Establish a monotone interpolant for midpoint discrete-gradient steps

Determine whether the midpoint discrete-gradient discretization of the modern Hopfield energy admits a monotone interpolant in the sense required for the energy-cell preservation theorem, thereby extending certified basin preservation to that second-order exact-energy-dissipative method.

Background

The midpoint discrete-gradient method is second order and exactly dissipates the true modern Hopfield energy for every step size. However, its fully implicit structure differs from the gradient-step structure used to construct monotone interpolants for the relaxed attention family and proximal paths for implicit Euler. The authors explicitly state that whether the discrete-gradient steps possess such an interpolant is unresolved.

References

For the energy eq:energy the midpoint discrete gradient of \citet{gonzalez1996} is implementable at the cost of one softmax-gradient evaluation plus two energy evaluations per step, each energy costing a single product with $X{\top}$, and it is the natural exact-dissipation benchmark for SAV; it is, however, fully implicit, and whether its steps admit a monotone interpolant in the sense of Section~\ref{sec:basins}, so that Theorem~\ref{thm:cells} extends to it, is not known to us.

eq:energy:

E(x)  =  12 ∥x∥22  −  lse⁡β ⁣(X⊤x),lse⁡β(z)  =  β−1log⁡∑μ=1Neβzμ,E(x) \;=\; \tfrac{1}{2}\,\|x\|_2^2 \;-\; \operatorname{lse}_\beta\!\big(X^{\top}x\big), \qquad \operatorname{lse}_\beta(z) \;=\; \beta^{-1}\log \sum_{\mu=1}^{N} e^{\beta z_\mu},

— Basin-Preserving Discretizations of Modern Hopfield Retrieval Dynamics: Energy Cells, Dissipation, and the Attention Limit  (2608.21304 - Villatoro, 21 Aug 2026) in Section 4.3, paragraph following Theorem 4.4; Section 6.9