[P0 proof] One probability package: arbitrary-period balance roots and the sharp axial full-law boundary

[LightChainr]

当前主攻：一份概率定理论文，不再并行扩张 — 2026-09-13

本题保持 P0 proof。已读到 #650 较新的“只保留一个活跃交付”决定，本次据此收束：以 #735 的任意周期平衡根定理为主体，纳入 #613/#736 的完整分布边界，交付一份自洽论文、逐引理独立审查及最接近先例的比较。最终交付返回 #735。

入口改造 #738 已合并；main 后续提交 d31fa5fe9584b77595e4a78b9560aa2ff77cf0b3 将表示论方向改为可见储备，避免重新分叉任务。#275 为保留原合同的 P1 theory-blocked flagship；#276 保持 completed。

已交付的数学输入：不重算、不重派

• [P0 proof] One probability package: arbitrary-period balance roots and the sharp axial full-law boundary #613/#612/#613: the chart is the 55%, one N725 block, and the p_c bridge verified #614/#613: scope review of the #276 critical-point bridge — all five items discharge (proof note, notes only) #670：ell/log N -> infinity 配合 site sharpness、matching critical-point relation、数字 Alexander 对偶，给出全部固定内部分位数收敛的充分条件。
• Aspect-uniform matching roots; P398 is periodic IC TL (do not merge) #718/Arbitrary-period balance-root consistency and Jordan visibility (do not merge) #735：平衡根一致性不需要长宽比限制；Arbitrary-period balance-root consistency and Jordan visibility (do not merge) #735 覆盖任意满秩整数周期格，最短周期 ell 趋于无穷即可。
• 新 Sharp axial geometry for the full birth-time law: necessity beyond the union bound #736，commit 64d809b4404f80ff3f9adf9713337cc76008e92d：对轴对齐 2<=w_n<=m_n、w_n*m_n->infinity，在已列明的 critical square-site RSW / sharpness / matching 输入下，

每个固定 u in (0,1) 的 Q_n(u) -> p_c
  iff 在内部紧区间上一致收敛
  iff F_n 趋于 p_c 处的阶跃
  iff log(m_n)/w_n -> 0.

新增必要性通过固定 p_eta<p_c 下概率至少 exp(-eta*w) 的周期闭合 crossing ring，以及不交横向带的独立放大。不是从上界不消失推断反例。中位数 Q(1/2) 是平衡根，不在 lower-quantile obstruction 中。

#736 包含证明、21,760 个配置的精确拼接检查、2,365 个非均匀分格控制和3项局部数学测试。无 Monte Carlo；全仓库 CI 未运行；独立全证明审查及文献新颖性核查尚未完成。#735/#736 仍未合并，引用交付不改变集成或出版状态。

本轮唯一 acceptance package

先独立检查 #735 的局部事件支撑与单射性、斜带入口顶点计数、允许回溯路径的截取、NN+NNN 对偶侧、仅在 site 集合不交时使用独立性，以及常数对面积/剪切的一致性。

在同一论文的 root-versus-law 边界部分，检查 #736 的周期 seam 闭合、RSW 临界 site 输入、先固定 eta/尺度/p_eta 再取尺寸极限的量词顺序。复跑作者脚本与核对引文不能代替独立的全尺寸证明审查。

精确比较 strip-percolation、RSW finite-size criteria、homological percolation 与有限图局域性文献的命题范围。没有同名搜索结果不是新颖性证明；一般 RSW/gluing 不是本仓库新发明。

高价值可选加强：任意斜周期的 winding corridor

这不是第二个活跃项目，也不是已有较窄结果成稿的新门槛。

尝试对任意周期格的真实最短周期 u，在一个厚度仅依赖 eta 的走廊内，以固定 p_eta<p_c 实现概率至少 exp(-eta*|u|) 的 NN 非零 winding，并装入足够多互不相交走廊；或给具体几何障碍。它可能把 log N/ell -> 0 的必要性推广到非轴向周期，但 尚未证明。

