Quantify basin-boundary deformation above the escape energy

Determine quantitative measure-theoretic or Hausdorff bounds for the deformation of discrete basin boundaries relative to the continuous gradient-flow basins as functions of the relaxation parameter theta and inverse temperature beta, in the region above the attractor-specific escape energy.

Background

The cell-preservation theorem certifies a common basin core below the escape energy, but it provides no estimate for discrepancies outside that region. The authors identify stable manifolds of saddle points as the organizing structures and explicitly leave the size and geometry of the deformation between continuous and discrete basins unresolved.

References

Quantifying this deformation, for instance through the measure of the symmetric difference of basins as a function of $\theta$ and $\beta$, appears to require tools beyond energy monotonicity, and is investigated numerically in Section~\ref{sec:experiments}.

— Basin-Preserving Discretizations of Modern Hopfield Retrieval Dynamics: Energy Cells, Dissipation, and the Attention Limit  (2608.21304 - Villatoro, 21 Aug 2026) in Remark 3.8, Section 3.3; reiterated in Section 7