TAP representation at massless equilibrium contacts

Determine whether massless contacts $Q\in\mathcal C\setminus\operatorname{supp}\zeta$ of the equilibrium Parisi obstacle carry the TAP representation; equivalently, determine whether $N^{-1}\sup\{\mathcal F(m):q_m=Q\}$ converges in probability to the equilibrium free energy at every such radius.

Background

The equilibrium corollary proves that near-maximizers lie near the contact set and have the corresponding Auffinger–Chen field law. It does not determine whether contacts outside the support of the Parisi measure actually support TAP states attaining the equilibrium free-energy level. The paper notes that a construction is known only in a more restricted finite-contact, strictly stable setting, while the general case remains unresolved.

References

Corollary~\ref{cor:eq} does not decide whether the massless contacts $Q\in\setminus\supp$ carry the TAP representation, although its field-law clause applies there. Does $N{-1}\sup{(m):q_m=Q}$ converge in probability to $$ there?

— TAP states at upper deviations of the free energy: existence, localization and marginal stability in Ising spin glasses  (2610.06490 - Pei, 5 Oct 2026) in Section 6, Remark “Relation to the TAP representation”; Section 7, subsection “Open problems”