High-capacity kernel Hopfield networks exhibit a \textit{Ridge of Optimization} characterized by extreme stability. While previously linked to \textit{Spectral Concentration}, its origin remains elusive. Here, we analyze the network dynamics on a statistical manifold, revealing that the Ridge corresponds to the Edge of Stability, a critical boundary where the Fisher Information Matrix becomes singular. We demonstrate that the apparent Euclidean force antagonism is a manifestation of \textit{Dual Equilibrium} in the Riemannian space. This unifies learning dynamics and capacity via the Minimum Description Length principle, offering a geometric theory of self-organized criticality.

翻译：高容量核Hopfield网络展现出以极端稳定性为特征的\textit{优化脊}。尽管先前已将其与\textit{谱集中}现象关联，但其起源仍不明确。本文在统计流形上分析网络动力学，揭示该脊对应于\textit{稳定边缘}——即费舍尔信息矩阵趋于奇异的关键边界。我们证明，表观的欧几里得力拮抗实质是黎曼空间中\textit{对偶平衡}的体现。这一发现通过最小描述长度原理统一了学习动力学与网络容量，为自组织临界性提供了一种几何理论框架。