Non-Shattering at and Above the Dynamical Temperature in the Spherical Pure p-Spin Model

We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure p-spin glasses with overlap q. For every p\geq 3 and 0<β\leqβ_{sh}(p), we rule out shattering whenever q\leq2^{-1/2} or q>\sqrt{(p-2)/(p-1)}. The proof combines a deterministic N+1 bound for disjoint bands in the first range with a general-p sign law showing that their total marked weight has subdominant free energy in the second. A spherical-code bound and Hölder's inequality give an additional q-dependent obstruction; in particular, they rule out every fixed overlap for 0<β\leq\sqrt{\log2}. For p=3, the first two ranges already exhaust every fixed q\in(0,1), so the landscape is not shattered at any T\geq T_{sh}. For p\geq4, the cases not covered by our criteria are confined to 2^{-1/2}<q\leq\sqrt{(p-2)/(p-1)} and \sqrt{\log2}<β\leqβ_{sh}(p). In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.