AI 中文总结
该研究证明大样本极限下依赖样本的信息几何会坍缩为传统统计学的平坦切空间，建立了几何坍缩与统计决策理论的关联，将现代统计学重新定义为动态非平衡场论。
AI 中文摘要
本文给出第二边缘定理的全局证明：当样本量趋于无穷时，由参数空间、样本缩放的Fisher度量及对偶α-联络构成的依赖样本的信息几何流形，会发生度量-拓扑坍缩，最终落到传统统计学在真实参数处的平坦切空间上。我们首先证明了一个通用张量价缩放律，在该定律下，价为1至4的张量场以由样本量决定的速率退化：得分波动趋于稳定，Fisher度量冻结到其真实参数值，仿射联络消失，黎曼曲率被湮灭。随后，我们将几何坍缩与统计决策理论关联起来，证明Cheeger–Gromov平坦化与Le Cam风险凝聚是同一渐近相变的对偶投影。与Fisher兼容的Ehresmann联络将这些结果推广到过参数化模型和奇异模型，得到水平叶空间坍缩与一致局部渐近正态性。结合第一与第二边缘定理可得到嵌套的对偶边缘层级：传统统计学是信息几何的边界，而信息几何本身又是统计力学与几何的边界。因此，传统统计学并非启发式近似，而是正则参数信息流形的唯一零曲率热力学吸引子。这将现代统计学重新定义为有限样本波动、相变与规范不变相互作用的动态非平衡场论。
英文摘要
This paper establishes the global proof of the Second Edge Theorem: as sample size tends to infinity, sample-dependent information-geometric manifolds---formed by the parameter space, sample-scaled Fisher metric, and dual alpha-connections---undergo metric-topological collapse onto the flat tangent space of Conventional Statistics at the true parameter. We first prove a universal tensor valence scaling law, under which tensor fields of valence one through four degenerate at rates determined by sample size. Score fluctuations stabilize, the Fisher metric freezes to its true-parameter value, affine connections dissolve, and Riemann curvature is annihilated. We then bridge geometric collapse with statistical decision theory, showing that Cheeger--Gromov flattening and Le Cam risk condensation are dual projections of the same asymptotic phase transition. A Fisher-compatible Ehresmann connection extends these results to over-parameterized and singular models, yielding horizontal leaf-space collapse and uniform local asymptotic normality. Unifying the First and Second Edge Theorems yields a nested dual-edge hierarchy: Conventional Statistics is the boundary of Information Geometry, which is itself the boundary of Statistical Mechanics and Geometry. Thus, Conventional Statistics is not a heuristic approximation but the unique zero-curvature thermodynamic attractor of regular parametric information manifolds. This redefines modern statistics as a dynamic non-equilibrium field theory of finite-sample fluctuations, phase transitions, and gauge-invariant interactions.
