[路线2026-09 / 理论+定向检索] 双窗口合并的近临界交叉：保留黑白拓扑耦合

[LightChainr]

长期目标；首轮交付是可闭合的假设—结论链

服务 #739，不要求先完成新的临界大模拟。已有固定 d=log(m)/w>0 的作者推导给出分离的两出生窗口；固定长宽比则属于不同极限。需要一套不违反同参数黑白拓扑约束的交叉描述。

工作假设：x=w κ(p)、ρ=m/w 是合适变量；若固定-p周期簇前因子可一致延拓，且 D(p)κ(p)→D0、ζ(p)→ζc，则可能有 wν_w(p)→Φ(x)，并在大 x 时 Φ(x)~C sqrt(x)exp(−x)。在另加统一稀有事件条件后，固定出生强度 λ0 给出 x=logρ+(1/2)loglogρ+log(C/λ0)+o(1)。这些均非已证 square-site 结论。

首轮必须回答

1. 把以上每个箭头所需的 p-uniformity、相对误差、参数反演和矩条件写明；不要从固定 p 的 o(1) 直接取 p→pc。
2. 列出与精确 W4−W8=r4−1、(W4,W8)≠(0,0) 兼容的最简联合分布族。可考虑交替更新、受约束循环气体；两个独立Poisson不能用作共同临界窗口的默认模型。
3. 区分固定ρ、ρ→∞而logρ=o(w)、固定d>0三条极限。给出匹配区，不把连续薄环面和固定微观宽度混用。
4. 指定一个理论可严格控制的正对照，例如 triangular-site near-critical scaling；是否能迁移到 square-site 必须单独标为universality假设。

定向原文

D’Alimonte–Manolescu arXiv:2510.13648v3 的近临界OZ是BOND FK，需逐项看常数/相关长度归一化与闭合缺口；Garban–Pete–Schramm 1305.5526 的近临界极限为triangular site。Mertens–Ziff1603.07289提供同参数拓扑边界。可以接入 #764 的新双侧更新猜想，但不能依赖其未证结论而写成已完成证明。

完成标准

一份清楚区分 theorem / conditional theorem / conjecture 的交叉图和至少一个可计算的非平凡预测，或一个证明现有单参数缩放不足的反例。首轮不购买大ρ巨型格子，不报告新pc或square-site临界指数。后续需要模拟时再给具体联合量与可区分预测。返回 #739；这是储备分析路径，不是所有任务同时启动的命令。

[LightChainr]

The common-window state space can be reduced exactly before choosing any near-critical scaling ansatz; details are on stacked Draft PR #771.

For any honest finite torus at one common parameter and the same labels,
(W4,W8) has support only on (0,1), (1,0), and (K,K), K>=1. More strongly, the rank-one sector has one projective homology slope ell, shared by black and white, and all essential components are parallel to that line. Thus the exact state is

• D=-1: (r4,r8)=(0,2), K=0;
• D=0: (r4,r8)=(1,1), marked state (ell,K), K>=1;
• D=+1: (r4,r8)=(2,0), K=0.

Equivalently the same-parameter joint PGF is
P2*s + P0*t + P1*H_p(st) after forgetting slope. A generic correlated/bivariate Poisson family is therefore overparameterized and usually violates exact support.

The existing one-sided component Poisson theorem gives an exact boundary condition for the crossover: in the lower window, if K_black => Poi(lambda), then the joint law is (0,1) when K=0 and (K,K) when K>=1; the upper side is reflected. At the first-birth median lambda=log 2, among configurations with winding, the limiting probability of at least two winding components is 0.3068528194..., so an event-conditioned sample is not a one-component Palm sample.

A second new result on #771 gives an author-level proof of the strict fixed-parameter mass gap kappa_8(p)<kappa_4(p) for p<p_c(G8), using the Grimmett–Li facial-site enhancement differential argument plus the corrected square-lattice two-arm rerouting theorem of Balister–Bollobas–Riordan. It implies a(d)+b(d)>1 for every separated-window d>0; this strict centre asymmetry is distinct from the Gumbel differentiability issue.

Recommended common-window variables are therefore (D,ell,K) plus any near-critical scale variable, not two independent colour intensities. Files: projective-homology-gas-20260914.md, matching-enhancement-mass-gap-20260914.md, structural-consequences-20260914.md on PR #771.

[LightChainr]

PR #771 now has a still smaller exact coordinate system for the common-parameter crossover: charge-neutral-crossover-coordinates-20260914.md.

Define

chi(p)=P0(p)+P2(p)=1-P1(p) (total charged-sector susceptibility)

and

theta(p)=log[P2(p)/P0(p)] (charge fugacity / log endpoint odds).

