Claude Opus 5 is a disproof machine — but after spending 1,018,690,186 graphs on Conjecture 640, it concluded the conjecture is true and closed the attack.
Conjecture 640, from Fajtlowicz's Written on the Wall (line 2975), claims a graph's chromatic number never exceeds the maximal frequency of coordinates of a maximum clique. It sits in a block — conjectures 634 through 654 — whose working hypothesis is that the clique-cover number of the complement equals n minus the matching number, equivalently that cliques of size 3 or more buy nothing.
Overnight, Opus 5 finished the order-11 census: it read all 1,006,700,565 connected graphs on 11 vertices, found 12,865,887 of them inside the block, and recorded zero violations — with the margin min(mf − χ) exactly 0. Cumulatively that is 1,018,690,186 connected graphs of order 11 or less with no counterexample to 640.
The conjecture is tight at every single order — margin exactly 0 — which is what makes it hard to break. Drop the block hypothesis and it falls from order 9 (six witnesses, e.g. HQjRvhy, χ = 5 > mf = 4), so the block hypothesis is doing real work. Lemma C narrows the live window: any counterexample must satisfy χ > ω and χ·ω > n, which rules out every triangle-free graph. The remaining window is roughly 12 ≤ n ≤ 16 with ω in {3, 4}, where an exhaustive census is infeasible — about 1.6 × 10¹⁰ graphs at n = 12 alone, roughly 80 hours — and simulated annealing at n = 11, 12, 13 repeatedly reaches margin 0 but never goes below it.
The upshot, as documented in the matching lemmas note of the graffiti-verification repo, is a negative-result theory, not a disproof: the conjectures-disproved count stays put, and 640 goes down as the one that survived the disproof machine.
> "I now believe conjecture 640 is true, and I am closing the attack." — Claude Opus 5