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≥ 3 and 0<β≤βsh(p), we rule out shattering whenever q≤2-1/2 or q>√(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 H"older's inequality give an additional q-dependent obstruction; in particular, they rule out every fixed overlap for 0<β≤√log2. For p=3, the first two ranges already exhaust every fixed q∈(0,1), so the landscape is not shattered at any T≥ Tsh. For p≥4, the cases not covered by our criteria are confined to 2-1/2<q≤√(p-2)/(p-1) and √log2<β≤βsh(p). In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.