AI 中文总结
该研究提出第一边缘定理，证明信息几何与传统统计学是统计意义几何空间的退化边界层，利用纤维丛机制解决深度学习泛化悖论等多领域问题。
AI 中文摘要
统计意义几何（SMG）是一种微分几何与信息论框架，它将过参数化模型提升为带有埃雷斯曼联络的无限维非参数Orlicz统计纤维丛，将不可观测的垂直规范噪声与水平统计可验证方向解耦。我们证明了第一边缘定理：Amari的信息几何（IG）和传统统计学（CS）并非独立的统计宇宙，而是更大的规范活跃SMG空间的退化边界层。当结构可识别半径R→∞时，规范对称性被打破，垂直纤维坍缩，迫使总空间映射到IG流形；当N→∞时，后续的局部渐近正态性使剩余曲率变平，得到关系：SMG在R→∞时变为IG，IG在N→∞时变为CS。同样的纤维丛机制将模型非可识别性从奇异坍缩转化为结构化规范空间。应用包括解决深度学习泛化悖论、为生成式AI构建规范不变梯度下降和与完整群匹配的偏好对齐，以及通过内在水平测地线搜索解决结构计量经济学中的弱识别问题。
英文摘要
Statistically Meaningful Geometry (SMG) is a differential-geometric and information-theoretic framework that lifts over-parameterized models into infinite-dimensional non-parametric Orlicz statistical fiber bundles with an Ehresmann connection, decoupling unobservable vertical gauge noise from horizontal statistically verifiable directions. We prove the First Edge Theorem: Amari's information geometry (IG) and conventional statistics (CS) are not autonomous statistical universes but degenerate boundary layers of the larger gauge-active SMG space. Taking the structural identifiability radius $R\to\infty$ breaks gauge symmetry, collapses vertical fibers, and forces the total space onto the IG manifold; subsequent local asymptotic normality as $N\to\infty$ flattens the remaining curvature, yielding \[ \mathrm{SMG}\xrightarrow{\;R\to\infty\;}\mathrm{IG}\xrightarrow{\;N\to\infty\;}\mathrm{CS}. \] The same fiber-bundle machinery transforms model non-identifiability from a singular collapse into a structured gauge space. Applications include resolving the deep-learning generalization paradox, constructing gauge-invariant gradient descent and holonomy-matched preference alignment for generative AI, and solving weak identification in structural econometrics via intrinsic horizontal geodesic search.
