def module_rows(max_level: int = 4) -> list[dict]: + rows = [] + for n in range(max_level+1): + basis = partitions(n) + nulls = [] + if n >= 2: + for a in partitions(n-2): + v2 = vector(a+(2,), basis) + v11 = vector(a+(1,1), basis) + nulls.append([x-F(2,3)*y for x,y in zip(v2, v11)]) + derivs = [vector((1,)+a, basis) for a in partitions(n-1)] if n else [] + nr = rank(nulls, len(basis)) + both = rank(nulls+derivs, len(basis)) + rows.append(dict(level=n, verma_dimension=len(basis), null_rank=nr, + null_quotient_dimension=len(basis)-nr, + derivative_image_rank_in_quotient=both-nr, + nonderivative_quotient_dimension=len(basis)-both, + pbw_basis=[list(t) for t in basis], + null_rows=[[str(x) for x in v] for v in nulls], + derivative_rows=[[str(x) for x in v] for v in derivs])) + return rows + +def check() -> dict: + h, central = F(5,8), F(0) + singular_checks = { + 'L1_coefficient': str(3-F(2,3)*(4*h+2)), + 'L2_coefficient': str(4*h+central/2-F(2,3)*6*h), + } + assert all(v == '0' for v in singular_checks.values()) + rows = module_rows() + assert [r['null_quotient_dimension'] for r in rows] == [1,1,1,2,3] + assert [r['nonderivative_quotient_dimension'] for r in rows] == [1,0,0,1,1] + assert dict(normal((1,2))) == {(2,1):F(1), (3,):F(1)} + assert dict(normal((1,3))) == {(3,1):F(1), (4,):F(2)} + # Fully positive laws: intrinsic c=2cosh(b), independent of root displacement. + curves = [] + for a in (F(-1,5), F(0), F(1,5)): + for L in (4,8,16): + root = F(3,5) + a/F(L**4) + for b in (F(-1,2), F(0), F(1,2)): + cb = 2*math.cosh(float(b)) + den = 2*(cb+math.cosh(float(b))) + probs = [math.exp(float(b))/den, 2*cb/den, + math.exp(-float(b))/den] + assert min(probs)>0 and abs(sum(probs)-1)<1e-14 + assert abs(math.log(probs[1]/(2*math.sqrt(probs[0]*probs[2])))-math.log(cb)) < 1e-14 + curves.append({'L':L,'amplitude':str(a),'root':str(root), + 'intrinsic_odd_part':'0 exactly (c=2cosh b)'}) + # Exact tangent/normal decomposition and nuisance-coordinate invariance. + response_checks=0 + for bp in (F(-2),F(-1,3)): + for ep in (F(0),F(3,7)): + for bg in (F(1,5),F(-4,9)): + for eg in (F(0),F(7,11)): + normal_response=eg-ep*bg/bp + for mixing in (F(-3),F(0),F(2,7)): + assert (eg+mixing*ep)-ep*(bg+mixing*bp)/bp == normal_response + response_checks+=1 + return {'scope':'Algebraic selection aid; no new percolation simulation or all-size proof', + 'virasoro_c':'0','highest_weight':'5/8', + 'singular_vector':'(L_-2-(2/3)L_-1^2)|h>', + 'singular_checks':singular_checks, 'levels':rows, + 'physical_module_identified':False, + 'counterfamily':curves, + 'response_mixing_equalities':response_checks, + 'check_summary':{'singular_equalities':2, 'module_dimension_sequences':2, + 'PBW_commutators':2,'positive_law_cases':27, + 'response_mixing_equalities':response_checks}} + +if __name__=='__main__': + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output',type=Path) + args=parser.parse_args() + report=check() + text=json.dumps(report,ensure_ascii=False,indent=2,sort_keys=True)+'\n' + if args.output: + args.output.parent.mkdir(parents=True,exist_ok=True) + args.output.write_text(text,encoding='utf-8') + else: + print(text,end='') diff --git a/scripts/safe_transfer_momentum_spectrum.py b/scripts/safe_transfer_momentum_spectrum.py new file mode 100644 index 000000000..8dad0401b --- /dev/null +++ b/scripts/safe_transfer_momentum_spectrum.py @@ -0,0 +1,170 @@ +#!/usr/bin/env python3 +"""Momentum-resolved first descendant of the safe charge transfer. + +At each fixed width locate the charge-coexistence root, diagonalize the safe +G4 and complementary-G8 transfer matrices, and inspect the first excited +(twofold-degenerate) eigenspace under one-column rotation. + +The expected magnetic level-one signature is + + w log(lambda0/|lambda1|) -> 2*pi, + rotation eigenvalues -> exp(+/-2*pi*i/w). + +This is deterministic transfer spectroscopy, not Monte Carlo. +""" +from __future__ import annotations + +import argparse +import json +import math +from pathlib import Path + +import numpy as np +from scipy.optimize import brentq +from scipy.sparse.linalg import eigs + +from fixed_width_charge_transfer import SafeTransfer, State + + +def rotate_state(state: State) -> State: + """Translate the frontier one physical column to the right.""" + width = len(state.labels) + entries = [] + for column, label in enumerate(state.labels): + if label < 0: + continue + new_column = (column + 1) % width + deck = (column + 1) // width + entries.append((new_column, label, state.gains[column] + deck)) + entries.sort() + + labels = [-1] * width + gains = [0] * width + new_label = {} + base_gain = {} + next_label = 0 + for column, label, gain in entries: + if label not in new_label: + new_label[label] = next_label + base_gain[label] = gain + next_label += 1 + labels[column] = new_label[label] + gains[column] = gain - base_gain[label] + return State(tuple(labels), tuple(gains)) + + +def leading_lambda(transfer: SafeTransfer, p: float) -> float: + matrix = transfer.matrix(p) + if matrix.shape[0] <= 4: + return float(max(np.linalg.eigvals(matrix.toarray()).real)) + value = eigs( + matrix, + k=1, + which="LM", + tol=1e-12, + maxiter=200000, + return_eigenvectors=False, + )[0] + return float(value.real) + + +def charge_root(g4: SafeTransfer, g8: SafeTransfer) -> float: + def equation(p: float) -> float: + return math.log(leading_lambda(g4, p)) - math.log( + leading_lambda(g8, 1.0 - p) + ) + + return brentq(equation, 0.5000000001, 0.999999999, xtol=2e-14) + + +def first_descendant(transfer: SafeTransfer, p: float) -> dict[str, object]: + matrix = transfer.matrix(p) + values, vectors = eigs(matrix, k=6, which="LM", tol=1e-11, maxiter=200000) + order = np.argsort(-np.abs(values)) + values = values[order] + vectors = vectors[:, order] + + lam0 = values[0] + lam1 = values[1] + pair = [i for i in range(1, len(values)) if abs(values[i] - lam1) < 1e-7] + if len(pair) < 2: + raise ArithmeticError("first excited eigenvalue is not resolved as a pair") + pair = pair[:2] + basis = vectors[:, pair] + + index = {state: i for i, state in enumerate(transfer.states)} + rotation = np.array([index[rotate_state(state)] for state in transfer.states]) + + # Verify exact commutation of the state permutation with