[路线2026-09 / 理论] 倾斜指数环面的出生中心：方向质量与最便宜同调类

[LightChainr]

问题

#739 的根/全分布几何结论已覆盖任意整数周期，但两中心公式 a=κ4^{-1}(d)、b=1−κ8^{-1}(d) 的作者证明是轴向指数长条。这里把剩下的方向问题单独做成可检查命题，不做宽度或角度普查。

设最短周期 u_n 的长度 ell_n→∞、方向 u_n/ell_n→e，横向高度 h_n=N_n/ell_n，且 log h_n/ell_n→d>0。工作猜想：当其他非平行周期在相关范数中代价严格更大时，两中心由 τ_{G,p}(e)=d 反演。τ 是平面两点连接的方向范数，不自动等于欧氏长度乘轴向κ。

必须先解决的几何点

1. 欧氏最短向量与相关范数最短向量可不同；证明指数伸长情况下何时只剩±u，或给出应使用 min_{v∈Λ{0}} τ_p(v) 的修正版。
2. 上界应保留整个周期的连接代价，不能用半径 ell/2 的臂界损失一个因子后仍声称精确中心。
3. 下界使用固定方向连接种子并显式闭合接缝；NN 物理相互作用不能随坐标变换任意旋转。
4. 方向只沿子序列收敛时应给子序列结论；两个非平行廉价类共存会受交数/秩约束，不能相加为独立Poisson。
5. 区分 Λ 中本原与环境 Z² 中本原；正常有限单元条件继续保留。

完成形式

给出可适用的方向中心定理及证明，或构造反例说明 d 与欧氏倾角还不足。先处理固定整数方向 u_n=n(a,b)，再处理变化方向，不要求一次宣布全部模空间结论。引用方向质量的存在/凸性时核对实际 SITE 或可证明的迁移，不把 bond OZ 全文当 site 定理。

复用 geometric-balance/manuscript.md、exponential-birth-centres.md 的已有论证；作者证明需在用到时逐条检查，不借重跑小整数替代全尺寸推导。检索可并入 #757/#740 的相应方向条目。返回 #739；无大模拟、GPU、默认新角度扫描。

[LightChainr]

PR #771 now has a direction-uniform enhancement consequence that simplifies this task further.

The bounded two-terminal pivotal conversion used to prove the axial strict mass gap is endpoint-direction independent. On every compact I subset (0,p_c(G8)) it gives one delta_I>0, uniform in direction, with

P_p^G8(0<->x) >= P_{p+delta_I}^G4(0<->x)

for all sufficiently distant x and p in I. Taking directional logarithmic rates yields

tau_8,p(e) <= tau_4,p+delta_I(e) for every e in S^1.

This does not yet silently claim tau_8,p(e)<tau_4,p(e) for all directions; that final strict same-p step needs a direction-uniform strict p-monotonicity statement for tau_4. The axial version is already available and closes the strict a(d)+b(d)>1 theorem.

Combined with the determinant/norm separation in structural-consequences-20260914.md, exponential elongation now has a cleaner proof architecture: the shortest period direction is the only competitive homology class, the directional norm sets the centre, and matching enhancement compares the two graph norms uniformly over direction. Multi-direction competition is no longer a default obstruction in this regime.

File: directional-enhancement-sandwich-20260914.md on Draft PR #771.

[LightChainr]

One more direction result is now on PR #771: dilute-directional-geodesic-entropy-20260914.md gives an elementary large-d asymptotic for every fixed rational direction, with no OZ input.

For u=(a,b), A=|a|, B=|b|:

NN:
tau_4,p(u) = (A+B) log(1/p) - (A+B) H_2(A/(A+B)) + o(1) as p->0.

Matching, with M=max(A,B), beta=min(A,B)/M:
tau_8,p(u) = M log(1/p) - M h_3(beta) + o(1), where h_3(beta)=inf_t[log(1+2 cosh t)-beta t] is the maximum entropy of a {-1,0,1} step with mean beta.

Proof is two elementary matching bounds: (i) an adaptive single forward path through fresh independent layers gives the connection-probability lower bound with the exact endpoint large-deviation entropy; (ii) all SA paths are bounded by all walks and a multivariate exponential tilt gives the matching support-function lower bound on tau.

For unit Euclidean direction e=(a,b)/r, this gives as d->infinity:

p_4,e(d) = exp[-H_2(A/(A+B))] exp[-(r/(A+B)) d] (1+o(1)),

p_8,e(d) = exp[-h_3(beta)] exp[-(r/M) d] (1+o(1)).

Examples:

• axis: p4 ~ e^-d, p8 ~ (1/3)e^-d;
• diagonal: p4 ~ (1/2)e^{-d/sqrt2}, p8 ~ e^{-sqrt2 d}.

So for every genuinely tilted fixed rational direction, p8/p4 decays exponentially in d; matching/NN centre separation is much stronger than the axial factor-three effect. This gives a sharp positive control for the directional centre theorem before any numerical directional mass inversion.

[LightChainr]

PR #771 now contains an author-level closure of the directional centre problem in the actual exponential-aspect geometry, including varying shortest-period directions.

The key new bridge is first-exit-torus-winding-upper-20260914.md. For any compact first-exit body T certified inside the subcritical exponential-moment domain, a minimum nonzero-homology torus witness can be chosen as a vertex-simple cycle. Applying the local first-exit skeleton to its lift gives disjoint quotient SITE witness sets, so BK is legitimate on the quotient and every factor is a genuine planar first-exit coefficient. This yields

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

Taking T=(1-eps)K_p via the large-box exhaustion theorem gives the full-period bound

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

This avoids the unsafe shortcut of comparing arbitrary quotient arcs directly with full-plane two-point events and retains the complete period-vector cost.

varying-direction-exponential-centres-20260914.md then proves the #765 theorem. If u_n is a shortest period, ell_n=|u_n|, e_n=u_n/ell_n -> e, h_n=N_n/ell_n, and log h_n/ell_n -> d>0, then for either NN or matching and fixed subcritical p,

(1/ell_n) log P_p(r>0) -> -max{tau_p(e)-d,0}.

The lower bound uses one fixed integer seed direction approximating e, repeated to within O(eta ell_n) of u_n, followed by a deterministic connector; finite-group packing gives h_n/poly(ell_n) disjoint attempts.

Exponential transverse height automatically eliminates nonparallel period competition: for every independent period lambda, |lambda| >= N_n/ell_n = h_n, so its correlation-norm cost is super-ell_n. In particular two nonparallel O(ell_n) periods would force N_n=O(ell_n^2), contradicting exponential elongation.

Consequences now on #771:

• T1 -> a_e(d) with tau_{4,a_e(d)}(e)=d;
• T2 -> b_e(d) with tau_{8,1-b_e(d)}(e)=d;
• a Friedgut--Kalai contradiction on these directional tori proves p -> tau_{G,p}(e) strictly decreasing for every e in S^1;
• the endpoint-direction-independent matching enhancement then upgrades to tau_{8,p}(e) < tau_{4,p}(e) for every direction and 0<p<pc(G8).

So the remaining content of #765 is proof review/acceptance, not a missing direction case. No angle scan or OZ prefactor was used.