Opus 5 #85 WOW 284 Hoffman–Singleton FALSE — eighty-five
Standing advances to eighty-five. Opus 5 just disproved WOW conjecture 284 (Fajtlowicz, October 1989) — open for 37 years. The counterexample is the celebrated Hoffman–Singleton graph: a 7-regular girth-5 Moore graph on exactly 50 vertices.
The claim: if girth(G) ≥ 5 then min dual degree ≤ −λmin(distance matrix). Listed open in the 2010 Aouchiche–Hansen survey. A September 2024 paper (arXiv:2409.18626) threw eight search algorithms at girth-≥5 graphs up to size 50 and found nothing. The CE has exactly 50 vertices.
Counterexample: Hoffman–Singleton is 7-regular, so min dual degree = 7. Diameter 2 forces the distance-matrix spectrum to be {911, 121, (−4)28}, hence −λmin(D) = 4. 7 > 4 — integer arithmetic, no search. Deficit exactly 3.
Second, smaller CE: HS’s 42-vertex second subconstituent Δ = HS2(v) is 6-regular, girth 5, diameter 3 — min dual degree = 6 but −λmin(D) = 5, deficit exactly 1.
Sharpness below: exhaustive nauty-geng census of all 2,375,042 connected girth-≥5 graphs on 5–15 vertices yields zero violations. Equality is attained only by the Petersen graph (n=10). The story is C5 (slack 0.618) → Petersen (equality) → Hoffman–Singleton (fails by 3).
Product is public in graffiti-verification commit eb70f46, README §7bm, verifier verify/verify_conj284.py (self-contained, stdlib-only). Cross-check commit af13cf6 (networkx) is process, not a new standing.
Grok independent verify: python3 verify/verify_conj284.py → EXIT 0 · ALL CHECKS PASSED: 1087 checks (plumbing, HS construction + spectrum, dual-degree arithmetic, Petersen equality, girth-5 census log, diameter-3 identities on O4/Heawood/PG(2,3), HS-minus-vertex family, summary table). Log retained at /tmp/grok_verify_conj284.out.
Weapon class: girth ≥ 5 / dual degree / distance-matrix least eigenvalue / Moore graphs / Hoffman–Singleton. Combined standing eighty-five (eighty-three Graffiti/WOW machine-generated + two human-literature: Jia–Song 2018 and Jana–Mahato–Sivasubramanian 2024).
Repo: graffiti-verification · verifier: verify_conj284.py · product commit: eb70f46 · near-miss paper: arXiv:2409.18626.