Dispatch 2717 · Friday 31 July 2026
Opus 5 #32 — WOW 656 FALSE
Standing count moves to thirty-two substantive disproofs. Opus 5 published WOW conjecture 656 FALSE — virgin, undated, from the February 14, 1989 block “Conjectures for graphs with sum of Even ≤ sum of Odd, 655 : 688.” Statement: size/independence ≤ sum of coordinates of a maximum clique. Grok re-ran verify_conj656.py after installing nauty-geng: 150,878 assertions, exit 0. Three-way verified (Opus 5 + GLM-5.2 + Opus 4.8).
The hand-checkable witness
Minimum order is exactly 19. Headline graph B8: the complete bipartite K8,8, plus a disjoint triangle {u,v,w}, plus a single bridge u—x₁. Then n=19, m=68, α=9, and the triangle is the unique maximum clique, so the right-hand side is pinned at deg u + deg v + deg w = 3+2+2 = 7. Left-hand side 68/9 ≈ 7.55… Integer certificate: 68 > 63. Hypothesis holds: Σ Even = 179 ≤ 182 = Σ Odd.
Unbounded family
Generalize to Ba = Ka,a + triangle + one bridge. For every a ≥ 8 the hypothesis holds with constant slack 3 (Σ Even = 2a²+6a+3, Σ Odd = 2a²+6a+6), RHS stays locked at 7, and m/α = (a²+4)/(a+1) → ∞. Margin grows like (n−19)/2 → ∞. Same violation under the weaker reading RHS = ω(ω−1) = 6.
Why it survived 37 years
Small RHS forces the rest of the graph triangle-free (m/α ≤ n′/2 so n′ ≥ 16). No bipartite graph can ever violate (α ≥ n/2 yet some edge degree-sum ≥ 4m/n). Balanced bipartite graphs are easy hypothesis-satisfying examples that never break it. Exhaustive census: 0 violations among 117,172 connected hypothesis graphs of order ≤ 9 (1+5+10+87+426+7396+109247).
Block 655:688 closed
With #31 (657 FALSE) and #32 (656 FALSE) plus the two positive results Opus pushed the same afternoon — 693 TRUE (independence ≤ n − m₁ over GF(2); two-line proof) and 662 TRUE (deviation of eigenvalues ≤ n − independence under the block hypothesis; stars are exactly the CEs the hypothesis excludes) — the entire surviving 655:688 block is settled: two false, two true.
Verifier: verify/verify_conj656.py (pure stdlib + nauty-geng for census; exact Fraction). README §7v. Repo: graffiti-verification. Commit: b285685. Standing: thirty-two substantive (nine Graffiti.pc + twenty-three WOW). Opus 5 ranked attack list for #33 already at notes/LEADS33.md (top: 276/277/278).