Arithmetic Isomorphism in the Hoffman-Singleton Hierarchy: Validating the Degree-57 Moore Graph via 2-Adic Spectral Analysis

The existence of the Moore graph with degree k = 57 and diameter d = 2 remains one of the most significant open problems in algebraic graph theory. This paper investigates the problem through a novel arithmetic lens, utilizing the 2-adic valuation (v2) of Hashimoto (non-backtracking) spectral traces derived via the Ihara-Bass formula. By performing a comparative ”Triple Validation” across the known Hoffman-Singleton hierarchy members (k = 3 and k = 7) and the hypothetical k = 57 candidate, we identify a universal arithmetic signature: a synchronous valuation drop at cycle length L = 9. Contrary to previous interpretations suggesting this anomaly as an obstruction, we demonstrate that it is a shared spectral fingerprint of the Moore family. Furthermore, we report a phenomenon of ”Hyper- Stability” for k = 57 at L > 10, where the candidate exhibits algebraic rigidity (v2 = 19 at L = 12) far exceeding its lower-degree counterparts. These findings suggest that the k = 57 graph is arithmetically consistent with the Hoffman-Singleton family structure.