[分析／出生窗口] 重建无前因子的 Poisson–Gumbel 证明，分离强结论与可保留弱结论

[LightChainr]

唯一科学问题

在实际 NN / matching 独立 SITE 的指数长条 log m/w→d>0 中，完整绕行簇 Poisson 近似是否足以支持作者交付中的真实有限中位数中心化 1/w Gumbel 联合极限？这是 #739 的可独立分析单元，不是固定-p 前因子 #740 的重复任务，也不应阻止较窄的几何论文先完成。

输入：poisson-birth-windows.md、exponential-birth-centres.md、two-birth-reduction.md，以及后续 cylinder-mass-and-network-bounds.md 的探索树提升引理。最后一项可能仅在所有者交付包中，先固定实际文件/commit。当前仓库参考 head 907a9d9。

独立重建的关键链

1. 每个完整绕行 component 唯一锚定；H=w² 局部化截去的是完整簇，不是任意路径。检查局部依赖邻域、多个簇和接缝。
2. 非单调的完整簇锚点不能直接套 BK。写出相异锚点到不交 increasing winding witnesses 的映射，并核算 Arratia–Goldstein–Gordon 的 b1、b2 及 TV 约定。
3. 黑、白两个窗口使用同一批 uniform labels。不同参数窗口的联合独立极限不能偷换为同一 p 下独立；后者被 W4−W8=r−1 的精确拓扑约束排除。
4. 核查圆柱上一致簇体积尾界、活动权重求导、E(K+|boundary|)=O(w)、半凸性以及移动参数点的导数收敛。区分方向解析性与占据概率 p 的正则性。
5. 弱收敛不自动给均值/方差/协方差渐近；分别验证 uniform integrability。有限中位数与无限中心的 log(w)/w 位移分开。

有价值的任何一种完成形式

A：给出可自包含检查的无前因子 Gumbel 证明及准确例外集。
B：若仿射化缺一项，保留严格的 intensity-clock Poisson/极值结论，列出最小附加正则性。
C：若具体引理失败，给出反例或修正界，并标明下游哪些式子必须退回条件结论。不得把失败扩散成“整个几何研究无效”。

不要求先证明 p 全域解析，不把 unit residue 或近临界 uniformity 加成新前提。有限模拟只能找错，不替代概率证明。交付正文修订和依赖表，不再产生同义复核工单。建议在几何主干有人整理时由另一位概率论审读者有界并行；无新大型采样或 GPU。返回 #739 / #650。

[LightChainr]

PR #771 now contains fixed-direction-poisson-gumbel-20260914.md, which extends the no-prefactor Poisson/Gumbel mechanism beyond the coordinate axis for every fixed primitive ambient integer direction.

The reduction is exact. Fix primitive u, choose v with det(u,v)=1, and use the integer coordinate map A=[u v] in SL_2(Z). Periods (nu,mv) become (n e1,m e2). The NN/matching graphs are not rotated into NN/matching; instead their step sets become the fixed finite-range anisotropic sets A^{-1} S4, A^{-1} S8. Since u,v are fixed, all interaction ranges are fixed constants independent of n,m.

Every ingredient of poisson-birth-windows.md survives this fixed finite-range change:

• complete winding components are anchored on the transformed cylinder; empty guard slabs of fixed thickness replace empty rows;
• subcritical arm decay gives uniform component-height and cluster-volume tails;
• H=n^2 localization is still superexponentially accurate;
• the component density has exponent -(1/n)log nu_n -> xi_u(p)=tau_p(u);
• AGG dependency neighbourhoods remain polynomial (O_u(n^3)), while two distinct anchors map to disjoint increasing winding witnesses and site BK gives the same b2 control;
• same-label lower/upper windows still decouple because the black/white enclosing events have opposite monotonicity;
• the component activity score gives E_*|C|, E_*|boundary C|=O(n) and therefore uniform semiconvexity of (1/n)log nu_n;
• at differentiability points, moving-parameter intensity derivatives converge to -xi_u'(p).

Thus for regular d, with true finite medians a_{n,m}, b_{n,m} and
v4,u=-xi_4,u'(a_u(d)), v8,u=-xi_8,u'(c_u(d)),

v4,u n (T1-a_{n,m}) and v8,u n (T2-b_{n,m})

converge jointly to the same independent median-centred Gumbel pair as the axial theorem:
P(X<=x)=1-2^{-exp(x)}, P(Y<=y)=2^{-exp(-y)}.

In physical width ell=n|u|, the scale is equivalently ell[-partial_p tau_p(e)].

Important boundary: this does not silently give the varying-direction window theorem. If u_n changes, straightening by SL_2(Z) can make the transformed step range grow with n, so guard widths, dependency neighbourhoods and cylinder block constants are not automatically uniform. The new varying-direction centre theorem remains valid because it uses coordinate-free first-exit/Wulff bounds, but the full Poisson/Gumbel window needs one additional local-uniformity argument in that regime.