From Cut Localization to Modulus Freeze: Reproducible Empirical Constraints and Theoretical Targets for the Erdős–Gyárfás Conjecture via Matching-Covered Theory

We present a reproducible audit of a critical set of 31 cubic vertex-transitive graphs (17 Route-3 elite and 14 RBC) for the Erdős–Gyárfás conjecture (EGC) through the lens of matching-covered theory. Our computational layers certify arithmetic non-degeneration of the conformal length spectrum, deep dyadic depth (a conformal alternating cycle of length 27 = 128 in each of the 31 graphs), robust conformality (with a deterministic repair rule), and a systematic failure of all natural geometric translations from θ-structures to nontrivial facet-relevant odd cuts. We isolate the lattice-to-cycle-length gap (“Z-span” vs. “2-term symmetric differences”) as the correct theoretical frontier and propose a modulus-freeze conjecture as a shared kernel for two proof programs: a Z2-obstruction route (Petersen-type torsion) and an abundance-versus-restriction route via flip-graph growth and perfect-matching counts. A supplementary bundle provides inputs, scripts, and summary artifacts sufficient to reproduce all reported counts.