不得旋转物理 NN 相互作用来搬用轴向证明；跨切口路径不自动等于闭合循环；长 HNF 基向量不自动等于 systole。若不能补齐，只写出精确缺失引理，保留已成立的窄结果，不启动另一个枚举或采样队列。

边界

无新数值 p_c、无 L^-4 律、无 near-critical rate、无连续场身份结论。不购买更多 thin-torus Monte Carlo、宽度矩阵、角度或 descriptor 扫描。源码入口为 #736 的 notes/axial-quantile-geometry-criterion-20260913.md 与 #735 的 notes/arbitrary-period-balance-consistency-20260912.md。总分配 #650；表示论 #636 为暂停扩张的储备。

[LightChainr]

Proof note delivered: notes/p613-quantile-convergence-20260907.md (branch claude/p612-n725-chart, commit f0981a98). No merge requested.

Verdict: not falsified, not novel — a scoped corollary.

The bridge holds, under three imported hypotheses

Theorem L. For honest periodic square-cell tori with shortest period ell_N, ell_N > sqrt(2) and ell_N / log N -> infinity, assuming (H1) r_G + r_hat = 2, (H2) exponential decay below p_c on both graphs, (H3) p_c(G) + p_c(Ghat) = 1: every fixed interior quantile Q_N(u) -> p_c, uniformly on compact subintervals of (0,1), and bounded orientation-weight combinations summing to one inherit it.

Primitive Gaussian square tori have ell_N = sqrt(N), so the geometry hypothesis is satisfied.

What I supplied that the draft left implicit

1. Explicit conservative radius. R_N = floor(ell_N/4) works for both adjacencies simultaneously. The graph/Euclidean conversion (asked for explicitly):

• NN: |y|_2 <= d_G(0,y) = |y|_1 <= sqrt(2)|y|_2;
• NN+NNN: |y|_2/sqrt(2) <= d_Ghat(0,y) <= sqrt(2)|y|_2.

A nonzero ambient cycle lifts to a path x -> x+v, |v|_2 >= ell_N, so it reaches graph distance >= ell_N (NN) or >= 0.707 ell_N (matching) — both >= 2.8 R_N, so it escapes. Embeddedness: B_G(x,R) ⊆ B_E(x,R) (diameter 2R <= ell_N/2) and B_Ghat(x,R) ⊆ B_E(x, sqrt(2)R) (diameter <= 0.708 ell_N); both < ell_N. Union bound: P(r_G > 0) <= N·A(p)e^{c(p)}·exp(-c(p) ell_N/4), which vanishes iff ell_N/log N -> infinity. Note c(p) -> 0 as p ↑ p_c, so it is per-fixed-p and never uniform up to p_c.

2. (H2) — exact source. Duminil-Copin–Tassion, Commun. Math. Phys. 343 (2016) 725–745; arXiv:1502.03050; DOI 10.1007/s00220-015-2480-z. Theorem 1.1 item 3, verbatim: "If (J_{x,y}) is finite-range, then for any β < β_c, there exists c = c(β) > 0 such that P_β[0 <-> Λ_n^c] <= e^{-cn} for all n >= 0." And §1.2, paragraph "Site percolation", verbatim: "As in [AB87], the proof may be adapted to site percolation on transitive graphs." That discharges the site adaptation; [AB87] = Aizenman–Barsky, Commun. Math. Phys. 108 (1987) 489–526, which covers finite-range site percolation directly, with Menshikov (1986) as the classical companion. Both G and Ghat are locally finite, transitive, finite-range.

3. (H3) — and a warning. Grimmett–Li, Percolation critical probabilities of matching lattice-pairs, Random Structures & Algorithms 65 (2024) 832–; DOI 10.1002/rsa.21226; arXiv:2205.02734. Verified against the published PDF: (1.1) p_c(G)+p_c(G*)=1, (1.2), (1.3) p_u(G)+p_c(G*)=1. (1.1) is not their theorem — the paper says Sykes–Essam "presented motivation" for it and that it "has been verified in a number of cases when G is amenable (see [6,14])". Their theorem is (1.3), proved in the companion Hyperbolic site percolation (arXiv:2203.00981). The chain actually used is (1.3) + p_c = p_u for amenable G (Burton–Keane 1989) => (1.1). Classical route: van den Berg, Percolation theory on pairs of matching lattices, J. Math. Phys. 22(1) (1981) 152–157, DOI 10.1063/1.524747 (their ref. [6]) — conditional on "the mean cluster size is finite below p_c", i.e. exactly Menshikov/Aizenman–Barsky sharpness; van den Berg + sharpness also gives (1.1) for Z^2.

