arXiv:2511.23083v5 Announce Type: replace
Abstract: 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.