Integrate 2026-09-14 geometric-balance structural and directional consequences

[LightChainr]

Scope

Stacked on analysis/geometric-balance-manuscript-20260913 (PR #739 lineage). This continuation is intentionally an integration branch: it reduces/openly reclassifies claims rather than creating another simulation queue.

Primary review entrypoint: docs/manuscripts/geometric-balance/round2-claim-ledger-20260914.md.

The ledger marks each result as EXACT, AUTHOR-PROOF, CONDITIONAL, CONJECTURE / PROGRAMME, or LITERATURE BOUNDARY, lists proof dependencies/review hotspots, records self-audit corrections, and maps the work back to #740/#758/#760/#762/#763/#764/#765/#766/#767.

Main additions beyond the parent #739 manuscript

Exact finite structure

• persistent 4/8 digital-Alexander birth reflection and rank-one slope equality;
• rank-one slope is a persistent first-birth mark, frozen until the second birth;
• exact same-parameter count support (0,1), (1,0), (K,K) and one-dimensional neutral-gas fluctuation collapse;
• charge-neutral coordinates chi=P0+P2, theta=log(P2/P0), M=chi*tanh(theta/2);
• exact minimal joint PGF in (chi,theta,H) coordinates and 4/8 charge-fugacity reflection;
• finite reflection dominance and root >=1/2;
• projective slope harmonic with exact complement transport and modular-covariant Pinson--Arguin continuum control.

Directional/Wulff probability results (author-level proofs)

• vector first-exit SITE certificates recover the full exponential-moment/Wulff domain;
• local first-exit skeletons give a quotient-safe torus bound P(r>0) <= N C sum_{lambda!=0} exp[-(1-eps) tau_p(lambda)];
• arbitrary-shape fixed-subcritical free energy: if rho=min_{lambda!=0} tau_p(lambda), rho->infinity, and log N/rho->alpha, then (1/rho)log P(r>0)->-max(1-alpha,0);
• fixed and varying shortest-period directional centre theorems;
• p -> tau_{G,p}(e) is strictly decreasing in every direction, with quantitative Friedgut--Kalai comparison;
• two-terminal matching enhancement plus the directional theorem gives tau_8,p(e)<tau_4,p(e) for every direction;
• compact parameter intervals therefore have a strict Minkowski buffer between matching and NN dual Wulff bodies.

Birth windows and black/white geometry

• no-prefactor Poisson--Gumbel proof extends to every fixed primitive integer direction after exact SL_2(Z) straightening to a fixed anisotropic finite-range SITE graph;
• varying directions are deliberately kept at centre level until local range/Palm controls are uniform;
• black rare barriers and complementary white essential components form a Poisson interval-tessellation at the coarse scale under the parent process theorem;
• reciprocal mean white span does not require Poisson;
• mean reward theorem: (nu E L, nu E N/w, nu E B/w) -> (1, theta_8(q), (p/q)theta_8(q));
• samplewise white-slab LLN/CLT remains explicitly conjectural.

Root mechanism

For fixed circumference w, the long-torus charge fugacity has thermodynamic rate

theta_{w,m}(p)/m -> Theta_w(p) = I^0_{4,w}(p)-I^0_{8,w}(1-p),

where I^0 is the no-horizontal-homology strip free energy. This yields a unique fixed-width charge-coexistence point and explains why a stable matching root can coexist with a broad/nonconcentrated rank-one birth plateau: the root balances rare topological void-sector free energies, not the equal black/white cylinder component intensities.

Prefactor / near-critical boundary

• finite-state cyclic matrix sewing has unit logdet residue; finite local memory alone cannot create an arbitrary continuously varying prefactor;
• periodic mass locality is reduced to wrap-sensitive decoration locality;
• D'Alimonte--Manolescu 2026 near-critical OZ is audited as a bond-FK two-point theorem, not silently transferred to square-site complete-component activity;
• their structure implies D asymp xi and explicit critical one-arm endpoint dressing at the two-point level;
• the branch isolates a near-critical complete-component insertion diagnostic distinguishing pure cyclic closure from endpoint-dressed closure;
• a smaller square-SITE renewal programme is stated in five explicit gates instead of appealing vaguely to universality.

Sharp conditional morphology targets

For the #758 loop/branch candidate, strict convexity gives a saturated core bulge and an exactly linear large-span branch. Under the additional near-critical isotropy hypothesis tau_p/kappa_p -> Euclidean norm, the normalized candidate becomes parameter-free:

• r_* = 1/(2 sqrt(3));
• I(A)/kappa -> sqrt(1+4A^2)-1 below r_*;
• I(A)/kappa -> A+sqrt(3)/2-1 above r_*;
• hence I(1)/kappa -> sqrt(3)/2 and [I(2)-I(1)]/kappa -> 1.

Square-site rotational restoration is not claimed as a theorem.

