Support accumulation for mixed covariances

Prove that zero is an accumulation point of the support of the Parisi measure for every covariance with b2_2>0 and be'''(0)=0, beyond the cases covered by the paper and the near-critical class K(b5_*), thereby extending the applicability of Theorems local and field.

Background

The local fifth-order law and the small-external-field asymptotics require condition (H), namely that zero is not an isolated point of the support of the Parisi measure. The paper verifies this condition for the Sherrington–Kirkpatrick model using external preprints and for a near-critical class of covariances, but does not establish it for general genuinely mixed covariances satisfying b2_2>0 and be'''(0)=0.

A proof of support accumulation in this broader class would make the local constrained-Parisi and small-field theorems applicable to every covariance covered by the stated analytic assumptions.

References

Beyond these cases and $K(\varepsilon_)$, with $\varepsilon_$ as in Theorem~\ref{thm:nearcrit}(a), we do not know a proof of \ for covariances with $\beta_2>0$ and $\xi'''(0)=0$ (Section~\ref{sec:intro-constrained}); any such proof makes Theorems~\ref{thm:local} and~\ref{thm:field} apply to the covariances it covers.

— Upper deviations of the Sherrington-Kirkpatrick free energy: the fifth-order law and the onset of constrained Parisi measures  (2610.06480 - Pei, 5 Oct 2026) in Section "Earlier work and open questions", subsection "Inputs and open questions", paragraph "What the results rest on"