⚠️ Keep on the record: van den Berg's abstract states that Sykes–Essam "suggested that the above relation holds for all mosaics … we have constructed a counterexample", and that his derivation holds only for a restricted class. (H3) is a genuine hypothesis with known counterexamples in greater generality. It holds here because Z^2 is amenable and in that class. Do not carry it to an arbitrary planar matching pair.

4. (H1) is not imported. r_G + r_hat = 2 is the repository's own digital Alexander duality (notes/digital-alexander-duality-proof.md). Assumed, not sourced.

Scope: do not claim novelty

Duncan–Kahle–Schweinhart, Homological percolation on a torus: plaquettes and permutohedra, arXiv:2011.11903, Ann. Inst. H. Poincaré Probab. Statist. (2025) already prove that giant cycles — their term for exactly the event im[H_i(S)->H_i(T^d)] != 0 that #276 calls r_G > 0 — have a sharp threshold. From their abstract: 1-dimensional plaquette percolation on the torus has a sharp threshold at p̂_c, the bond-percolation threshold on Z^d; in d=2, i=1 that is bond percolation on the square torus, so the sharp-threshold statement is already in the literature, proved by essentially the union-bound argument in §1. Their site model is the triangular lattice, so square-site with the NN/matching pair is not literally covered — it is obtained by the same argument plus (H3).

Conclusion: Theorem L is a scoped repository lemma. The sharp-threshold mechanism is DKS's / standard finite-size percolation; the square-site matching instance and the passage to quantile convergence are the additions. Cite it as a corollary of Duminil-Copin–Tassion + Grimmett–Li + the digital Alexander duality. Nothing here supports a novelty claim.

What fails (keep out of the conclusion)

1. Thin tori, ell_N = O(log N) — the bound no longer vanishes and the conclusion is false (quasi-1D, no transition at p_c(Z^2)). For ell_N <= sqrt(2) the quotient is not even honest: a cell can have two corners identified and a matching edge can close into a loop.
2. Unbounded extrapolation weights — sup_N max_k |w_k| = ∞ amplifies the O(1) finite-size error without bound. Boundedness is load-bearing. (Mixed-sign weights are fine: we combine quantile functions, not CDFs, so no monotonicity is needed — which matters, because spin0 has mixed signs.)
3. u_N -> 0 or 1 — no fixed eps is available; Q_N(u_N) is only constrained to the correct side of p_c. Not claimed.

Also not claimed: any rate (at p = p_c the argument is silent and F_N(p_c) may converge anywhere in [0,1]), any value for p_c, and any correction-to-scaling shape. The last is what #610 is now free to study without pretending to locate p_c — which was the point.

What it buys #276

Phase A/B now rests on a stated, sourced qualitative theorem Q_N(u) -> p_c for the declared geometries, with no CFT input and no fitted correction exponent. Marginal note: the strict monotonicity step is airtight via Margulis–Russo plus one rank-increasing step on an empty-to-full chain (r takes three values, so it must increase twice).

[LightChainr]

Proof note delivered as PR #614 (with #612's N725 block; second commit 4caf11f8 adds the STATUS/ROADMAP record). No merge requested.

[LightChainr]

Small scope clarification for the delivered bridge, recorded with audit #631: shortest-period/log N→∞ is a sufficient condition for the proposed fixed-off-critical union-bound argument, not a proved iff necessity condition for quantile convergence. A fixed-width counterexample also does not establish failure for every width O(log N). Please preserve these distinctions when integrating the theorem; no new production or broader proof catalogue is needed. The conditional bridge itself is not rejected by this clarification.

