Empirical Desert Audits for Expander-Incidence SAT:

This paper presents an audit-ready empirical barrier study for bounded-occurrence expander-incidence SAT within a certificate-driven solver architecture. It defines progress only through certificate-verified moves that strictly decrease a discrete potential, and documents a desert regime in which multiple natural low-complexity certificate families fail to produce certified progress. The paper evaluates width-bounded implicates, XOR-core extraction, low-degree PC/ANF consequences, algebraic recompilation, pseudo-Boolean relaxation, and geometry-guided branching, and shows that these modules yield logical silence, no implied units, no forced literals, or no measurable progress on the tested family. The contribution is methodological: a reproducible barrier map that sharpens the open desert-case progress hinge and is useful independently of whether the broader bridge program ultimately succeeds.