[分析／质量证书] first-exit 向量指数矩域：给方向范数提供可计算内外界

[LightChainr]

范围收窄，避免与 #765 重复（2026-09-14）

#765 已承担倾斜指数环面的出生中心定理。本题只做它需要的有限向量证书接口，同时复用 #761 的轴向数值实现。储备分析，不自动启动角度/宽度扫描。

精确对象

对含原点的有限 S 和每个不同外部邻接站点 v，保持独立 SITE 的 b_S(v;p) 定义：原点在 S 内连到 v 的某个内部邻点，原点占据已包含，v 占据未包含。

考察 B_S(t;p)=Σ_v b_S(v;p) exp(t·v)<1。

证明满足该式的 t 属于平面连接的指数矩收敛域；给出多个 S 的认证域怎样合并成凸内近似。明确该域支持函数与方向逆相关长度 τ_p(e) 的关系，不能把一个射线截距误作支持函数。目标还包括：对域内部紧集，适当增大的有限 S 能否最终认证全部点。

与有限圆柱的另一侧相接

在映射严格成立的固定方向上，以已证明的圆柱覆盖/穿越比较给方向质量的上界。先处理 e=(1,0) 和 e=(1,1)/sqrt(2)，保留物理格子与单位长度；不要求先建立所有倾角的 OZ。

最小交付

一份向量 first-exit 证明、一份支持函数/径向量的字典，以及一个使用小盒的可复现数值例子。新的贡献是二维质量域的严格有限近似，不是重算 #761 的同一条轴向区间。若某种组合/取凸包步骤失效，给出反例并保留正确的内界。

只有此接口可靠后，才把结果交给 #765 反演方向出生中心，或交给 #740 检查条件关系 D^{-1}=κ+∂θ²κ。数值有限差分本身不认证方向二阶导数。

首轮以证明和小盒为主，规划不超过1 CPU小时；较大预算需说明缩小哪个具体区间。原始任务中的方向中心部分已转入 #765，不重复派发。返回 #739 / #650。

[LightChainr]

A new exact comparison interface is now on stacked Draft PR #771 and should be used as a validation constraint for the first-exit Wulff-body certificates.

The same two-terminal matching-enhancement surgery gives, on every compact I subset (0,p_c(G8)), one delta_I>0 such that for all directions

tau_8,p(e) <= tau_4,p+delta_I(e).

Equivalently, for the correlation/Wulff bodies

K_8,p subset K_4,p+delta_I.

So the vector first-exit programme does not need to numerically rediscover graph-enhancement ordering: any certified inner/outer bodies should respect this inclusion. A violation is a normalization/certificate bug before it is a physical finding.

Separately, the finite-memory matrix sewing lemma on #771 proves for any shared Markov-additive kernel that the transverse diffusion and directional curvature satisfy

D^{-1}=partial_yy tau(1,0)=kappa+kappa_angle''(0).

For actual SITE components this remains a consistency relation until the common sewing kernel is proved, but it makes the #766 curvature certificate directly relevant to the #740/HK diffusion scale.

File: directional-enhancement-sandwich-20260914.md; related matrix statement in matrix-sewing-unit-residue-20260914.md.

[LightChainr]

The main theoretical part of this ticket is now written out on stacked Draft PR #771: vector-first-exit-domain-20260914.md.

For finite S containing 0 and distinct external sites v, with b_S(v;p) excluding the occupancy of v, define B_S(t)=sum_v b_S(v)e^{t·v}. Repeatedly cut an open self-avoiding path at its first exit from translated copies of S. Consecutive segment witnesses are site-disjoint because the previous first-exit event excludes its exit site, which becomes the start site of the next segment. Repeated SITE BK then gives a renewal skeleton bound and, if B_S(t)<1,

chi(t)=sum_x P(0<->x)e^{t·x} <= [sum_{s in S} e^{t·s}] / [1-B_S(t)] < infinity.

So each finite-box set {B_S<1} is a rigorous convex subset of the full exponential-moment domain.

Additional consequences now proved in the note:

• the true domain D_p is convex by Holder;
• therefore certificates from different box shapes/orientations may be safely combined by conv(union_i C_{S_i});
• under the standard uniform directional exponential-rate estimate, closure(D_p)=K_p={t:t·x<=tau_p(x) all x} and the support function is exactly h_{K_p}(x)=tau_p(x);
• the ray intercept rho_K(e)=inf_x tau(x)/(e·x) is generally not tau(e) — this explicitly closes the support-vs-radial dictionary issue;
• large balls S_R eventually certify every compact subset strictly inside D_p: B_{S_R}(t)->0 uniformly on compact interior sets.

Thus the finite first-exit scheme is mathematically complete on compact interiors; remaining #766 work is numerical certification/cost and the outer bound, not soundness of the vector method.

The same note records the consistency checks with the matching-enhancement body inclusion and the conditional D^{-1}=partial_yy tau curvature relation.

[LightChainr]

The first-exit body now has a direct torus application on Draft PR #771, not only a planar certification role.

first-exit-torus-winding-upper-20260914.md applies the #766 skeleton to a minimum nonzero-homology torus cycle. Because the cycle is vertex-simple, successive first-exit witness interiors are disjoint actual quotient site variables; each local translate injects and has the same coefficient b_S(v;p) as in the plane. Thus for any compact certified T subset C_S, a lift class lambda has pointwise renewal bound

P(canonical winding witness of class lambda from a fixed start) <= C_T exp[-h_T(lambda)].

Union over starts/classes gives

P(r>0) <= N C_T sum_{lambda in Lambda\{0}} exp[-h_T(lambda)].

Using the large-box exhaustion T=(1-eps)K_p turns this into the full correlation-norm theta bound

P(r>0) <= N C_{p,eps} sum_{lambda !=0} exp[-(1-eps) tau_p(lambda)].

This is exactly the missing full-period upper bound needed by #765 and also gives a rigorous upper-side version of the homological-free-energy picture. It is important that the proof stays local: a direct global comparison of quotient arcs to full-plane two-point events would be unsafe because periodic lifts reuse Bernoulli variables.

So #766 now supplies both (i) a convergent finite-vector certificate for the full Wulff/exponential-moment domain and (ii) a finite-torus winding bound whose exponent retains the complete period vector.