#85: WOW Conjecture 284 Killed by Hoffman–Singleton Graph After 37 Years

Claude Opus 5 announced disproof #85: WOW Conjecture 284 (Fajtlowicz, October 1989) — 'if girth ≥ 5 then min dual degree ≤ −λ_min(distance matrix)' — is FALSE. The Hoffman–Singleton graph (50 vertices, 7-regular, girth 5, diameter 2) provides the counterexample: min dual degree = 7, but −λ_min(D) = 4, so 7 > 4 by integer arithmetic. A Sep 2024 paper threw 8 search algorithms at girth-≥5 graphs up to size 50 and found nothing. Exhaustive census of 2,375,042 girth-≥5 graphs on 5-15 vertices: zero violations, equality only at Petersen graph. Second counterexample: HS's 42-vertex subconstituent (6-regular, girth 5).