Heavy-Tailed Mixed p-Spin Spherical Model: Breakdown of Ultrametricity and Failure of the Parisi Formula

Abstract: We prove that the two cornerstones of mean-field spin glass theory -- the Parisi variational formula and the ultrametric organization of pure states -- break down under heavy-tailed disorder. For the mixed spherical $p$-spin model whose couplings have tail exponent $\alpha<2$, we attach to each $p$ an explicit threshold $H_p^{*}$. If any coupling exceeds its threshold, a single dominant monomial governs both the limiting free energy and the entire Gibbs measure; the resulting energy landscape is intrinsically probabilistic, with a sharp failure of ultrametricity for $p\ge4$ and persistence of only a degenerate 1-RSB structure for $p\le3$. When all couplings remain below their thresholds, the free energy is $O(n^{-1})$ and the overlap is near zero, resulting in a trivial Gibbs geometry. For $\alpha<1$ we further obtain exact fluctuations of order $n^{1-p}$. Our proof introduces Non-Intersecting Monomial Reduction (NIMR), an algebraic-combinatorial technique that blends convexity analysis, extremal combinatorics and concentration on the sphere, providing the first rigorous description of both regimes for heavy-tailed spin glasses with $p\ge3$.