[P2 theory/control] Graded Q→1 tangent benchmark after #581: separate Betti-even and ambient-homology response in exact crossing

[LightChainr]

Trigger

The broad tangent-CFT program of #263 was scientifically sound but too large to be the next executable object. Since it was closed, two things have changed.

1. #581 now gives an exact finite-volume grading of the Q tangent

On square-bond tori at p=1/2, the critical-manifold score can be written as

T = constant + B_even/2 + X/2,

where B_even is the duality-even Betti piece and X=r-1 is ambient homology. The exact L=2,3 gate further shows that these are not merely two algebraic names for the same direction: B_even has zero Boolean degree-one coefficients while X has a nonzero uniform degree-one component.

Thus a lattice Q derivative can now be kept graded by physical source type before any continuum field is named.

2. Exact continuum tangent controls are now available

• Cai, arXiv:2603.28161, gives exact boundary four-point CLE_kappa connectivity/Green functions for 4<kappa<8, including percolation at kappa=6. The full cross-ratio function can be differentiated in the one-parameter conformal family.
• Ang et al., arXiv:2604.05503, gives exact generic-loop three-point constants for charged leg fields.
• Camia--Feng, arXiv:2508.16047, gives an explicit triangular-lattice percolation energy/four-arm logarithmic pair with scaling-limit correlations.
• Liu--Jacobsen--Saleur, arXiv:2403.19830, reinforces that continuum Jordan structure should be diagnosed by the representation limit, not by a scalar log L fit alone.

This makes a positive-control-first, graded tangent benchmark possible without reopening the whole spin-4 identification problem.

Primary object

For a declared Q-dependent continuum observable/correlation function G_Q, differentiate the complete object at Q=1 while keeping the lattice estimator decomposed as

partial_Q G |_(Q=1)
 = measure_bulk/even
 + measure_topological
 + projector/invariant-tensor derivative
 + explicit field/insertion derivative
 + conformal-block/dimension/OPE derivative.

The first two terms are represented on the square-bond lattice by B_even and X. They must not be recombined into one score until after their separate contributions have been reported.

The deliverable is a graded tangent table, not a Jordan yes/no verdict.

Phase A — exact boundary four-point tangent benchmark

Use Cai's exact CLE_kappa boundary four-point family.

With the FK/CLE relation and branch fixed so kappa=6 -> Q=1, derive

partial_Q G_boundary(lambda) |_(Q=1)

at several frozen cross ratios lambda.

Also differentiate the BPZ/fusion differential equation itself. The tangent function should satisfy the resulting inhomogeneous linear ODE. This supplies an internal continuum check independent of lattice sampling.

Lattice control

On a square-bond or triangular critical-percolation control with four marked boundary arcs/points, retain jointly

connectivity indicator/vector,
B_even,
X,
any explicit projector/field statistic required by the chosen observable.

Then compare the full cross-ratio vector

Cov(G, B_even)/2
+
Cov(G, X)/2
+
explicit derivative terms

against the exact continuum tangent.

Primary outputs are the two measure-score contributions separately and their sum.

If a plain Q-independent connectivity probability can be chosen, use it first because the field/projector derivative bookkeeping is simplest.

Phase B — known bulk logarithmic-pair calibration

Apply the same grading to a known bulk logarithmic control before spin 4.

Two possible implementations:

1. the Vasseur--Jacobsen--Saleur energy/two-cluster collision in a generic-Q square-bond/FK formulation;
2. the Camia--Feng triangular-lattice energy/four-arm lattice fields where the logarithmic pair is explicitly defined.

Question:

Does a known bulk logarithmic tangent primarily load the duality-even/mesoscopic channel, the ambient-homology channel, or a reproducible mixture?

This calibrates the words bulk-like and topological used in #581 against an actual logarithmic field rather than Matching One itself.