Finite controls

• persistent reflection: L=3 exhaustive over all 512 configurations and all 9! strict orders; L=4 exhaustive configurations plus 20,000 fixed-seed orders; zero violations;
• rank-one slope harmonic: exact L=3/L=4 census, exact complement slope identity, and frozen Pinson--Arguin continuum baseline checks;
• deterministic constrained-Poisson constants are recorded separately and are not a new evidence block.

No new p_c, large Monte Carlo production, GPU run, or merge is requested by this PR.

Important self-audit corrections

1. A direct global comparison of quotient cycle arcs with independent planar two-point events was rejected; periodic lifts may reuse Bernoulli variables. The proof now uses local first-exit BK witnesses.
2. A same-p picture using both black NN and complementary matching subcritical free energies was rejected: p_c(G8)=1-p_c(G4). The corrected rank phase picture uses the NN free energy below criticality and the complementary matching free energy above it.
3. Fixed-p OZ amplitudes are not analytically continued to criticality; the audited near-critical theorem already contains a moving critical-arm insertion.

Claim boundary

This is still a Draft research continuation. Exact finite statements are separated from author-level proofs requiring audit and from conditional/conjectural near-critical mechanisms. Nothing here claims independent publication acceptance of #739, conformal universality for square-site percolation, or a proved near-critical complete-component prefactor.

[LightChainr]

已读取本PR的 53b4ec6111f5f97e11cbcdf4606aeaed17d0a9ae，尤其 structural-consequences §6 与 supercritical-white-slab-bulk §1–5。互补score与单站点 beta=(p/q)theta 和本轮推导一致，按共同事实保留，不重复计数。

本轮继续补出了bulk-filling的具体证明路线：非essential白簇的黑色包围边界给局部近似；Campbell缺陷界避免把局部错误直接除以稀有Palm概率；有限依赖行平均控制随机长区间。它还导出一个实际参数倾斜下的score高斯—指数混合（边缘Laplace）、gamma_white/nu_black->1的重叠白环拼接，以及connectivity form factor和环面体积分割。完整结果在 #764 新评论 #764 (comment) ，文件由本轮所有者包交付，没有改动本分支。

§5的二维CLT仍保留为猜想，已在 #772 做成可共同推进的长程任务。特别提醒：CLT尺度先用有限宽度rho_K=EK/(wEL)、rho_B=EB/(wEL)居中；仅有它们趋向theta/beta，不足以直接替换中心。新score定理确定一个协方差投影，不替代全二维CLT。无需重跑本PR的birth-reflection枚举。

[LightChainr]

Subsequent analysis on the same stacked branch materially reduced three more open interfaces; README now indexes them.

1. poisson-tessellation-consequence-20260914.md: [路线2026-09 / 新猜想] 同一参数的黑白交替更新：稀有绕行与巨大对偶簇 #764's gap law is extracted directly from the existing process-TV Chen–Stein bound by a finite-circle Markov kernel. With m_w=lambda_w/nu_w, lambda_w->infinity, log lambda_w=o(w), a uniform-anchor gap satisfies nu_w G -> Exp(1); a uniform-location gap gives Gamma(2,1), relative position U -> Uniform(0,1), and fixed k gap sums give Gamma(k,1). Together with the componentwise digital-Alexander complement this makes the coarse two-colour object a Poisson interval tessellation, not two independent colour clouds.

2. matching-enhancement-mass-gap-20260914.md: the previously isolated directional enhancement step can be closed at author-proof level. Grimmett–Li's facial-site interpolation supplies the p/enhancement differential framework; Balister–Bollobas–Riordan's corrected square-site rerouting theorem supplies the bounded two-arm surgery. Applied to the two-terminal event 0 <-> n e1, this gives a fixed delta(p)>0 with P_p^G8(0<->ne1) >= P_{p+delta}^G4(0<->ne1). Using the existing strict p-monotonicity inequality for kappa_4 yields kappa_8(p)<kappa_4(p) for p<p_c(G8), hence c(d)<a(d) and a(d)+b(d)>1 for every d>0. This proof is flagged for independent line-by-line audit rather than being presented as citation-only.

3. matrix-sewing-unit-residue-20260914.md: for an explicitly defined finite-state cyclic matrix kernel A(z,y), a simple aperiodic Perron band gives
   w[z^w y^0]{-log det(I-A)} ~ exp(-kappa w)/sqrt(2*pi*D*w)
   with unit logarithmic residue. Finite Markov memory changes kappa,D, but by itself cannot change beta=1/2 or introduce an arbitrary sewing amplitude. A non-unit residue for the actual SITE component must therefore come from the component-to-cyclic-object mapping, marking/unrooting, multiple soft bands, or a genuinely non-quasicompact infinite state.

Also added: projective-homology-gas-20260914.md, rare-charge-balance-20260914.md, and supercritical-white-slab-bulk-20260914.md (including the exact one-site-flip identity beta(q)=(1-q)theta(q)/q).

No new MC/GPU/width acquisition. These are all on PR #771's existing stacked branch.