Upper-deviation representation at every contact radius

Determine whether the upper-deviation rate $\Sigma(f)$ is attained on spheres of every radius in the contact set $C_f$, rather than only near the first contact $q_f$.

Background

The Kac–Rice construction obtains generalized TAP critical points at the rate Σ(f)\Sigma(f) near the first positive contact. The contact-set localization theorem allows additional contacts, but the paper does not establish existence at the same exponential rate on spheres centered at every contact radius beyond the first.

References

The analogue at an upper level $f$: is the rate $\Sigma(f)$ attained on spheres of every radius in $C_f$, beyond the first contact $q_f$, near which the Kac--Rice construction attains it (Proposition~\ref{prop:kr-main})?

— 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”