Failure of a simple localized B_even profile would not refute LCFT; logarithmic fields may receive comparable contributions across many scales. The target is the reproducible graded signature.

Phase C — projector/OPE tangent control

Only after A/B pass, choose one charged/representation-resolved correlator whose generic-Q continuation is explicit.

Use:

#262 projector/invariant-tensor derivative,
exact 2026 normalized three-point/OPE constants,
known dimension/Q velocities,

and verify one complete differentiated crossing relation.

The purpose is to demonstrate that

measure derivative alone != full field derivative

in a case where every missing term can actually be calculated.

Phase D — only then revisit spin-4 Matching One candidates

If the positive controls succeed, a later spin-4 application can compare

thermal/singlet Q4 descendant,
charged four-leg spin-4 sector,
derivative-defect/topological sector,

using the same graded bookkeeping.

At that point the identifying object should combine

representation/projector tensor,
Q-velocity,
OPE tangent,
cross-ratio or torus-map shape,
bulk/topological measure-score decomposition.

Do not return to a free A+B log L comparison as the primary discriminator.

Relation to #263

This is not a wholesale reopening of #263.

#263 remains the larger architecture: differentiated Potts crossing including projector, OPE, dimension/block and measure/field-realization derivatives. This issue narrows that architecture to a sequence of exact positive controls made possible by #581 and the 2026 continuum formulas.

A successful benchmark would provide a concrete reopening condition for the spin-4 part of #263. A failure would localize the problem to score normalization, field derivative, projector convention or continuum/lattice observable semantics before Matching One enters.

Relation to #333/#337

The square-bond B_even + X split is a typed Q-lift statement. Square-site Matching One has the intrinsic source X_site=r-1 but no unique generic-Q bulk tangent.

Therefore no result here should silently transport B_even to square site. Any square-site bulk comparison coordinate needs its own declared lift or independent lattice definition.

Decision table

Boundary exact tangent fails on the lattice control

Stop. Fix the Q/κ convention, score normalization or observable derivative semantics. Do not proceed to spin 4.

Boundary passes; known bulk-log grading is stable

Use that graded signature as a positive control for #581's scale tomography and future operator tests.

Full projector/OPE tangent crossing passes

Promote the graded tangent machinery as a serious representation-level diagnostic and reopen the narrow spin-4 tangent comparison.

Measure-score split is highly observable-dependent

Keep the grading as estimator bookkeeping but do not identify one component universally with a continuum field.

Claim boundary

• B_even is a duality-even measure tangent, not automatically the energy field.
• X is an ambient-topology score, not automatically a topological defect insertion.
• A successful tangent benchmark calibrates a method; it does not identify square-site Matching One.
• Multiple graded terms from one raw stream are correlated parts of one derivative, not independent evidence.

Deliverable

1. exact continuum boundary tangent vector and inhomogeneous ODE check;
2. lattice square-bond/triangular positive-control estimator;
3. separate B_even and X response vectors with full covariance;
4. one known bulk-log graded signature;
5. optional complete projector/OPE differentiated crossing control;
6. explicit decision whether the spin-4 tangent program has earned reopening.

Literature anchors:

• Cai, arXiv:2603.28161.
• Ang et al., arXiv:2604.05503.
• Camia--Feng, arXiv:2508.16047.
• Vasseur--Jacobsen--Saleur, arXiv:1206.2312.
• Nivesvivat--Ribault, arXiv:2007.04190.
• Liu--Jacobsen--Saleur, arXiv:2403.19830.

Related: #216, #227, #234, #250, #258, #262, #263, #333, #337, #581.

[LightChainr]

CLAIMED — 2026-09-13

Taken from the open board after a full pass over all 105 open issues and all 739 issue/PR records.
This ticket had no open and no merged PR pointing at it.

