Understand GOE conditioned on SK(W)=c as a quiet planting distribution

Characterize the conditional distribution of Gaussian Orthogonal Ensemble matrices W given SK(W)=c (or an approximate or lower-bounded value), including its typical structure and algorithmic indistinguishability properties, to determine whether this natural construction serves as a valid quiet planting for certifying upper bounds in the Sherrington–Kirkpatrick model.

Background

In the quiet planting approach for certifying algorithmic upper bounds in the SK model, one constructs distributions that are computationally indistinguishable from GOE yet have guaranteed large optimization values. The authors discuss several candidate constructions, including conditioning GOE on observables related to SK(W).

They note that the specific conditional distribution obtained by drawing W from GOE conditioned on SK(W)=c (or approximately/at least c) is natural but remains poorly understood, leaving open whether it achieves the necessary indistinguishability and structural properties to qualify as quiet planting.