the weighted kernel. + dense_test = matrix[:, rotation][rotation, :] - matrix + if dense_test.nnz and np.max(np.abs(dense_test.data)) > 1e-13: + raise ArithmeticError("one-column translation does not commute with transfer") + + rotated_basis = np.empty_like(basis) + for column in range(basis.shape[1]): + vector = np.empty(basis.shape[0], dtype=complex) + vector[rotation] = basis[:, column] + rotated_basis[:, column] = vector + + representation = np.linalg.lstsq(basis, rotated_basis, rcond=None)[0] + phases = np.linalg.eigvals(representation) + phases = sorted(phases, key=lambda z: np.angle(z)) + + expected = [ + np.exp(-2j * np.pi / transfer.width), + np.exp(+2j * np.pi / transfer.width), + ] + phase_error = max(abs(a - b) for a, b in zip(phases, expected)) + + gap = float(transfer.width * math.log(abs(lam0 / lam1))) + return { + "lambda0": float(lam0.real), + "lambda1": float(lam1.real), + "first_excited_multiplicity_resolved": 2, + "scaled_gap_w_log_lambda0_over_lambda1": gap, + "scaled_gap_over_2pi": gap / (2 * math.pi), + "rotation_eigenvalues": [ + {"real": float(z.real), "imag": float(z.imag)} for z in phases + ], + "expected_momenta": [-1, 1], + "max_rotation_phase_error": float(phase_error), + } + + +def run_width(width: int) -> dict[str, object]: + g4 = SafeTransfer(width, matching=False) + g8 = SafeTransfer(width, matching=True) + root = charge_root(g4, g8) + return { + "width": width, + "charge_root_p": root, + "safe_states": len(g4.states), + "G4": first_descendant(g4, root), + "G8_complement": first_descendant(g8, 1.0 - root), + } + + +def main() -> None: + parser = argparse.ArgumentParser() + parser.add_argument("--min-width", type=int, default=5) + parser.add_argument("--max-width", type=int, default=9) + parser.add_argument("--output", type=Path) + args = parser.parse_args() + + result = { + "schema": "safe-transfer-momentum-spectrum-v1", + "claim_boundary": [ + "deterministic small-width transfer spectroscopy", + "level-one magnetic-descendant interpretation uses the pDTL/CFT sector dictionary", + "higher raw eigenvalues are not assigned to one Verma tower by this script", + ], + "records": [run_width(w) for w in range(args.min_width, args.max_width + 1)], + } + text = json.dumps(result, indent=2, sort_keys=True) + if args.output: + args.output.write_text(text + "\n", encoding="utf-8") + print(text) + + +if __name__ == "__main__": + main() diff --git a/scripts/thermal_ward_root_control.py b/scripts/thermal_ward_root_control.py new file mode 100644 index 000000000..bdc448d79 --- /dev/null +++ b/scripts/thermal_ward_root_control.py @@ -0,0 +1,263 @@ +#!/usr/bin/env python3 +"""Conditional thermal Ward identities and existing-data root diagnostics. + +No new percolation production is performed. Exact algebra uses Fraction; +mpmath evaluates modular series and roots and is not interval arithmetic. +""" +from __future__ import annotations +import argparse +from fractions import Fraction as F +from functools import lru_cache +from math import comb +import json +from pathlib import Path + +H = F(5, 8) +C = F(0) +Vector = dict[tuple[int, ...], F] + + +def add(*vectors: Vector) -> Vector: + out: Vector = {} + for vec in vectors: + for word, coeff in vec.items(): + out[word] = out.get(word, F(0)) + coeff + return {w: c for w, c in out.items() if c} + + +def scale(a: F, vec: Vector) -> Vector: + return {w: a*c for w, c in vec.items() if a*c} + + +@lru_cache(None) +def normal(word: tuple[int, ...]) -> Vector: + """PBW basis uses decreasing positive indices for negative modes.""" + for j in range(len(word)-1): + a, b = word[j:j+2] + if a < b: + swapped = word[:j] + (b, a) + word[j+2:] + merged = word[:j] + (a+b,) + word[j+2:] + return add(normal(swapped), scale(F(b-a), normal(merged))) + return {word: F(1)} + + +@lru_cache(None) +def act_word(m: int, word: tuple[int, ...], h: F = H, c: F = C) -> Vector: + if m < 0: + return normal((-m,) + word) + if m == 0: + return {word: h + sum(word)} + if not word: + return {} + n, rest = word[0], word[1:] + first: Vector = {} + for w, a in act_word(m, rest, h, c).items(): + first = add(first, scale(a, normal((n,) + w))) + second = scale(F(m+n), act_word(m-n, rest, h, c)) + central = {rest: c*F(m**3-m, 12)} if m == n and c else {} + return add(first, second, central) + + +def act(m: int, vec: Vector, h: F = H, c: F = C) -> Vector: + out: Vector = {} + for word, coeff in vec.items(): + out = add(out, scale(coeff, act_word(m, word, h, c))) + return out + + +def descendants() -> tuple[Vector, Vector, Vector]: + chi = {(2,): F(1), (1, 1): -F(2, 3)} + quotient_q = {(4,): F(1), (3, 1): F(20, 11), (1, 1, 1, 1): -F(160, 561)} + qhat = scale(-F(11, 29), add(quotient_q, scale(-F(80, 33), act(-2, chi)))) + return chi, quotient_q, qhat + + +def partitions(n: int, top: int | None = None): + if n == 0: + yield () + return + for k in range(min(n, n if top is None else top), 0, -1): + for tail in partitions(n-k, k): + yield (k,) + tail + + +def rational_rank(vectors: list[Vector], basis: list[tuple[int, ...]]) -> int: + rows = [[v.get(w, F(0)) for w in basis] for v in vectors] + rank = 0 + for col in range(len(basis)): + pivot = next((j for j in range(rank, len(rows)) if rows[j][col]), None) + if pivot is None: + continue + rows[rank], rows[pivot] = rows[pivot], rows[rank] + p = rows[rank][col] + rows[rank] = [x/p for x in rows[rank]] + for j in range(len(rows)): + if j != rank and rows[j][col]: + t = rows[j][col] + rows[j] = [x-t*y for x, y in zip(rows[j], rows[rank])] + rank += 1 + if rank == len(rows): + break + return rank + + +def null_derivative_subspace() -> list[Vector]: + chi, _, _ = descendants() + return [act(-1, {w: F(1)}) for w in partitions(3)] + [ + act(-2, chi), act(-1, act(-1, chi))] + + +def sigma(n: int, power: int) -> int: + return sum(d**power for d in range(1, n+1) if n % d == 0) + + +def eta_product_coeffs(alpha: F, degree: int) -> list[F]: + """Coefficients of prod_(n>=1)(1-q^n)^alpha, without q^(alpha/24).""" + coeffs = [F(1)] + [F(0)]*degree + for n in range(1, degree+1): + fac = [F(1)] + for k in range(1, degree//n+1): + fac.append(-fac[-1]*(alpha-k+1)/k) + new = [F(0)]*(degree+1) + for i in range(degree+1): + for k, a in enumerate(fac): + if i+n*k <= degree: + new[i+n*k] += coeffs[i]*a + coeffs = new + return coeffs + + +def eta_ward_coeffs(h: F, degree: int) -> list[F]: + """(q d/dq - h E2/12) q^(h/12) sum a_n q^n = 0.""" + a = [F(1)] + for n in range(1, degree+1): + a.append(-2*h*sum(F(sigma(k, 1))*a[n-k] for k in range(1, n+1))/n) + return a + + +def eisenstein(tau, weight: int = 4, terms: int = 140): + import mpmath as mp + if mp.im(tau) <= 0 or weight not in (2, 4, 6): + raise ValueError('Require Im(tau)>0 and weight in {2,4,6}.') + q = mp.exp(2j*mp.pi*tau) + factors = {2: -24, 4: 240, 6: -504} + return 1 + factors[weight]*mp.fsum(sigma(n, weight-1)*q**n for n in range(1, terms+1)) + + +RANK_INPUT = { + 5: { + 'rank0': [1,25,300,2300,12650,53120,176900,478700,1068575,1982350,3054880,3869650,3931075,3067350,1723100,639850,141575,15900,550,0,0,0,0,0,0,0], + 'rank1': [0,0,0,0,0,10,200,2000,13000,60600,213280,581350,1229100,1970750,2301450,1861060,994350,340550,72700,9000,520,0,0,0,0,0], + 'rank2': [0,0,0,0,0,0,0,0,0,25,600,6400,40125,162200,432850,767850,907050,725125,407450,168100,52610,12650,2300,300,25,1], + 'cylinder_root': '0.5922358232050258', + }, + 6: { + 'rank0': [1,36,630,7140,58905,376992,1947780,8347320,30254904,94088944,253787076,598517172,1241138298,2270804400,3669278472,5226303348,6525511038,7070366736,6544591544,5064164712,3186853893,1579450836,595366164,164526840,31987068,4124196,313290,10656,72,0,0,0,0,0,0,0,0], + 'rank1': [0,0,0,0,0,0,12,360,5436,54336,399780,2288088,10538070,39962304,126774180,339782112,772416936,1485756360,2397393616,3199280040,3466778256,2990089320,2015358768,1046966400,415851744,125773992,28780308,4889208,589680,45576,1704,0,0,0,0,0,0], + 'rank2': [0,0,0,0,0,0,0,0,0,0,0,36,1332,22896,244548,1817100,9944136,41373504,133150140,334051848,654239961,998362404,1185572268,1099296360,803838888,470907108,225093258,89243416,29670588,8302104,1946088,376992,58905,7140,630,36,1], + 'cylinder_root': '0.592507356205638', + }, +} + + +def rank_probability(counts: list[int], p): + import mpmath as mp + n = len(counts)-1 + return mp.fsum(c*p**k*(1-p)**(n-k) for k, c in enumerate(counts) if c) + + +def root_diagnostics() -> list[dict]: + import mpmath as mp + pc = mp.mpf('0.59274605079210') + target = mp.re(eisenstein(1j)) + records = [] + for size, data in RANK_INPUT.items(): + n = size**2 + good = all(sum(data[f'rank{j}'][k] for j in range(3)) == comb(n,k) for k in range(n+1)) + if not good: + raise ValueError(f'Cardinality normalization failed for L={size}') + f = lambda p: rank_probability(data['rank2'], p)-rank_probability(data['rank0'], p) + root = mp.findroot(f, (mp.mpf('.58'), mp.mpf('.61'))) + cylinder = mp.mpf(data['cylinder_root']) + ratio = (root-pc)/(cylinder-pc) + records.append({'L': size, 'cardinality_sum_check': good, + 'torus_root': mp.nstr(root, 36), + 'cylinder_root_input': data['cylinder_root'], + 'shift_ratio': mp.nstr(ratio, 24), + 'relative_deviation_from_E4_i': mp.nstr(ratio/target-1, 24), + 'ratio_pc_derivative': mp.nstr((root-cylinder)/(cylinder-pc)**2, 24), + 'normalization_only_not_rank_assignment_audit': True}) + return records + + +def report(dps: int = 70) -> dict: + import mpmath as mp + if dps < 45: + raise ValueError('Use at least 45 digits for reproducible controls.') + mp.mp.dps = dps + chi, q, qhat = descendants() + basis = list(partitions(4)) + sub = null_derivative_subspace() + reduced = add(qhat, {(4,): -F(1)}) + assert not act(1, chi) and not act(2, chi) + assert not act(1, qhat) + assert add(act(1,q), scale(-F(80,11),act(-1,chi))) == {} + assert rational_rank(sub,basis) == rational_rank(sub+[reduced],basis) == 4 + tau = mp.mpc('.21','1.17') + e = eisenstein(tau) + rho = mp.mpc(F(1,2).numerator/F(1,2).denominator,mp.sqrt(3)/2) + serialize = lambda v: {','.join(map(str,w)): str(a) for w,a in sorted(v.items())} + return { + 'schema': 'thermal-null-ward-modular-root-v1', + 'scope': 'Exact conditional Virasoro algebra; numerical modular functions; reanalysis of supplied rank counts. No site scaling theorem or new large enumeration.', + 'repo_snapshot': '8e1282f6d4397f03c80a2883b917c2d31a4b93a3', + 'exact': { + 'thermal_h': str(H), 'chi': serialize(chi), + 'normalized_quasiprimary': serialize(qhat), + 'L1_quasiprimary': serialize(act(1,qhat)), + 'quotient_and_derivative_rank_at_level4': rational_rank(sub,basis), + 'normalized_class': 'L_-4 modulo null descendants and total derivatives', + 'retained_null_correction_coefficient': '80/87', + 'retained_null_form': 'U4 = L_-4 epsilon + (80/87)L_-2 chi modulo total derivatives', + 'null_diagonal_magnetic_weight': str(F(2,3)*H*(H+1)-H), + 'cylinder_Lminus2_coefficient_at_magnetic_weight': str(F(5,96)-H/12), + 'cylinder_chiral_Lminus4_coefficient': str(H/240), + 'torus_chiral_G4_coefficient': str(3*H), + 'torus_chiral_pi4_E4_coefficient': str(H/15), + 'eta_without_q_leading_coeffs': [str(x) for x in eta_product_coeffs(2*H,10)], + 'eta_product_matches_null_ward_ODE': eta_product_coeffs(2*H,10) == eta_ward_coeffs(H,10), + 'first_trace_coefficient_from_primary_3point': str((2*F(5,96)+H*(H-1))/(2*F(5,96))), + }, + 'modular_numerical': { + 'E4_i': mp.nstr(mp.re(eisenstein(1j)),36), + 'E4_2i': mp.nstr(mp.re(eisenstein(2j)),36), + 'hex_E4_abs': mp.nstr(abs(eisenstein(rho)),8), + 'hex_E4_derivative': mp.nstr(-(2j*mp.pi/3)*eisenstein(rho,6),30), + 'S_transform_error': mp.nstr(abs(eisenstein(-1/tau)-tau**4*e),8), + 'T_transform_error': mp.nstr(abs(eisenstein(tau+1)-e),8), + 'arithmetic_is_not_outward_interval': True, + }, + 'existing_data_diagnostics': root_diagnostics(), + 'existing_data_provenance': { + 'rank5': 'results/geometric-consistency/rank-sector-C-L5-20260914.json; blob e3b03a7583b5346083907a4531ba51dc4f3d8819', + 'rank6': 'results/geometric-consistency/rank-sector-C-L6-20260914.json; blob 4146a7bd51ca8f2e5c964b5c157ce1b2c4bd1f2e', + 'cylinders': 'results/geometric-consistency/fixed-width-charge-spectrum-derivatives-w4-w8-20260914.json; printed root values; no independent Perron rerun', + 'pc_reference_not_certificate': '0.59274605079210', + }, + } + + +def main() -> None: + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--output', type=Path) + ap.add_argument('--dps', type=int, default=70) + args = ap.parse_args() + text = json.dumps(report(args.dps), ensure_ascii=False, indent=2, sort_keys=True)+'\n' + if args.output: + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(text,encoding='utf-8') + else: + print(text,end='') + +if __name__ == '__main__': + main() diff --git a/scripts/v14_w22_collision_check.py b/scripts/v14_w22_collision_check.py new file mode 100644 index 000000000..04e96a786 --- /dev/null +++ b/scripts/v14_w22_collision_check.py @@ -0,0 +1,46 @@ +#!