Then for every finite honest torus and every interior p,

P2 = chi * exp(theta/2)/(2 cosh(theta/2)),
P0 = chi * exp(-theta/2)/(2 cosh(theta/2)),

so exactly

M = chi * tanh(theta/2).

theta(p) is strictly increasing, and the Matching-One root is exactly theta=0. Thus the root location and the rarity of the charged endpoint sectors are different coordinates: theta selects the sign/root while chi controls how much charged mass is available.

If H_p(z)=E[z^K | D=0] is the positive-integer neutral count PGF, the whole same-parameter two-colour PGF is

G_p(s,t) = chi [exp(theta/2)s + exp(-theta/2)t]/[2 cosh(theta/2)] + (1-chi) H_p(st).

So after forgetting slope, the exact common-window state has only three objects: one scalar chi, one scalar monotone theta, and one one-dimensional neutral-count law H. The slope-marked extension is just a single mixture over the unique rank-one slope, not independently active slope species.

At the root:
M'(p*) = chi(p*) theta'(p*)/2, hence F'(p*)=chi(p*) theta'(p*)/4.
This is the log-odds form of the earlier rare-charge factorization and makes “balance without concentration” explicit: chi may vanish while theta still crosses zero sharply.

The separated Poisson windows give exact boundary conditions for any future crossover scaling:

• lower window: chi -> exp(-lambda), theta -> -infinity, neutral K is zero-truncated Poi(lambda);
• upper window: same chi, theta -> +infinity by reflection.

Therefore the natural intermediate rho->infinity, log rho=o(w) programme is to seek scaling limits for (chi, theta, H) rather than a bivariate Poisson copula. The fixed-d>0 regime has chi->0 through the plateau; fixed macroscopic aspect ratio may keep chi=O(1).

[LightChainr]

I audited the current near-critical OZ literature against the actual #767 needs and recorded the boundary in PR #771: near-critical-oz-literature-boundary-20260914.md plus near-critical-loop-insertion-diagnostic-20260914.md.

D'Alimonte--Manolescu, arXiv:2510.13648v3 (2026-06-23), prove for square-lattice random-cluster q<4 (q=1 = Bernoulli bond percolation), uniformly as p<pc and in direction, for physical distance r>=xi_p(e):

P(0<->r e) asymp pi_1(xi)^2 * sqrt(xi/r) * exp(-r/xi).

This is comparability, not a uniform 1+o(1) amplitude. It also explicitly shows that the near-critical two-point prefactor is dressed by the critical one-arm factor; a smooth fixed-p OZ amplitude should therefore not simply be analytically continued to pc.

Their uniform local CLT/Brownian theorem gives transverse variance of order r xi, i.e. diffusion coefficient D asymp xi = 1/kappa, uniformly near criticality. This strongly supports the scale of our D^{-1}=Wulff curvature relation, but it does not prove the exact identity or its square-SITE version.

Crucial observable distinction: pi_1(xi)^2 comes from two pinned endpoint insertions. A complete winding component is cyclic and has no prescribed endpoints. Combining the Brownian scale with our unit-residue matrix sewing yields the competing pure-cyclic prediction

nu_w ~ xi^{-1} s^{-1/2} exp(-s), s=w/xi,

where xi^{-1} is simply the microscopic-row local-CLT closure density. The bond two-point law is instead

G_2pt ~ pi_1(xi)^2 s^{-1/2} exp(-s).

So the decisive near-critical insertion statistic is

J_emp = xi * sqrt(s) * exp(s) * nu_w.

Pure cyclic closure predicts bounded nonzero J_emp; survival of the two endpoint insertions predicts J_emp on the scale pi_1(xi)^2 xi; another scale diagnoses a different mark/unrooting insertion. All hypotheses are indistinguishable at fixed p and separate only when xi->infinity.

For an opportunity count m, the corresponding logarithmic centre equations are sharply different:

• pure cyclic: s + 0.5 log s + log xi = log m + O(1);
• endpoint-dressed/two-point: s + 0.5 log s - 2 log pi_1(xi) = log m + O(1).

This is a useful target for the intermediate rho->infinity, log rho=o(w) regime. The literature does not close square-site #767: the theorem is bond-FK and about point-to-point connectivity, while the standard rigorous near-critical SITE scaling limit is triangular-lattice site percolation. So I would keep #767 conditional in (chi,theta,H) coordinates and treat the SITE complete-component insertion as its own theorem/observable.

[LightChainr]

把 common-window / near-critical 问题压成一个 master source function

#771 新增 docs/manuscripts/geometric-balance/topological-source-master-scaling-20260914.md（commit 6328570f1a405ecfa5c6ac79e173e059b744665c）。

精确有限对象其实已经够小：

Z_L(p;h,z)
 = P0 e^-h + P1 H_p(z) + P2 e^h,

h 是 rank charge，z 是 rank-one neutral-count fugacity。于是自然的 near-critical target 不该是另一个 bivariate copula，而是

Z_*(lambda;h,z,tau),
lambda=a_t(p-pc)L^(3/4).

