[路线2026-09 / 理论] 保持周期同调的 site 网络上界：闭合主环—长枝大偏差

[LightChainr]

要解决的缺口

服务 #739，不重复既有确定性最优化或概率下界。当前作者交付 cylinder-mass-and-network-bounds.md / loop-branch-variational-frontier.md 已提出并论证构造侧：完整 NN 绕行簇在 L≥Aw 条件下，可以按主环加竖直枝的相关范数代价出现。仍缺反方向：没有遗漏更便宜的簇形状。

令 tau_p 为严格亚临界 NN SITE 平面连接代价范数，kappa=tau_p(1,0)=tau_p(0,1)，完整簇按最低行锚定、每簇一次的 Palm 分布计数。候选率为

I_p(A)=min_{0≤r≤A}{tau_p(1,2r)-kappa+kappa(A-r)}。

目标是证明 liminf_w [-log Pr_Palm(L≥ceil(Aw))/w] ≥ I_p(A)，或给出一个严格更便宜的拓扑网络/概率机制。与既有构造侧合起来才是完整 LDP；本单不预设猜想正确。

建议攻法

先限制固定 p 的紧亚临界区间、A 在固定正紧区间，NN 图先行。做保留非零 deck 位移的粗骨架；用 site BK 或已证明的替代界给出骨架权重；证明粗骨架数量只产生可吸收的熵；单独处理 vertex-Palm 到 component-Palm 的归一化与异常大簇。不能把 root-cluster 条件法直接当完整绕行簇分布。

主参考：Kovchegov–Sheffield arXiv:math/0310256v2 的平面 BOND 网络 LDP；CIV math/0610100 的开放连接骨架和装饰控制；0707.1089 的 site 工具。两类论文都不是现成的 site-periodic-Palm 定理。原文有益的局部引理可以单独迁移，必须写出边、点和同调处理。

完成形式

一份逐步骤的上界证明；或一个可实现的反例/修正率函数；或只关闭一个明确的关键引理并列清剩余依赖。区分固定 A、A→∞ 和 H→∞ 先于 w→∞。不从小宽度尾质量直接宣布统一 LDP。Matching 图的推广是后续附录，不得因图名相近而自动包含。

理论为主，无新 Monte Carlo 或大枚举。允许使用已存在的稀有完整簇采样器寻找反例，但那只是探索，不是上界证明。结果返回 #739；可并行思考，不要求独占机器。

[LightChainr]

PR #771 now has a sharper falsification consequence of the candidate rate, independent of its unknown intercept.

From strict convexity, the candidate minimizer r_*(p) is unique and square symmetry gives 0<r_*<1/2. Therefore every A>=1/2 is already in the exactly linear branch regime:

I_p(A)=kappa A + c_*.

So if the actual Palm span LDP equals the candidate, then for any A2>A1>=1/2,

-(1/w) log [ P(L>=A2 w) / P(L>=A1 w) ] -> kappa(p) (A2-A1).

In particular the #762 planned thresholds A=1 and A=2 imply the simple exponent test

-(1/w) log [P(L>=2w)/P(L>=w)] -> kappa(p).

This cancels the unknown loop intercept, Palm normalization and prefactor. It is a substantially cleaner first falsification of the mechanism than fitting the whole rate curve.

With a refined local tail asymptotic, the same linear branch predicts microscopic overshoot ratios P(L>=Aw+k)/P(L>=Aw) -> exp(-kappa k) for A>1/2.

File: loop-branch-linear-tail-tests-20260914.md on PR #771. It keeps the operational #762 core distinct from the variational/OZ loop coordinate until a map between them is proved.

[LightChainr]

PR #771 sharpens the loop/branch candidate into a much more falsifiable shape.

First, for the candidate

I_p(A)=min_{0<=r<=A}{tau_p(1,2r)-kappa_p+kappa_p(A-r)},

strict convexity implies a fixed saturation point r_*(p): below it the optimal winding core bulges with r=A; above it the core bulge is frozen at r_* and all extra macroscopic span goes into branches. Therefore the tail is exactly linear after saturation, not merely asymptotically close to linear.

Second, if the small-A branch is matched to the Brownian-range moderate-deviation law, the quadratic coefficients force the consistency relation

D_p^{-1}=partial_yy tau_p(1,0)=kappa_p+kappa_angle''(0).

Third, near-critical-isotropic-loop-branch-rate-20260914.md records one explicit conditional universality target. If the normalized near-critical Wulff norm becomes Euclidean,

tau_p(x)/kappa_p -> |x|,

then the entire normalized candidate becomes parameter-free:

r_* -> 1/(2 sqrt(3)) = 0.288675...,

I(A)/kappa -> sqrt(1+4A^2)-1 for A<=1/(2sqrt3),

I(A)/kappa -> A+sqrt(3)/2-1 above it.

So I(1)/kappa -> sqrt(3)/2, I(2)/kappa -> 1+sqrt(3)/2, and [I(2)-I(1)]/kappa -> 1.

The square-site Wulff-rounding hypothesis is not promoted as a theorem; it is intentionally isolated as the one universality input. Everything after that hypothesis is elementary convex optimization.

[LightChainr]

我直接用已提交的 span-spectrum-20260913.json 做了一个不重跑 transfer 的 projected two-pole/Prony 审计，并开了小型 stacked Draft #800。这个结果会撤掉一个可能的证明捷径，但不反驳本题 LDP 候选。

对 NN p=1/4，late d_h 的两个最慢可见 pole 快速合并，而且 residues 近乎反号：

w   xi_rel=1/log(rho1/rho2)    xi_rel/w    c2/c1
3      12.0                       4.0       -1.0049
4      33.4                       8.3       -1.0204
5      84.1                      16.8       -1.0153
6     203.7                      34.0       -1.0091
7     629.4                      89.9       -1.0038

NN p=1/8 w6 给 xi_rel~5288，matching p=1/16 w5/6 也给 ~321/~1209，仍是近反号 doublet。直接 tail hazard 也显示 h=w,2w 尚未进入 fixed-w ultimate Perron regime。

因此此前一种 bridge 假设 xi_rel=o(w) 与当前趋势明显不相容；不要用“先 fixed-w h->infty，再把 Perron slope代入 h=A w”来证明 linear branch。

但这不 falsify simultaneous h=A w LDP。若 gamma1,2=gamma_bar∓Delta/2、residues≈±A，则

d_h ≈ exp(-gamma_bar h)[B cosh(Delta h/2)+2A sinh(Delta h/2)].

当 Delta w->0 时，h=A w 的指数率仍可由 common mass gamma_bar 控制，而 near-cancellation 主要改变 subexponential factor。建议本题上界直接在 simultaneous limit 做 coarse network/BK，不再要求 slow doublet先混成单 Perron mode。

PR #800: span-visible-pole-splitting-20260914.md + extractor + machine table。下一步最值钱的是从 tagged operator直接取两个 visible eigenvectors/source overlaps，辨认这两个 metastable sectors。