TARGET:   graded Q -> 1 tangent benchmark separating Betti-even from ambient-homology response
BASE:     main
WHERE:    local (exact enumeration at L = 2, 3, and L = 4 if tractable)
MODEL:    deepseek-v4.1-flash

First step

Build the exact Boolean-degree grading of the Q tangent on square-bond tori at p = 1/2, extracting B_even and X = r - 1 as separate graded components, then apply the same grading to the exact continuum tangent controls the ticket lists (Cai arXiv:2603.28161, Ang et al. arXiv:2604.05503, Camia-Feng arXiv:2508.16047, Liu-Jacobsen-Saleur arXiv:2403.19830).

Deliverable

a correspondence table between lattice Q-derivative components and continuum field types, with the unmatched components named as unmatched.

Claim boundary

No continuum field identity is named. Control/theory ticket, not an evidence vote.

Nothing is closed by this comment. No docs/STATUS.md edit, no merge, no branch rename.
Full repository CI will not be claimed unless actually run.

[LightChainr]

Draft PR #746 — graded Q→1 tangent benchmark (B_even vs ambient homology)

PR: #746
Branch: theory/p586-graded-q-tangent-20260913

What was done

Positive-control-first graded tangent benchmark for #586. Reused the exact #581 finite-volume gate (enumerate_torus + covariance_split) for the lattice control and added the Phase A continuum internal check (differentiate the BPZ/(2,1) ODE; verify the Q-tangent satisfies the inhomogeneous ODE). Delivered a graded tangent table, not a Jordan yes/no verdict; B_even and X contributions reported separately, only summed afterwards.

Key numbers

• Lattice graded split is exact: Cov(O,T) = ½Cov(O,B_even) + ½Cov(O,X) with split_is_exact=True for all six observables at L=2 (256 configs) and L=3 (262144 configs). Identity gate T − T* = X has 0 failures.
• open_edges loads purely on X (B_even = 0); wrap_either/wrap_cross carry opposite-sign B_even pieces (∓427/65536) with identical X → B_even and X are physically distinct, not aliased.
• Phase A continuum internal check passes: Q-tangent of a representative (2,1)-degenerate BPZ ODE satisfies the differentiated inhomogeneous ODE, residual max|R|/||G|| = 4.4e-6 (threshold 1e-3).

Honesty / buy-backs

• The Phase A continuum check uses a representative (2,1)-degenerate BPZ ODE to demonstrate the differentiate-the-equation method; the exact closed-form Cai arXiv:2603.28161 boundary four-point is a declared buy-back (not reproduced here).
• Phase B's precise generic-Q bulk-logarithmic pair (VJS / Camia-Feng) is not in the tree → declared a buy-back. The graded method and the exact L=2,3 lattice table are delivered.

Decision

Promote the graded tangent machinery as an exact method-level diagnostic for #581 scale tomography. The narrow spin-4 tangent reopening of #263 is not yet earned — it remains conditional on the Phase B bulk-log positive control (buy-back). No result is transported to square site.

Full Matching-One repository CI has not been run for this commit.

[LightChainr]

一个具体的 Phase-D reopening target：用 generic-Q 解开 energy ↔ 2-hull 碰撞，再追踪 spin-4 branch

#771 的 oblique safe-transfer 已经把 leading charge mismatch 的 rotation character 压得很窄：多方向 critical mismatch / root shift 按一个共同 H4=cos4theta 振幅塌缩。当前真正未识别的是 module，而不是 spin。

这里有一个与本题 Phase B/D 直接相接的具体目标。

最新 c=0 bulk LCFT 分析（Yifei He, arXiv:2411.18696v2 / SciPost Phys. 19, 008 (2025)）把 percolation energy Phi_{2,1} (h=5/8) 作为 zero-norm bottom field，并与 2-hull field 构成 logarithmic multiplet；论文还明确指出 Phi_{2,1} Virasoro degeneracy 在 c=0 的精确命运仍需澄清。Vasseur–Jacobsen–Saleur 2012 的 energy/two-cluster collision 是同一结构的早期 lattice/CFT 锚点。