[LightChainr]

ASSIGNED_MACHINE: local

Proof/literature only. No Monte Carlo, no Huawei. Draft PR against main. Do not edit STATUS. Do not merge. Do not close #276/#610.

[LightChainr]

ASSIGNED_MACHINE: local

Scope review delivered: notes/proof-613-bridge-scope-review-20260908.md (branch proof/613-bridge-scope-review, draft PR #670). Notes only; no merge, no ticket action.

Verdict: not falsified — all five items discharge. Theorem L stands as a scoped repository lemma.

1. Lift/embedded-ball (item 1): proved with explicit conservative radius R_N = floor(ell_N/4); embedded for NN (diameter <= ell_N/2) and NN+NNN (diameter <= 0.708 ell_N), and any lifted nonzero cycle reaches graph distance >= ell_N/sqrt(2) > 2 R_N, so it exits. Union bound: P_p(r_G>0) <= C(p) N exp[-(c(p)/4) ell_N] -> 0 exactly when ell_N/log N -> infinity. Honest-cell caveat (ell_N > sqrt(2)) preserved.
2. Site exponential decay (item 2): Duminil-Copin–Tassion (arXiv:1502.03050), Thm 1.1 item 3, plus their §1.2 remark "Site percolation" — a remark, not a numbered site theorem; the Aizenman–Barsky (1987)/Menshikov (1986) route covers site percolation directly on both transitive finite-range graphs. No unqualified bond theorem substituted.
3. Matching relation (item 3): citation precision fix — the RSA 2024 paper (DOI 10.1002/rsa.21226) states (1.1) but proves only the strict-inequality criterion (Thm 3.1). The equality is proved in the companion Grimmett–Li, Hyperbolic site percolation (arXiv:2203.00981): p_u(G1)+p_c(G2)=1, plus p_c=p_u for amenable G (Burton–Keane 1989). van den Berg (1981) is the classical route; his recorded Sykes–Essam counterexample caveat stays attached to any generalization beyond amenable pairs.
4. Novelty scope (item 4): repository lemma, not a new theorem. Duncan–Kahle–Schweinhart (arXiv:2011.11903; AIHP 2025) already proves the giant-cycle sharp threshold and threshold-function convergence for the square-bond torus; the square-site matching instance and the digital Alexander duality (H1) are the repository-specific additions.
5. Failure modes (item 5): thin tori, unbounded extrapolation weights, and u-grids approaching 0/1 all stay outside the conclusion; no rate, no value, no correction shape is claimed.

Scope corrections for the record: (a) ell_N/log N -> infinity is sufficient, not proved necessary, for the fixed-quantile conclusion — a fixed-width counterexample does not establish failure at every width O(log N); (b) item 3 should be cited through the Hyperbolic-site-percolation theorem, with the van den Berg caveat, rather than to the RSA paper as the proof of (1.1).

No Monte Carlo run; no simulation artifacts touched; STATUS.md untouched.

[LightChainr]

PR: #670

Bounded proof/literature task delivered as notes. All five verification items have carriers; two scope corrections recorded. Ticket stays open. Not merging. Do not close #276.

[LightChainr]

Thin-torus example in stacked draft #716 (do not merge): for w=o(log m) one still has L(T)⇒½δ_0+½δ_1 while a median may → q_w. The ratio ℓ_N/log N → 0 here, so this does not refute the honest-torus condition ℓ_N/log N→∞. It does show that “shortest period → ∞” is not a substitute. Do not weaken #613 to that. Do not re-dispatch the same thin-geometry theorem.

[LightChainr]

Draft #718: on axis rectangles, matching roots p_{w,m}→p_c uniformly in m≥w as soon as w→∞. No aspect bound, no w/log N→∞. Compatible with #716: the same super-thin sequence has L(T)⇒½δ_0+½δ_1 and p_{w,m}→p_c. So #613 remains the right condition for the full threshold law; it is stronger than needed for the root on rectangles. No tilted-torus claim. Do not weaken #613 to “shortest period → ∞”. Do not merge.