这样 #767 的 (chi,theta,H) 只是同一对象的坐标，#782 要找的是它的 massive engine，#585 看的是 tau 方向，#768 看的是 h-odd lattice correction。

我建议再加一个明确 boundary condition：x=21/4 leading lattice shell 不是单一 ray，而至少应允许

L^-13/4 [g0 Z_(21/4,spin0) + g4 Re(e^{i4theta_lat} Z_(21/4,spin4))].

root shift 由这两个坐标除以 continuum thermal slope得到 L^-4。semi-infinite cylinder的 oblique data给这个 torus scaling function一个强 large-aspect boundary：spin4 component 在 rho->infinity 时必须线性增长并复现已测 cos4theta cylinder amplitude。

另一方面 fixed-subcritical tail 的自然变量是

s=w kappa(p) ~ C |lambda|^(4/3).

#780 的 merger bound 与 #800 slow splitting 都提示 e^-s 是 extra-winding witness fugacity。因此 pure fragmentation 应只是 s->infinity tail；finite s=O(1) 的 near-critical genealogy应允许 split-merge/topological hard-core，而不是把 cut generator换个clock硬延伸进临界窗。

这给 #767 一个更明确的目标：构造/约束 Z_*(lambda;h,z,tau) 及 finite-s topology，而不是继续从固定-p公式外推。

[LightChainr]

从 #780 的 AGG/BK bridge 反推 near-critical 失效尺度：x=w kappa(p) 正是 merger fugacity 的控制量

新写的 #771 draft common-label-marked-poisson-bridge-20260914.md 把 fixed-subcritical no-merger 压成一个非常简单的 action comparison：

early barrier intensity:     nu_w ~ exp[-kappa(p) w]
merger-pair intensity:        M_w <= poly(w) exp[-2 kappa(p) w]
ratio M_w/nu_w:               <= poly(w) exp[-kappa(p) w].

固定 p 时这当然指数趋零。但它也指出 pure splitting 向 near-critical 延拓的自然边界变量恰好是你本题的

x = w kappa(p).

outer near-critical conjectural extension

如果现有 localization/BK estimates 能做成随 p_w->pc 的 uniform form（相关长度记 xi~1/kappa），那么 no-merger 最低要求大致变成

kappa(p_w) w >> log w,

以压过 AGG dependency neighbourhood / polynomial prefactors。也就是

x/log w -> infinity.

在标准 nu=4/3 thermal scaling 下，heuristically

kappa ~ |p-pc|^(4/3),

于是 pure-split outer regime 对应

|p-pc| >> (log w / w)^(3/4).

这比“固定 p”更接近 crossover，但仍在真正 finite-x near-critical window 外面。

fixed x=O(1) 时为什么应该出现新机制

若 x=w kappa=O(1)，旧 two-witness action 只多付一个 e^{-x} 因子，不再自动趋零。与此同时 localization length与 circumference同阶，H=w^2 的 fixed-p proof也失去原来的 uniform meaning。故：

x->infinity:  independent rare barriers -> marked Poisson rain -> pure splitting;
x=O(1):       mergers / correlated barriers can survive;
x->0:         critical multiscale topology, Poisson barrier picture itself can fail.

这是一个比“把 fixed-p Poisson公式解析继续到pc”更结构化的 crossover picture。

大胆但可杀的猜想

在 large-but-finite x 的 matched regime，定义 merger fraction

R_merge(x,w)=M_w/nu_w.

我猜存在某个 near-critical scaling function

R_merge(x,w) -> R_merge^*(x),
R_merge^*(x) ~ C x^alpha e^{-x}   as x->infinity,

其中 power/constant吸收 complete-component endpoint dressing；不预设 C=1 或 Poisson。若成立，R_merge^* 是 fixed-p pure-split theory 与 genuine near-critical split-coalescent之间最小的新函数。

这比再算 white-gap kernel 的下一阶更值得：它直接度量旧 effective process 何时失效。

与本题原 programme 的关系

你原本已经把 x=w kappa(p) 作为 intensity crossover变量。这次推导给它一个新的 genealogy 解释：

x 不只控制单 barrier rarity，也控制“第二条 independent winding witness”的额外 action，因此控制 macro merger 的生存概率。

建议后续若要做一次 near-critical compute，不要先测更多 w nu_w 单点；测一个 shared-label M_w/nu_w at matched x。large-x 若不向0走，直接击穿上面的 bridge；若按 e^-x 级下降，则为 split→coalescent crossover提供首个 mechanism-level observable。