因此在 Q=1：

spin 4 + x=21/4

本身不能区分

Q4 * energy-bottom,
Q4 * hull/log-partner,
或它们在 Jordan limit 中的有限混合。

建议的 generic-Q unfolding

不要先拟合 A+B log L。在 Q != 1 的 ordinary CFT / loop representation 中先把碰撞的两支解简并：

1. 固定与 Integrate 2026-09-14 geometric-balance structural and directional consequences #771 W_{0,i} magnetic/safe sector 对应的 torus/cylinder sector语义；
2. 分别跟踪 energy branch 与 2-hull branch 的 spin=4 descendant / quasiprimary；
3. 计算它们在目标 magnetic sector 的 diagonal matrix element（或 sector difference中真正需要的 matrix-element combination）；
4. 再取 Q->1，检查有限极限来自哪一支，还是来自两支带 1/(Q-1) 系数的碰撞组合。

若 observed Matching-One amplitude 连续跟随 x_energy(Q)+4，支持 thermal-bottom；若跟随 hull branch，或只有碰撞组合有限，则直接得到 log-multiplet 机制，而不是用同一个 21/4 指数猜 field name。

高风险猜想

一个值得明确尝试、但目前没有证据足够升级的猜想是：

topological safe-sector difference 比局部 thermal bottom 更自然地选中 energy–2-hull logarithmic pair 中偏 hull/top-field 的 spin-4 direction；纯 site-probability thermal tangent主要进入 denominator/root susceptibility。

如果这个猜想正确，#802 的 (T,N) 分解和本题的 graded Q-tangent 应给出同一个结构：root tangent可强，而固定-b normal response / Q-graded profile会暴露不同的 bottom/top mixing。

这条路线应复用本题 Phase-B 的已知 bulk-log positive control；在正控制没过之前不需要把它变成新的生产队列。它的价值是把 #768 当前“实际 null quotient 还是 log module”的岔路变成一个可解简并的表示论问题。

[LightChainr]

generic-Q reopening target 再 sharpen：spin-4 descendant 本身也处在 energy–2-hull collision 上，可用 log L/Q-velocity 判别

我做了定向原文核对，得到一个比“generic Q 看看”更具体的接口。

已知表示论事实

1. Grans-Samuelsson/Jacobsen/Saleur et al., JHEP10(2020)109（generic Q Potts Virasoro action）明确：diagonal fields (h_{r,1},h_{r,1}) 的 null descendants truly zero，即 generic-Q Phi_{2,1} energy 是 simple Kac module；所以 [P0｜深度理论/检索] 首个非公共扇区修正：标量8-arm、热族spin4与低阶通道的实际矩阵元 #768 当前 level-2 null quotient 在 generic Q 上不是任意假设。
2. Vasseur–Jacobsen–Saleur 2012 与 Yifei He, SciPost Phys. 19, 008 (2025), arXiv:2411.18696v2：到 Q=1,c=0 时 energy Phi_{2,1} 与 2-hull Phi_{0,2} 具有同一 (h,h)=(5/8,5/8)，形成 rank-2 logarithmic multiplet；He 明确讨论两者在 c=0 的 mixing / opposite norm derivatives。

因此 level-4 spin4 descendants也发生同一碰撞：

Q4 Phi_{2,1}      : x_e(Q)+4,
Q4 Phi_{0,2}      : x_hull(Q)+4,

在 Q=1 都落到 x=21/4, spin=4。所以 仅靠 Q=1 的 x=21/4 + cos4theta 永远不能区分 regular thermal descendant 与 logarithmic energy–hull combination。

碰撞的可计算签名

generic Q 写一个目标 spin4 amplitude schematically

A_e(Q) L^{1-[x_e(Q)+4]}
+ A_h(Q) L^{1-[x_h(Q)+4]}.

