Self-Organization and Spectral Mechanism of Attractor Landscapes in High-Capacity Kernel Hopfield Networks

Kernel-based learning methods can dramatically increase the storage capacity of Hopfield networks, yet the dynamical mechanisms behind this enhancement remain poorly understood. We address this gap by combining a geometric characterization of the attractor landscape with the spectral theory of kernel machines. Using a novel metric, Pinnacle Sharpness, we empirically uncover a rich phase diagram of attractor stability, identifying a Ridge of Optimization where the network achieves maximal robustness under high-load conditions. Phenomenologically, this ridge is characterized by a Force Antagonism, in which a strong driving force is counterbalanced by a collective feedback force. We theoretically interpret this behavior as a consequence of a specific reorganization of the weight spectrum, which we term Spectral Concentration. Unlike a simple rank-1 collapse, our analysis shows that the network on the ridge self-organizes into a critical regime: the leading eigenvalue is amplified to enhance global stability (Direct Force), while the trailing eigenvalues remain finite to sustain high memory capacity (Indirect Force). Together, these results suggest a spectral mechanism by which learning reconciles stability and capacity in high-dimensional associative memory models.