Retrieval Metastability and Centroid Hopping in Continuous Log-Sum-ReLU Dense Associative Memory

We study the retrieval state of Dense Associative Memory with the log-sum-ReLU (LSR) energy on the $N$-sphere as a function of the load $\alpha$ and temperature $T$. The compact support of the LSR kernel partitions the retrieval region into three regimes. At low $\alpha$, even the nearest spurious pattern is separated from the target by a free-energy barrier above $NT$, and retrieval is stable on any accessible timescale. Above a threshold ${\alpha }_{\mathrm{th}}$, the state thermally wandering near the retrieval minimum already fills a spurious pattern basin; the two basins merge into a centroid and retrieval is lost. Between these limits lies the metastable regime: spurious pattern basins exist nearby but are separated from the retrieval basin by an entropic barrier. To escape, the state must drift beyond its typical thermal range and reach a spurious basin, a rare event whose rate we identify as a multi-channel Kramers process. After escape, the system hops between two-pattern centroids. Monte Carlo simulations validate this mechanism. The disorder-averaged escape rate, with a single calibrated prefactor, reproduces the measured escape probabilities across the simulated range of load and temperature. At large $N$ the exponential growth of the number of escape channels is exactly compensated by their entropic barriers, so the escape rate falls as $e^{-N({\alpha }_{\mathrm{th}}-\alpha )}$ at every temperature, and the metastable regime extends to ${\alpha }_{\mathrm{th}}$. Simulations at fixed load and increasing system size show this suppression. The apparent phase boundary observed in finite-$N$ simulations lies inside the metastable regime: it is a kinetic crossover, reached where the escape time equals the simulation time, not a thermodynamic transition. Since the escape time grows exponentially with $N$, the metastable state becomes indistinguishable from thermodynamic stability at system sizes of practical interest. The kinetic boundary is relevant only for the moderate $N$ accessible to Monte Carlo simulations.