Establish convexity of the P-Hop interaction energy

Establish whether the interaction term $E_2(\varphi)$ is globally convex for the P-Hop kernel, thereby determining whether its phase-space iteration admits the same genuine CCCP and monotone energy-descent interpretation as the S-Hop kernel.

Background

The paper proves global strict convexity of the interaction term E2(φ)E_2(\varphi) for the S-Hop kernel and uses that result to interpret the update as a genuine CCCP iteration with monotone energy descent. The broader kernel-family analysis, including the P-Hop instantiation, establishes local contraction and storage-capacity results under structural assumptions but does not provide the analogous global convexity result for P-Hop.

Resolving this issue would determine whether P-Hop has a global energy-descent guarantee comparable to S-Hop, rather than relying solely on local contraction arguments within invariant attraction balls.

References

For the general P-Hop kernel, the convexity of~$E_2$ is not established, and the convergence analysis relies entirely on the contraction arguments in Sections~\ref{subsec:local-approx}--\ref{subsec:Jacobian}.

— Phase Kernel Lifts Capacity of Dense Associative Memory  (2610.21493 - Zhao et al., 15 Oct 2026) in Appendix, Section 1, subsection “Strict CCCP Descent for S-Hop” (discussion following Lemma/Corollary Zangwill global convergence for S-Hop)