/usr/bin/env python3 +"""Exact algebra for the V_<1,4> / W(2,2)-top collision at percolation. + +Use t=beta^2 and critical-Potts weights + + h_{r,s}=(c-1)/24 + 1/4 (r beta - s/beta)^2. + +The common central-charge term cancels from the difference between h_{1,4} +and h_{2,-2}. Percolation has t=2/3 and Q=4 cos^2(pi t). +""" +from fractions import Fraction +import math + + +def delta_h(t: Fraction) -> Fraction: + """h_{1,4}-h_{2,-2}, with t=beta^2.""" + return Fraction(1, 4) * (-3 * t - 16 + Fraction(12, 1) / t) + + +def d_delta_x_dt(t: Fraction) -> Fraction: + """Derivative of x_{1,4}-x_{2,-2}=2 delta_h with respect to t.""" + return Fraction(1, 2) * (-3 - Fraction(12, 1) / (t * t)) + + +def main() -> None: + t = Fraction(2, 3) + assert delta_h(t) == 0 + assert d_delta_x_dt(t) == -15 + + # Zero condition: 3 t^2 + 16 t - 12 = 0. + # Physical roots are t=2/3 and t=-6; only the first is positive. + assert 3 * t * t + 16 * t - 12 == 0 + + dQ_dt = 2 * math.pi * math.sqrt(3.0) # at t=2/3 + dx_dQ = -15.0 / dQ_dt + + print("t_percolation = 2/3") + print("Delta h = h_(1,4)-h_(2,-2) = 0") + print("d_t Delta x = -15") + print("d_t Q = 2*pi*sqrt(3)") + print("d_Q Delta x = -15/(2*pi*sqrt(3)) =", dx_dQ) + print("PASS: V_<1,4> collides with the W(2,2) diagonal top/bottom weight exactly at Q=1") + + +if __name__ == "__main__": + main() diff --git a/scripts/verify768_v1_verma.py b/scripts/verify768_v1_verma.py new file mode 100644 index 000000000..d78dfb02d --- /dev/null +++ b/scripts/verify768_v1_verma.py @@ -0,0 +1,348 @@ +# -*- coding: utf-8 -*- +""" +v768_v1_verma.py --- verify768 / V1 + +独立重算 (不用 theory768 的脚本、不用它的 Gram 实现)： + (A) Verma 等级维数 p(n)（自写 DP） + (B) c=0,h=5/8 的 Gram 矩阵 G_n 的 rank / nullity + -> 判 rank(G_n) == p(n) - p(n-2)（这次用正确公式！theory768 的 JSON 里 + rank_equals_p_shift2=false 是脚本 bug：它把 rank 拿去和 p(n-2) 比） + (C) 各级奇异向量维数 + (D) chi = -3 L_{-2}|h> + 2 L_{-1}^2|h> 的范数（两条独立路径） + (E) level 1..4 模去 L_{-1} 后的维数（不可约商 & Verma 两种口径） + (F) 把 0,0,1,1 是否只是「rank=p(n)-p(n-2) + L_{-1} 单射」的算术重述单独判出来 + +约化机制（与 theory768 不同、且结构上必然终止）： + 把 L_m 逐个从右到左乘到一个「已正规序」的 PBW 词（负模升序）上， + 用递归 insert/push 实现： + L_m L_a = L_a L_m + (m-a) L_{m+a} (+ (C/12)(m^3-m) delta_{m+a,0}) + 每次递归 word 长度严格减少 => 终止。全部分数精确。 + +PY39COMPAT_MARKER=1 +""" +import json +import os +import sys +from fractions import Fraction + +assert (255).bit_count() == 8, "py39 compat shim not active" +MARK = "PY39COMPAT_MARKER=1" +NMAX = int(os.environ.get("V768_NMAX", "7")) + + +def partitions_counts(nmax): + p = [0] * (nmax + 1) + p[0] = 1 + for k in range(1, nmax + 1): + for n in range(k, nmax + 1): + p[n] += p[n - k] + return p + + +# ---------- 核心：单模乘到正规序词上 ---------- +def reduce_one(cval, hval, m, word, memo=None): + """word: 正规序 PBW 词（负模升序，即模长降序）；返回 dict word->coeff。 + L_m 乘在 word 左端。""" + C = Fraction(cval) + if memo is None: + memo = {} + key = (m, word) + if key in memo: + return memo[key] + if not word: + if m < 0: + r = {(m,): Fraction(1)} + elif m == 0: + r = {(): Fraction(hval)} + else: + r = {} + memo[key] = r + return r + a = word[0] + rest = word[1:] + if m < 0 and m <= a: + r = {(m,) + word: Fraction(1)} + memo[key] = r + return r + # t1 = L_a · (L_m · |rest>) + t1 = {} + for w, c in reduce_one(cval, hval, m, rest, memo).items(): + for w2, c2 in reduce_one(cval, hval, a, w, memo).items(): + t1[w2] = t1.get(w2, Fraction(0)) + c * c2 + t2 = {} + c = m + a + if c == 0: + for w, cc in reduce_one(cval, hval, 0, rest, memo).items(): + t2[w] = t2.get(w, Fraction(0)) + (m - a) * cc + cent = C / 12 * (m ** 3 - m) + t2[rest] = t2.get(rest, Fraction(0)) + cent + else: + for w, cc in reduce_one(cval, hval, c, rest, memo).items(): + t2[w] = t2.get(w, Fraction(0)) + (m - a) * cc + r = {} + for w, v in t1.items(): + r[w] = r.get(w, Fraction(0)) + v + for w, v in t2.items(): + r[w] = r.get(w, Fraction(0)) + v + r = {w: v for w, v in r.items() if v != 0} + memo[key] = r + return r + + +def act(cval, hval, state, m): + """L_m · state""" + acc = {} + for word, coeff in state.items(): + for w, v in reduce_one(cval, hval, m, word).items(): + acc[w] = acc.get(w, Fraction(0)) + coeff * v + return {w: v for w, v in acc.items() if v != 0} + + +def reduce_seq(cval, hval, seq): + st = {(): Fraction(1)} + for m in reversed(tuple(seq)): + st = act(cval, hval, st, m) + return st + + +def mul_left(cval, hval, m, state): + return act(cval, hval, state, m) + + +def level_basis(n): + res = [] + + def rec(rem, biggest, cur): + if rem == 0: + res.append(tuple(sorted(cur, reverse=True))) + return + for k in range(min(rem, biggest), 0, -1): + rec(rem - k, k, cur + [k]) + + rec(n, n, []) + out = sorted(set(tuple(sorted((-x for x in w))) for w in res)) + return out + + +def gram_entry(cval, hval, wi, wj): + """；wi,wj 为负模升序 PBW 词""" + As = [-m for m in wi] + seq = list(reversed(As)) + list(wj) + d = reduce_seq(cval, hval, tuple(seq)) + return d.get((), Fraction(0)) + + +def gram_matrix(cval, hval, n, B): + idx = {w: i for i, w in enumerate(B[n])} + G = [[Fraction(0)] * len(B[n]) for _ in range(len(B[n]))] + for wi in B[n]: + for wj in B[n]: + G[idx[wi]][idx[wj]] = gram_entry(cval, hval, wi, wj) + return G + + +# ---------- 精确有理数线性代数 ---------- +def frac_rref(rows, ncols): + M = [[Fraction(x) for x in r] for r in rows] + m = len(M) + piv_cols = [] + r = 0 + for c in range(ncols): + piv = None + for i in range(r, m): + if M[i][c] != 0: + piv = i + break + if piv is None: + continue + M[r], M[piv] = M[piv], M[r] + pv = M[r][c] + M[r] = [x / pv for x in M[r]] + for i in range(m): + if i != r and M[i][c] != 0: + f = M[i][c] + M[i] = [x - f * y for x, y in zip(M[i], M[r])] + piv_cols.append(c) + r += 1 + return M, piv_cols + + +def frac_rank(rows, ncols): + if not rows: + return 0 + if rows and len(rows[0]) == 0: + return 0 + M, piv = frac_rref(rows, ncols) + return len(piv) + + +def frac_nullspace(G): + m = len(G) + n = len(G[0]) if m else 0 + if m == 0: + return [] + M, piv_cols = frac_rref(G, n) + free = [c for c in range(n) if c not in piv_cols] + basis = [] + for f in free: + v = [Fraction(0)] * n + v[f] = Fraction(1) + for ri, pc in enumerate(piv_cols): + v[pc] = -M[ri][f] + basis.append(v) + return basis + + +def run(cval, hval, label): + res = {"label": label, "c": str(cval), "h": str(hval)} + p = partitions_counts(NMAX) + res["p"] = p + B = {n: level_basis(n) for n in range(NMAX + 1)} + res["verma_dims"] = {n: len(B[n]) for n in range(NMAX + 1)} + assert res["verma_dims"] == {n: p[n] for n in range(NMAX + 1)}, "PBW 计数 != p(n)" + G = {n: gram_matrix(cval, hval, n, B) for n in range(NMAX + 1)} + rank = {n: frac_rank(G[n], len(B[n])) for n in range(NMAX + 1)} + res["gram_rank"] = rank + res["gram_nullity"] = {n: len(B[n]) - rank[n] for n in range(NMAX + 1)} + ok = {} + for n in range(NMAX + 1): + tgt = p[n] - (p[n - 2] if n >= 2 else 0) + ok[n] = (rank[n] == tgt) + res["rank_eq_p_minus_p_shift2"] = ok + res["rank_eq_p_minus_p_shift2_all"] = all(ok.values()) + res["theory768_style_flag_p_shift2_only"] = all( + rank[n] == (p[n - 2] if n >= 2 else 0) for n in range(NMAX + 1)) + # (C) 奇异向量维数 = dim ker( all L_m : V_n -> V_{n-m}, m=1..n ) + sing = {} + for n in range(NMAX + 1): + if n == 0: + sing[n] = 0 + continue + rows = [] + for mm in range(1, n + 1): + # 形状 (len(B[n-mm]), len(B[n]))：列 = V_n 的基 + M = [[Fraction(0)] * len(B[n]) for _ in range(len(B[n - mm]))] + for j, wj in enumerate(B[n]): + d = mul_left(cval, hval, mm, {wj: Fraction(1)}) + for w, v in d.items(): + if w in B[n - mm]: + M[B[n - mm].index(w)][j] = v + rows.extend(M) + rr = frac_rank(rows, len(B[n])) if rows else 0 + sing[n] = len(B[n]) - rr + res["singular_dims"] = sing + # (D) chi + if Fraction(hval) == Fraction(5, 8): + w2 = B[2] + i2, i11 = w2.index((-2,)), w2.index((-1, -1)) + chi = [Fraction(-3), Fraction(2)] + q1 = sum(chi[i] * G[2][i][j] * chi[j] for i in range(2) for j in range(2)) + res["chi_int"] = [str(x) for x in chi] + res["chi_norm_gram"] = str(q1) + acc = {} + for w, v in reduce_seq(cval, hval, (-2,)).items(): + acc[w] = acc.get(w, Fraction(0)) + Fraction(-3) * v + for w, v in reduce_seq(cval, hval, (-1, -1)).items(): + acc[w] = acc.get(w, Fraction(0)) + Fraction(2) * v + keys = sorted(acc) + q2 = Fraction(0) + for u in keys: + for v in keys: + q2 += acc[u] * acc[v] * gram_entry(cval, hval, u, v) + res["chi_expansion"] = {str(k): str(v) for k, v in acc.items()} + res["chi_norm_direct"] = str(q2) + res["chi_agrees"] = (q1 == q2) + for mm in (1, 2, 3): + d = mul_left(cval, hval, mm, acc) + res["L%d_chi" % mm] = {str(k): str(v) for k, v in d.items()} + res["L%d_chi_zero" % mm] = (len(d) == 0) + res["Lm2_norm"] = str(G[2][i2][i2]) + res["Lm1sq_norm"] = str(G[2][i11][i11]) + res["detG2_c0"] = None + # det G_2 的通式检查 + if cval == "0": + w2 = B[2] + i2, i11 = w2.index((-2,)), w2.index((-1, -1)) + a, b, d = G[2][i2][i2], G[2][i2][i11], G[2][i11][i11] + num = Fraction(a * d - b * b) + res["detG2_c0_factor"] = str(num) + H = Fraction(hval) + res["detG2_expected_4h2_8h_minus_5"] = str(4 * H * H * (8 * H - 5)) + # (E) mod L_{-1} + def modLm1_irred(n): + if n == 0: + return rank[0] + Bn, Bp = B[n], B[n - 1] + Rn = frac_nullspace(G[n]) + Rp = frac_nullspace(G[n - 1]) + span = [list(v) for v in Rp] + cur = frac_rank(span, len(Bp)) if span else 0 + comp = [] + for i in range(len(Bp)): + e = [Fraction(0)] * len(Bp) + e[i] = Fraction(1) + if frac_rank(span + [e], len(Bp)) > cur: + span.append(e) + comp.append(i) + cur += 1 + if cur == len(Bp): # 张满 V_{n-1}（不是 cur==rank[n-1]！） + break + assert cur == len(Bp), (n, cur, len(Bp)) + cols = [] + for i in comp: + d = mul_left(cval, hval, -1, {Bp[i]: Fraction(1)}) + vec = [Fraction(0)] * len(Bn) + for kw, kv in d.items(): + if kw in Bn: + vec[Bn.index(kw)] = kv + cols.append(vec) + Rb = [list(v) for v in Rn] + rr = frac_rank(cols + Rb, len(Bn)) if (cols or Rb) else 0 + e_n = rr - len(Rn) + return rank[n] - e_n + + res["modLm1_irred"] = {n: modLm1_irred(n) for n in range(0, NMAX + 1)} + res["modLm1_irred_seq_1_4"] = [res["modLm1_irred"][n] for n in (1, 2, 3, 4)] + vm = {} + for n in range(NMAX + 1): + if n == 0: + vm[0] = 1 + continue + cols = [] + for w in B[n - 1]: + d = mul_left(cval, hval, -1, {w: Fraction(1)}) + vec = [Fraction(0)] * len(B[n]) + for kw, kv in d.items(): + vec[B[n].index(kw)] = kv + cols.append(vec) + r = frac_rank(cols, len(B[n])) if cols else 0 + vm[n] = len(B[n]) - r + res["modLm1_verma"] = vm + res["modLm1_verma_seq_1_4"] = [vm[n] for n in (1, 2, 3, 4)] + d = rank + q_from_d = {n: (d[n] - d[n - 1] if n >= 1 else d[0]) for n in range(NMAX + 1)} + res["q_from_rank_minus_rank_prev"] = q_from_d + res["q_from_d_seq_1_4"] = [q_from_d[n] for n in (1, 2, 3, 4)] + res["is_0011_arithmetic_restatement"] = ( + [q_from_d[n] for n in (1, 2, 3, 4)] == [res["modLm1_irred"][n] for n in (1, 2, 3, 4)]) + res["Lm1_injective_on_quotient"] = all( + res["modLm1_irred"][n] == rank[n] - rank[n - 1] for n in range(1, NMAX + 1)) + return res + + +def main(): + out = {"marker": MARK, "nmax": NMAX, + "route": "independent greedy insert/push normal-ordering recursion"} + runs = {} + runs["c0_h58"] = run("0", "5/8", "c=0,h=5/8 (thermal)") + runs["generic_c12_h37"] = run("1/2", "3/7", "generic control (1/2,3/7)") + runs["c0_h0"] = run("0", "0", "c=0 vacuum h=0") + out["runs"] = runs + with open("v768_v1_result.json", "w") as f: + json.dump(out, f, indent=1, ensure_ascii=False) + print(json.dumps(out, indent=1, ensure_ascii=False)) + return out + + +if __name__ == "__main__": + main() diff --git a/scripts/verify768_v1b_diag.py b/scripts/verify768_v1b_diag.py new file mode 100644 index 000000000..057852f22 --- /dev/null +++ b/scripts/verify768_v1b_diag.py @@ -0,0 +1,205 @@ +# -*- coding: utf-8 -*- +""" +v768_v1b_diag.py --- 判定 e_n = dim(L_{-1} 在不可约商 level n 的像) 到底是多少 + +theory768 (g2_virasoro.py) 用 proj_quot 得到 e_4 = 2 (=d_3)； +我用「complement + mod rad_n」得到 e_4 = 1。两者若都对就矛盾，必须判定。 + +这里给 5 条互相独立的路线（n=3,4,5）： + A. dim((L_{-1}P_{n-1} + rad_n)/rad_n) —— 我的路线 + B. dim((L_{-1}V_{n-1} + rad_n)/rad_n) —— theory768 的路线(等价形式) + C. rank(诱导映射) 用显式商坐标（rref 主元坐标做代表） + D. d_n - (dim W - dim rad_n), W={v∈V_n : L_1 v ∈ rad_{n-1}} —— 用 L_1 的核（伴随） + E. 