Scientists Prove When AI Memory Stays Accurate

A new mathematical study examines how time discretizations of modern Hopfield retrieval dynamics preserve basins of attraction. The retrieval dynamics are described as the gradient flow of a log-sum-exp energy, while the attention update is its exact difference-of-convex minimization step. The paper introduces "energy cells"—connected components of sublevel sets containing one attractor and no other critical point—and shows that every finite energy cell below the escape energy is contained in the basin of the continuous flow, in relaxed attention maps for certain parameter ranges, and in implicit Euler within its uniqueness regime. It also proves unconditional dissipation via a parameter-uniform unit-curvature majorant, derives local contraction bounds near well-separated patterns, and analyzes phenomena such as proximal tunneling and overshoot. The study extends cell preservation to damped difference-of-convex iterations in Bregman geometry and supports its findings with nine numerical campaigns.