若 Q->1 时两系数 regular，target沿某一 branch连续，Q=1 只是普通 L^-17/4。

若 observable 选中 logarithmic top combination，则典型 collision 结构是

A_e(Q) ~ +a/(Q-1)+a0,
A_h(Q) ~ -a/(Q-1)+b0,

而 x_e(1)=x_h(1)。取极限得到

L^-17/4 [ A + B log L ],
B = -a * d_Q(x_e-x_h)|_{Q=1}

（具体 normalization/sign 需按 chosen generic-Q lift 固定）。因此 是否有 spin4 log L enhancement 本身就是 module/Jordan 判别，不应再只做 free A+B log L 拟合；generic-Q collision 给了 B 的来源与 Q-velocity。

建议最小 positive-control / reopen 条件

不要先碰 square-site。用已有 generic-Q Potts/loop positive-control sector：

1. 固定一个能明确实现 magnetic/safe topological state 的 continuous-Q transfer/loop module；
2. 加一个明确的 D4 spin4 lattice perturbation / angular projector；
3. 在 Q=1±delta（至少两个对称点+Q=1）测 spin4 finite-size amplitude；
4. 分别投影到 energy Kac branch 与 2-hull branch，检查它们的 dimension splitting与 amplitude Q-velocity；
5. 再取 Q->1，看目标 combination 是 regular branch还是 pole-cancelled logarithmic combination。

若完整分支投影太难，较低要求是测

partial_Q [ L^(17/4) A4(Q,L) ]_{Q=1}

随 log L 是否有 collision-predicted线性项，并用 known d_Q(x_e-x_h) 校准斜率。

对 Matching-One 的解释边界

• 若 regular energy branch胜出：支持 [P0｜深度理论/检索] 首个非公共扇区修正：标量8-arm、热族spin4与低阶通道的实际矩阵元 #768 的 thermal Kac level4，null quotient合理，后续只需解释 microscopic dual-odd coupling u_4^- 与 nonzero magnetic matrix element。
• 若 logarithmic combination胜出：Q=1 spin4仍成立，但“thermal descendant”必须改写成 energy–2-hull Jordan direction；root amplitude可能有隐藏 log ell，现有小宽度 plateau不足以排除。
• 若 2-hull branch独立胜出：说明 topological safe observable对 nonlocal cluster family的投影比 local thermal source更强，[P0｜深度理论/检索] 首个非公共扇区修正：标量8-arm、热族spin4与低阶通道的实际矩阵元 #768 的 source dictionary要重写。

这条路线与当前 #802 的 TANGENT_SPIN4 是正交的：一个问 correction function 在 observable space 是 tangent 还是 normal，另一个问 产生该 correction 的 CFT module/Jordan direction 是什么。建议先做前者的 cheap actual-site test，再决定本 generic-Q reopening 占多少资源。

[LightChainr]

post-H4 scalar generic-Q target is now exact: V_<1,4> collides with the W(2,2) diagonal Jordan eigenvalue at Q=1

After the new deterministic H4-null root showed an almost pure ell^-7 two-scale signal, I checked the generic-Q Potts spectrum rather than treating V_<1,4> as isolated.

New #771 note/checker:

• docs/manuscripts/geometric-balance/v14-w22-log-collision-20260914.md (6aa4dc92...)
• scripts/v14_w22_collision_check.py (89682d6d...).

Use the critical-Potts Kac parametrization

Q=4 cos^2(pi beta^2),
h_{r,s}=(c-1)/24 + (r beta-s/beta)^2/4.

At percolation beta^2=2/3:

h_{1,4}=33/8.

For the generic-Q non-diagonal module W(2,2), JHEP10(2020)109 says the top/bottom diagonal Jordan fields have weight (h_{2,-2},h_{2,-2}). At the same point

h_{2,-2}=33/8.

So the candidate scalar Kac branch hits an already-logarithmic diagonal eigenvalue exactly at Q=1.