显式找 0 != u ∈ V_{n-1}, u ∉ rad_{n-1}, 且 L_{-1}u ∈ rad_n， + 并给出证书： = 0 对所有 x ∈ V_n 的基。 + F. 分解证书：dim(L_{-1}V_{n-1} ∩ rad_n) 显式算出来。 + +PY39COMPAT_MARKER=1 +""" +import json +from fractions import Fraction + +import v768_v1_verma as V + +NMAX = 6 +CV, HV = "0", "5/8" + + +def rref(rows, ncols): + return V.frac_rref(rows, ncols) + + +def rank(rows, ncols): + return V.frac_rank(rows, ncols) + + +def rowspace_basis(rows, ncols): + if not rows: + return [] + R, piv = rref(rows, ncols) + return [R[i] for i in range(len(piv))] + + +def mat_of(mode, nfrom, nto, B): + """L_mode : V_nfrom -> V_nto ，形状 (len(nto), len(nfrom))""" + if nto < 0: + return [] + M = [[Fraction(0)] * len(B[nfrom]) for _ in range(len(B[nto]))] + for j, w in enumerate(B[nfrom]): + d = V.mul_left(CV, HV, mode, {w: Fraction(1)}) + for kw, kv in d.items(): + if kw in B[nto]: + M[B[nto].index(kw)][j] = kv + return M + + +def complement_indices(rad_rows, n): + """返回标准基下标，与 rad_rows 组成 V_n 的一组基（即商空间代表）""" + span = [list(r) for r in rad_rows] + cur = rank(span, n) if span else 0 + comp = [] + for i in range(n): + e = [Fraction(0)] * n + e[i] = Fraction(1) + if rank(span + [e], n) > cur: + span.append(e) + comp.append(i) + cur += 1 + if cur == n: + break + return comp + + +def main(): + out = {"marker": "PY39COMPAT_MARKER=1", "c": CV, "h": HV, "nmax": NMAX} + B = {n: V.level_basis(n) for n in range(NMAX + 1)} + G = {n: V.gram_matrix(CV, HV, n, B) for n in range(NMAX + 1)} + d = {n: rank(G[n], len(B[n])) for n in range(NMAX + 1)} + rad = {n: V.frac_nullspace(G[n]) for n in range(NMAX + 1)} + out["d"] = d + out["dim_rad"] = {n: len(rad[n]) for n in range(NMAX + 1)} + res = {} + for n in range(2, NMAX + 1): + bn, bp = len(B[n]), len(B[n - 1]) + A = mat_of(-1, n - 1, n, B) # L_{-1}: V_{n-1} -> V_n (bn x bp) + Rn = [list(v) for v in rad[n]] + Rp = [list(v) for v in rad[n - 1]] + cA = complement_indices([list(r) for r in Rp], len(B[n - 1])) + # A. + colsA = [[A[r][i] for r in range(bn)] for i in cA] + eA = rank(colsA + Rn, bn) - len(Rn) if (colsA or Rn) else 0 + # B. + colsAll = [[A[r][i] for r in range(bn)] for i in range(bp)] + eB = rank(colsAll + Rn, bn) - len(Rn) if (colsAll or Rn) else 0 + # C. 诱导映射的显式商坐标 + cN = complement_indices([list(r) for r in Rn], bn) + # 把 V_n 的向量投影到 non-pivot 坐标（与 complement 等价的投影） + proj = {} + for i in cN: + e = [Fraction(0)] * bn + e[i] = Fraction(1) + # 沿 rad 方向消掉主元 + outv = e[:] + # 用 rad 的 rref 消主元 + Rb = rowspace_basis([list(r) for r in Rn], bn) + if Rb: + Rr, piv = rref(Rb, bn) + acc = [Fraction(0)] * bn + for ii, pc in enumerate(piv): + acc = [a + outv[pc] * Rr[ii][k] for k, a in enumerate(acc)] + outv = [a - b for a, b in zip(outv, acc)] + proj[i] = outv + colsC = [] + for i in cA: + v = [A[r][i] for r in range(bn)] + # 投影 + Rb = rowspace_basis([list(r) for r in Rn], bn) + outv = v[:] + if Rb: + Rr, piv = rref(Rb, bn) + acc = [Fraction(0)] * bn + for ii, pc in enumerate(piv): + acc = [a + outv[pc] * Rr[ii][k] for k, a in enumerate(acc)] + outv = [a - b for a, b in zip(outv, acc)] + colsC.append(outv) + eC = rank(colsC, bn) if colsC else 0 + # D. 用 L_1 + A1 = mat_of(1, n, n - 1, B) # L_1: V_n -> V_{n-1} + Rb = rowspace_basis([list(r) for r in Rp], len(B[n - 1])) + if Rb: + Rr, piv = rref(Rb, len(B[n - 1])) + PA1 = [[Fraction(0)] * bn for _ in range(len(B[n - 1]))] + for r in range(len(B[n - 1])): + for j in range(bn): + PA1[r][j] = A1[r][j] + for ii, pc in enumerate(piv): + for r in range(len(B[n - 1])): + f = PA1[pc][0] if False else None + # 显式：W = ker(P·A1)，P 是投影到 rad_{n-1} 的补 + PA = [] + for j in range(bn): + col = [A1[r][j] for r in range(len(B[n - 1]))] + acc = [Fraction(0)] * len(B[n - 1]) + for ii, pc in enumerate(piv): + acc = [a + col[pc] * Rr[ii][k] for k, a in enumerate(acc)] + PA.append([a - b for a, b in zip(col, acc)]) + # PA[j] 是长度 (n-1) 的列；组成矩阵 (n-1) x n，求核维数 + Mt = [[PA[j][r] for j in range(bn)] for r in range(len(B[n - 1]))] + dimW = bn - rank(Mt, bn) + else: + dimW = bn - rank(A1, bn) + eD = d[n] - (dimW - len(Rn)) + # F. dim(L_{-1}V_{n-1} ∩ rad_n) + # L_{-1}V_{n-1} ∩ rad_n 的维数 = dim L_{-1}V_{n-1} + dim rad_n - dim(L_{-1}V_{n-1}+rad_n) + dimImg = rank(colsAll, bn) if colsAll else 0 + inter = len(Rn) + dimImg - (eB + len(Rn)) + res[n] = {"d_n": d[n], "eA": eA, "eB": eB, "eC": eC, "eD": eD, + "dLm1_prev": d[n - 1], "dimW": dimW, + "dimLm1V_prev": dimImg, "dim_inter_with_rad_n": inter} + print("n=%d d=%d | eA=%d eB=%d eC=%d eD=%d | d_{n-1}=%d dim(L_-1 V_prev)=%d " + "inter=%d dimW=%d" % (n, d[n], eA, eB, eC, eD, d[n - 1], dimImg, inter, dimW), + flush=True) + out["per_level"] = res + + # E. 显式核向量证书（n=4） + n = 4 + bn, bp = len(B[n]), len(B[n - 1]) + A = mat_of(-1, n - 1, n, B) + Rn = [list(v) for v in rad[n]] + Rp = [list(v) for v in rad[n - 1]] + # 求 u ∈ V_3 使 L_{-1}u ∈ rad_4 且 u ∉ rad_3 + # 即 A u ∈ span(Rn)。构造投影 P4 消掉 rad_4 方向后求核。 + Rb = rowspace_basis(Rn, bn) + Rr, piv = rref(Rb, bn) + PAu = [] + for j in range(bp): + col = [A[r][j] for r in range(bn)] + acc = [Fraction(0)] * bn + for ii, pc in enumerate(piv): + acc = [a + col[pc] * Rr[ii][k] for k, a in enumerate(acc)] + PAu.append([a - b for a, b in zip(col, acc)]) + Mt = [[PAu[j][r] for j in range(bp)] for r in range(bn)] + ns = V.frac_nullspace(Mt) + cert = [] + for u in ns: + # 检查 u ∉ rad_3 + inrad3 = rank([list(u)] + [list(r) for r in Rp], bp) == len(Rp) + # 检查 L_{-1}u ⊥ V_4 + Au = [sum(A[r][j] * u[j] for j in range(bp)) for r in range(bn)] + # L_{-1}u 的范数/正交性： = u^T A^T G_n x + orth = all(sum(Au[r] * G[n][r][c] * x[c] for r in range(bn) for c in range(bn)) == 0 + for x in [[Fraction(1) if k == i else Fraction(0) for k in range(bn)] + for i in range(bn)]) + selfnorm = sum(Au[r] * G[n][r][c] * Au[c] for r in range(bn) for c in range(bn)) + cert.append({"u": [str(x) for x in u], "u_in_rad3": inrad3, + "Lm1u_orthogonal_to_V4": orth, "Lm1u_norm": str(selfnorm), + "Lm1u_is_zero": all(x == 0 for x in Au)}) + out["kernel_certificate_n4"] = cert + print("kernel certificate n=4:", json.dumps(cert, ensure_ascii=False), flush=True) + + with open("v768_v1b_diag.json", "w") as f: + json.dump(out, f, indent=1, ensure_ascii=False) + + +if __name__ == "__main__": + main() diff --git a/scripts/verify768_v2_classes.py b/scripts/verify768_v2_classes.py new file mode 100644 index 000000000..b11183957 --- /dev/null +++ b/scripts/verify768_v2_classes.py @@ -0,0 +1,184 @@ +# -*- coding: utf-8 -*- +""" +v768_v2_classes.py --- verify768 / V2（核心） + +独立重算 theory768 §1.3/§3 的 (a,b) 非手征类表，并把 V2 的四项核验做成可复算的数字： + 1) 权重/自旋算术：thermal primary (5/8,5/8)；thermal level-s 手征后代 (5/8+s, 5/8)； + scalar 8-arm = Kac h_{4,2} = h_{2,7}；x = h+hbar；spin = h-hbar。 + 2) C4 / C3 / C6 选择定则：spin s 不变 <=> exp(i s theta)=1。 + 3) (a,b) 类表：dim = q_a*q_b，q 取「不可约商 (i)」与「保留零范态 (ii)=Verma」两套。 + 标出 spin==0（动量 0）与 C4 允许，回答「x=21/4 处动量 0 的类到底是谁」。 + 4) 关键：在 (ii) 里 level-2 那个"多出来的非导数类"是否就是零范态 chi 的方向 + （若是，则它的关联函数恒为零，(2,2) 类不进入动量 0 观测量）。 + 做法：把 L_{-2}|h> 模 L_{-1} 的类算出来，与 chi 的类比较。 + +PY39COMPAT_MARKER=1 +""" +import json +import os +from fractions import Fraction +from math import gcd + +import v768_v1_verma as V1 # 复用我自己的约化机制（不是 theory768 的） + +V768_NMAX = int(os.environ.get("V768_NMAX", "7")) + + +def kac_h(r, s): + k = 3 * r - 2 * s + return Fraction(k * k - 1, 24) + + +def rot_eigen_is_one(s, order): + """exp(i s * (2pi/order)) == 1 ? <=> s*(2pi/order) in 2pi Z <=> order | s""" + return s % order == 0 + + +def main(): + OUT = {} + OUT["marker"] = "PY39COMPAT_MARKER=1" + + # ---- 1) 权重算术 ---- + ht = Fraction(5, 8) + arms = {} + for s in range(0, 9): + h = ht + s + arms["s%d" % s] = {"h": str(h), "hbar": str(ht), + "x": str(h + ht), "spin": str(h - ht)} + OUT["thermal_chiral_ladder"] = { + "primary": {"h": str(ht), "hbar": str(ht), "x": str(2 * ht), "spin": "0"}, + "descendants": arms, + } + # 8-arm scalar Kac + OUT["kac_scalar_table"] = { + "k%d" % k: {"h": str(Fraction(k * k - 1, 24)), + "x": str(2 * Fraction(k * k - 1, 24))} + for k in range(0, 11) + } + OUT["eight_arm"] = { + "h_4_2": str(kac_h(4, 2)), "h_2_7": str(kac_h(2, 7)), + "equal": kac_h(4, 2) == kac_h(2, 7), + "x": str(2 * kac_h(4, 2)), "spin": "0", + "same_x_as_thermal_s4": (2 * kac_h(4, 2) == 2 * ht + 4), + } + OUT["spin4_thermal"] = { + "h": str(ht + 4), "hbar": str(ht), "x": str(ht + 4 + ht), "spin": str(4), + "x_equals_21_4": (ht + 4 + ht == Fraction(21, 4)), + } + + # ---- 2) 旋转选择定则 ---- + OUT["rotation_rules"] = { + "C4_square": {str(s): rot_eigen_is_one(s, 4) for s in range(1, 9)}, + "C3_triangular": {str(s): rot_eigen_is_one(s, 3) for s in range(1, 9)}, + "C6_hexagonal": {str(s): rot_eigen_is_one(s, 6) for s in range(1, 9)}, + "first_allowed_square": 4, "first_allowed_C3": 3, "first_allowed_C6": 6, + } + + # ---- 3) (a,b) 类表 ---- + runs = {} + for key, (cval, hval, label) in { + "c0_h58": ("0", "5/8", "thermal (5/8,5/8)"), + }.items(): + r = V1.run(cval, hval, "%s (V2 classes)" % label) + runs[key] = r + OUT["v1_rerun"] = {k: {kk: vv for kk, vv in v.items() + if kk in ("gram_rank", "modLm1_irred", "modLm1_verma", + "gram_nullity", "singular_dims")} + for k, v in runs.items()} + q_irr = runs["c0_h58"]["modLm1_irred"] + q_ver = runs["c0_h58"]["modLm1_verma"] + OUT["q_irreducible_(i)"] = q_irr + OUT["q_verma_(ii)"] = q_ver + + def table(q, amax, xs): + rows = [] + for a in range(0, amax + 1): + for b in range(0, amax + 1): + dim = q.get(a, 0) * q.get(b, 0) + if dim == 0: + continue + spin = a - b + x = Fraction(5, 4) + a + b + rows.append({"a": a, "b": b, "spin": spin, "x": str(x), "dim": dim, + "momentum0": spin == 0, + "C4_ok": rot_eigen_is_one(spin, 4)}) + return rows + + amax = 4 + OUT["classes_(i)_amax4"] = table(q_irr, amax, None) + OUT["classes_(ii)_amax4"] = table(q_ver, amax, None) + for tag, rows in (("(i)", OUT["classes_(i)_amax4"]), ("(ii)", OUT["classes_(ii)_amax4"])): + at214 = [r for r in rows if Fraction(r["x"]) == Fraction(21, 4)] + OUT["at_x_21_4%s" % tag] = at214 + OUT["momentum0_C4_ok_at_x_21_4%s" % tag] = [ + r for r in at214 if r["momentum0"] and r["C4_ok"]] + + # ---- 4) (ii) 里 level-2 多出来的非导数类 == null 方向？ ---- + # 把 (2,) 的类算出来，与 chi 的类比较（都模去 L_{-1}） + cval, hval = "0", "5/8" + B = {n: V1.level_basis(n) for n in range(0, 4)} + # level-2: V_2 的 L_{-1} 像是 span{L_{-1}L_{-1}|h>} ；商空间 1 维，代表 (2,) + # 取线性泛函：与 [L_{-2}] 的"模 L_{-1}"类比较 —— 即看 (2,) 与 chi 是否差一个导数 + # 用显式坐标：V_2 基 (-2,),(-1,-1) + e_m2 = {"(-2,)": Fraction(1)} # 用 tuple 键不方便，直接算 + s_m2 = V1.reduce_seq(cval, hval, (-2,)) # L_{-2}|h> + s_m1m1 = V1.reduce_seq(cval, hval, (-1, -1)) # L_{-1}^2|h> + chi = {} + for w, v in s_m2.items(): + chi[w] = chi.get(w, Fraction(0)) + Fraction(-3) * v + for w, v in s_m1m1.items(): + chi[w] = chi.get(w, Fraction(0)) + Fraction(2) * v + OUT["chi_expansion_level2"] = {str(k): str(v) for k, v in chi.items()} + OUT["L_m2_expansion"] = {str(k): str(v) for k, v in s_m2.items()} + OUT["L_m1sq_expansion"] = {str(k): str(v) for k, v in s_m1m1.items()} + # s_m2 = L_{-2}|h> 本身就是一个 PBW 词 -> 它的类是 [L_{-2}] + # chi = -3 L_{-2} + 2 L_{-1}^2 -> 模 L_{-1} 的类是 -3[L_{-2}] != 0 + # 结论：level-2 商的唯一类就是 chi 的方向 => (ii) 的额外类 = 零范态方向 + OUT["level2_quotient_class_is_chi_direction"] = True + # 数值核对：chi 的 norm 与 L_{-1} 像的正交性 + g2 = V1.gram_matrix(cval, hval, 2, B) + B2 = B[2] + i2, i11 = B2.index((-2,)), B2.index((-1, -1)) + v = [Fraction(-3), Fraction(2)] + OUT["chi_norm_via_gram"] = str(sum(v[i] * g2[i][j] * v[j] for i in range(2) for j in range(2))) + # L_{-1}^2|h> 的范数（非零 ⇒ 它本身不是 null，(2,) 方向才与 null 相关） + OUT["Lm1sq_norm"] = str(sum( + Fraction(1) * g2[i][j] * Fraction(1) for i in (i11,) for j in (i11,))) + # (2,) 单独不是 null： + OUT["Lm2_norm"] = str(g2[i2][i2]) + + # ---- 5) 动量守恒的精确陈述（符号级自证）---- + # [L_0 - Lbar_0, phi_{h,hbar}] = (h - hbar) phi；真空/动量 0 态被 L_0-Lbar_0 湮灭 + # 这里做可复算的形式核对：在 Verma 里 L_0|h> = h|h> + OUT["momentum_rule_check"] = {} + for hh in ("5/8", "21/8", "37/8"): + # L_0 作用在 |h> 上 + d = V1.reduce_seq("0", hh, (0,)) + OUT["momentum_rule_check"][hh] = {str(k): str(v) for k, v in d.items()} + OUT["momentum_rule_statement"] = ( + "[P,phi]=(h-hbar)phi, P|vac>=0 => =0 whenever h!=hbar (exact, " + "只要求沿紧致方向平移不变 + 初末态动量 0)") + + OUT["verdicts"] = { + "V2_1_delta_h_hbar": "成立（精确、初等）——条件是紧致方向平移不变(周期)、" + "初末态动量 0、phi 有确定 (h,hbar)。开放边界下不成立。", + "V2_2_observable_is_momentum0": "Θ_w/Δ_w 作为「每行扇区能量差」是动量 0 的；" + "但 note 把它当『一点函数』——无限周期圆柱上主场的" + "一点函数恒为 0（无论 h 是否 = hbar），" + "δ_{h,hbar} 其实是 torus/modular 迹的陈述。另：" + "『矩阵元为 0』只在一阶成立（#802 明确要求区分一/二阶）。", + "V2_3_eight_arm_weight": "(21/8,21/8)、spin 0、x=21/4 成立；但动量规则只能杀 spin!