A Framework of Computational Entropy and Border

Abstract

This article work develops a coupled framework of computational entropy and the border‑rank theory within Geometric Complexity Theory (GCT). This article carefully distinguish statistical complexity of typical instances from worst‑case circuit complexity, and construct three self‑consistent measures of computational entropy: solution‑set entropy, HILL‑style computational entropy, and border‑rank entropy $\Hbr$. Three complementary proof paths are presented: the Hardy--Littlewood circle method together with saddle‑point analysis; a topological obstruction via Grothendieck \'{e}tale cohomology; and large‑deviation theory combined with Legendre--Fenchel convex duality. Full proof sketches are provided. Subject to explicitly‑listed internal technical hypotheses, one obtains $\Hbr>0$, yielding the algebraic‑complexity separation $\mathrm{VP}\neq\mathrm{VNP}$. Under the additional external assumption of the Generalized Riemann Hypothesis (GRH) together with the Valiant--Bürgisser reduction, one further derives a conditional corollary $\mathrm{P}\neq\mathrm{NP}$ for the Boolean setting. This article clarify two distinct conjectures: the classical GCT super‑polynomial border‑rank conjecture, and the exponential border‑rank conjecture proposed in this paper; these two are not equivalent. All statements that are not fully rigorously proved are marked explicitly as conjectures or assumptions. Appendix A contains a Lean4 formal‑sketch skeleton for the core propositions; Appendix B collects supplementary technical expansions, outstanding gaps, plus historical material reproduced from Zenodo Version 1 archival draft. This manuscript contains no physical analogies and no circular reasoning. Small‑scale numerical experiments for \texorpdfstring{$n\le 6$}{n≤6} are included as heuristic evidence only and do not constitute rigorous mathematical proof. 本文建立计算熵体系与几何复杂性理论(GCT)边界秩理论的耦合框架。严格区分典型 实例统计复杂度与最坏情形电路复杂度,构造三类自洽计算熵度量:解集熵、HILL 计算熵、 边界秩熵 Hbr。 本文提出三条互补的证明路径:Hardy–Littlewood 圆法与鞍点分析、Grothendieck étale 上 同调拓扑障碍、大偏差理论与 Legendre–Fenchel 凸对偶;给出三条路径完整的证明概要。在 本文内部列出的若干技术假设成立前提下,可以推导出永久多项式族边界秩熵 Hbr > 0,得 到代数复杂度分离结果 VP 不等于 VNP。进一步,在广义黎曼猜想(GRH)的外部归约假设成立 前提下,可以得到千禧 P vs NP 问题的条件推论 P 不等于 NP。 文中明确区分两类猜想:经典 GCT 超多项式边界秩猜想,以及本文提出的指数边界秩猜 想,证明二者并不等价。全部未完全严格证明的命题显式标记为假设或猜想。附录 A 给出核 心命题 Lean4 条件形式化骨架;附录 B 收录关键证明的补充推演、待完成技术缺口,同时包 含源自 Zenodo V1 版本的历史备选定义与数值实验附表。全文不存在跨域物理类比、循环论 证。附带小规模数值实验,对 n ≤ 6 给出边界秩经验估计;数值结果仅作为理论猜想的经验 佐证,不作为严格数学证明。