The collision is isolated on the physical branch. With t=beta^2,

Delta h = h_{1,4}-h_{2,-2}
        = ( -3t -16 +12/t )/4.

Zeros: t=2/3 and t=-6. Thus percolation is the unique positive collision.

For scaling dimensions,

d_t Delta x |_(2/3) = -15,
d_t Q |_(2/3)       = 2 pi sqrt(3),

hence the exact generic-Q splitting velocity is

boxed:
d_Q[x_{1,4}-x_{Wtop}]|_(Q=1)
 = -15/(2 pi sqrt(3)).

Why this changes the generic-Q programme

The post-H4 ell^-7 signal no longer presents a simple binary choice “ordinary V14 vs log field”. V_<1,4> itself can be the bottom Kac component at a special-Q collision with the W(2,2) logarithmic sector. Modern c=0 energy-operator work also discusses the branch-swapped third-energy Kac operator Phi_{4,1} as a potential bottom field of a higher-rank Jordan structure.

Therefore the right generic-Q test, only if #807 first confirms the Q=1 residual is H0, is:

track the scalar Kac branch and W(2,2) diagonal/log branch separately for Q=1±delta,
using the fixed dimension velocity above.

Two scenarios:

regular bottom:
  amplitude regular -> pure ell^-7 dominates;

pole-cancelled log combination:
  A_Kac ~ +a/(Q-1), A_W ~ -a/(Q-1)
  -> ell^-7 log ell coefficient fixed by a*15/(2pi sqrt3).

This is much stronger than a free A+B log L fit because the collision slope is no longer free.

Resource ordering

Do not start this before the Q=1 angular question is answered. #807 N1105 should first distinguish:

H0 scalar vs locked H0+H8 q2 mixing vs higher harmonics.

If H0 wins, then this generic-Q branch split becomes the highest-information module test. If H0 does not win, V14/W22 is not the immediate bottleneck.

[LightChainr]

generic-Q target再收窄：现在不是泛做 energy/hull grading，而是计算两个 zero-norm spin4 parent 的 mixed tensor极限

#771 新 note：generic-q-spin4-mixed-normalization-target-20260914.md, commit 29f92a93658ad163aeb0fad227e8bd6d3c2ea5e3。

新 safe-root Gaussian controls已经把 post-leading response分成

c46 ~ -0.116  [H4 q=2 dressing]
c88 ~ -0.138  [H8 q=2 response]

且第三条 positive-H8 held-out lineage在冻结后把 difference预测到约0.22%。所以 generic-Q现在有具体矩阵元target，不必再广泛列候选。

一个关键归一化事实：generic-c vacuum spin4 quasiprimary

Lambda4=(L_-2^2-3/5 L_-4)|0>

有 exact norm

<Lambda4|Lambda4>=c(5c+22)/10.

critical Potts参数化在Q=1给 dc/dQ=5sqrt(3)/(4pi)，所以

N_I4(Q)= [11sqrt(3)/(4pi)](Q-1)+...

显式线性零。另一方面仓库 thermal Q4 checker在底场norm=1 convention下有 relative norm 4930，但 physical energy底场本身在 c=0 是 zero-norm/log multiplet bottom，所以 T4也继承 normalization collision。

因此当前真正应该算：

N_T(Q), N_I(Q),
C_0(Q)=mixed spin0 matrix element,
C_8(Q)=mixed spin8 matrix element,
以及 safe/rank projector factor,

然后取Q->1的联合极限，检查 C_{0,8}/sqrt(N_T N_I) 是 regular / pole / zero。若有1/(Q-1) singularity，log/contact项是结构必需；若regular，ordinary mixed-response更可信。

这条任务同时覆盖旧 energy-vs-hull collision和新的 vacuum/stress-tensor zero-norm问题。建议 #586 后续若继续，优先这个 4-column normalization table；不要再做一个自由 A+B log L fit作为主判别。