=0，" + "永远选不出标量；k=2..7 标量 arm 未排除 ⇒ 不唯一。", + "V2_4_third_scenario": "note 的附加结论（(2,2) 的 spin-0 类在 (ii)/(iii) 下进入动量 0）" + "与它自己 §1.2『(i)(ii) 关联函数相同』矛盾：(ii) 下该额外类" + "正是零范态 chi 的方向 ⇒ 关联函数恒为 0；(iii) 下该类是否" + "为真算子不由 Virasoro 代数决定 ⇒ 未建立。", + } + print(json.dumps(OUT, indent=1, ensure_ascii=False)) + with open("v768_v2_result.json", "w") as f: + json.dump(OUT, f, indent=1, ensure_ascii=False) + return OUT + + +if __name__ == "__main__": + main() diff --git a/tests/test_coalescent_bulk_clock.py b/tests/test_coalescent_bulk_clock.py new file mode 100644 index 000000000..187dfb1a4 --- /dev/null +++ b/tests/test_coalescent_bulk_clock.py @@ -0,0 +1,68 @@ +import sys +import unittest +from fractions import Fraction as F +from pathlib import Path +sys.path.insert(0,str(Path(__file__).resolve().parents[1]/'scripts')) +import coalescent_bulk_clock as c + +class CoalescentBulkClockTests(unittest.TestCase): + def test_01_cycle_histories(self): + for n in range(2,7): + result=c.enumerate_histories(n) + self.assertEqual(result['multiplicity_per_history'],2**(n-1)) + def test_02_uniform_full_history_transitions(self): + self.assertEqual(c.enumerate_histories(6)['ranked_histories'],2700) + def test_03_eppf(self): + self.assertEqual(c.enumerate_histories(6)['eppf_equalities'],203) + def test_04_two_parallel_edges(self): + self.assertEqual(c.cycle_history((0,1),(0,1)),c.cycle_history((0,1),(1,0))) + def test_05_fixed_order_is_not_uniform_pairs(self): + first=set() + from itertools import permutations + for perm in permutations(range(4)): + first.add(c.cycle_history((0,1,2,3),perm)[1]) + self.assertEqual(len(first),4) # not choose(4,2)=6 + def test_06_pair_survival_two_methods(self): + for n in range(2,13): + for p in (F(0),F(1,7),F(1,3),F(1,2),F(4,5),F(1)): + self.assertEqual(c.pair_separated_by_edges(n,p),c.pair_separated_by_levels(n,p)) + def test_07_jet_inverse(self): + x={(0,0):F(3),(1,0):F(2),(0,1):F(-1),(1,1):F(5)} + self.assertEqual(c.mul(x,c.inv(x)),{(0,0):F(1)}) + def test_08_score_fourth_moments(self): + for k,a in ((F(1),F(1)),(F(16),F(1,8)),(F(256),F(1,32)),(F(65536),F(1,512))): + self.assertEqual(c.score_moments(k,a)['fourth'],18) + def test_09_transform_same_time(self): + for r in (F(0),F(4,19),F(8)): + v=c.six_variable_transform(1,1,r,F(1,3),F(1,5),F(1,2),F(3,4),F(2,3),F(-1,4)) + expected=(1+F(1,3)+F(1,5)+r*(1-F(3,8))+(F(2,3)-F(1,4))**2/2)**-2 + self.assertEqual(v,expected) + def test_10_transform_marginals(self): + for k,a in ((F(2),F(1,2)),(F(4),F(1,3))): + v=c.six_variable_transform(k,a,8,0,0,1,1,F(2,3),0) + self.assertEqual(v,(1+F(2,3)**2/2)**-2) + v=c.six_variable_transform(k,a,8,0,0,1,1,0,F(2,3)) + self.assertEqual(v,(1+F(2,3)**2/2)**-2) + def test_11_transform_negative_cross(self): + self.assertGreater(c.six_variable_transform(2,F(7,10),0,0,0,1,1,10,-7),0) + def test_12_bad_covariance_rejected(self): + with self.assertRaises(ValueError): c.score_moments(4,1) + def test_13_nested_labels(self): + self.assertEqual(len(c.label_covariance_checks()),9) + def test_14_retention_NB_Beta(self): + for k in (2,3,4,8): + self.assertAlmostEqual(c.retention_moment_mixture(k,1),c.retention_mean(k),places=12) + def test_15_retention_has_atom(self): + self.assertEqual(c.retention_mean(1),1) + self.assertTrue(0=a[j] for j in range(c.N))) + for j in range(1,8): + self.assertGreater(c.beta_prime(r*F(j,8),r),0) + self.assertLessEqual(c.beta_prime(r*F(j,8),r),c.N) + def test_common_label_mass_transport(self): + for p,r in ((F(1,6),F(1,3)),(F(1,4),F(1,2))): + beta,loss,pairs,n=c.direct_pair_values(p,r) + self.assertEqual(n,3**c.N) + self.assertEqual(beta,c.beta(p,r)) + self.assertEqual(beta+loss,c.nu(p)) + self.assertLessEqual(loss,pairs) + def test_inverse_clock(self): + r,p0=F(3,4),F(1,4) + for s in (F(1,2),F(1),F(2)): + target=s*c.beta(p0,r) + lo,hi=c.invert_beta(target,r) + self.assertLessEqual(c.beta(lo,r),target) + self.assertLessEqual(target,c.beta(hi,r)) + self.assertLessEqual(hi-lo,r/F(2**48)) + def test_nonlinear_tangent_and_normal(self): + result=c.alignment_controls() + self.assertEqual(result['exact_controls'],20) + def test_higher_spin_alias(self): + result=c.harmonic_controls() + self.assertEqual(result['hidden_H8_into_scalar'],F(429,625)) + self.assertEqual(result['hidden_H8_into_spin4'],F(196,625)) + def test_two_harmonic_recovery_and_alias(self): + h=F(-527,625) + for b0,b4,b8 in ((F(2),F(3),F(0)),(F(0),F(0),F(1)),(F(2),F(-3),F(5))): + ya=b0+b4+b8 + yb=b0+b4*h+b8*(2*h*h-1) + fit4=(ya-yb)/(1-h) + fit0=(yb-h*ya)/(1-h) + self.assertEqual(fit0,b0+F(429,625)*b8) + self.assertEqual(fit4,b4+F(196,625)*b8) + def test_invalid_parameters(self): + with self.assertRaises(ValueError): c.beta(F(3,4),F(1,2)) + with self.assertRaises(ValueError): c.clock_bernstein(F(1)) + with self.assertRaises(ValueError): c.invert_beta(F(-1),F(1,2)) +class PhysicalSourceChecks(unittest.TestCase): + def test_torus_empty_full(self): + self.assertEqual(c.torus_rank(0),0) + self.assertEqual(c.torus_rank(511),2) + def test_pair_source_equivalence(self): + self.assertEqual(c.rank_response_controls()['normalized_measure_equalities'],4608) + def test_normalization_repair(self): + r=c.repaired_table() + for row in r['rows']: + self.assertGreater(row['physical_black_N'],0) + self.assertLess(abs(row['pair_normal_difference']),F(2,10**15)) + def test_spinful_zero_mode(self): + self.assertEqual(c.spin_zero_mode_control()['chiral_spin4_I3_eigenvalue'],F(-55,9216)) + +if __name__=='__main__': unittest.main() diff --git a/tests/test_query_cavity_controls.py b/tests/test_query_cavity_controls.py new file mode 100644 index 000000000..14607cbc7 --- /dev/null +++ b/tests/test_query_cavity_controls.py @@ -0,0 +1,60 @@ +from fractions import Fraction as F +from pathlib import Path +import importlib.util +import unittest + +P=Path(__file__).resolve().parents[1]/'scripts'/'query_cavity_controls.py' +spec=importlib.util.spec_from_file_location('cav',P) +cav=importlib.util.module_from_spec(spec);spec.loader.exec_module(cav) + +class QueryCavityTests(unittest.TestCase): + def test_wired_all_black_has_hidden_centre(self): + self.assertEqual(cav.wired_statistics(0),(0,8,1)) + def test_hidden_centre_colour_is_not_queried(self): + i=list(cav.product(range(-1,2),repeat=2)).index((0,0)) + self.assertEqual(cav.wired_statistics(1<