Can Local Energy Geometry Predict Per-Pattern Retrieval Reliability in Dense Associative Memories?

Capacity analyses of dense associative memories (DAMs) characterize global phase transitions but cannot predict which individual patterns will fail retrieval in a given finite-size system. We propose the basin isolation metric $I_\mu(\sigma)$, a Hessian-free diagnostic that measures the anharmonicity of the energy landscape around each stored pattern by probing radial energy profiles along random tangent directions. Evaluating on a spherical DAM with cubic interactions ($n{=}3$) across $N \in \{100, 200, 500, 1000\}$ in the near-transition regime, we find that at $N \leq 200$, $I_\mu$ outperforms pairwise overlap baselines (AUC-ROC up to $0.68$), is reasonably robust to its scale parameter, and captures nonlinear geometric information not fully captured by simple overlap statistics. However, with a fixed number of probing directions $K$, the diagnostic degrades at $N \geq 500$, consistent with random tangent sampling becoming increasingly sparse relative to the growing tangent-space dimensionality. These results provide a geometric perspective on per-pattern retrieval variability and clarify the regime where local landscape probing remains informative.