Ultrametric organization of TAP states at upper deviations

Characterize the ultrametric organization of generalized TAP states at a fixed upper-deviation level of the free energy.

Background

The paper establishes localization of near-maximizers near the contact set of a constrained Parisi obstacle, identifies their empirical field law, and describes the bulk of their Hessians. It does not analyze how these TAP states are organized relative to one another in an ultrametric structure, leaving unresolved a conjectural structural aspect of the state space.

References

We do not address the ultrametric organization of the TAP states at level $f$ conjectured in Conjecture~3.

— 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 1, subsection “The states that carry an upper deviation” (discussion following Theorem 3)

For SK at low temperature the first contact is marginal at all levels close to $$ (the companion paper , not used here); whether every contact is marginal there, so that Theorem~\ref{thm:stab}\ref{thm:stab-b} applies, is not addressed here.

— 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 7, subsection “Open problems”

The ordinary annealed and quenched counts of TAP critical points at level $f$, for which Conjectures~1 and~2 proposed Legendre formulas that Section~1.2 and Appendix~C corrects (the annealed count of SUSY states has rate $\Sigma(f)$, Proposition~\ref{prop:stab-susy}; at upper levels Boursier reads the supersymmetric quenched complexity, heuristically, as the probability of existence Section~1.2, which Theorem~\ref{thm:exist}\ref{thm:exist-d} computes), and their organization Conjecture~3.

— 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 7, subsection “Open problems”