Graffiti 722 is FALSE — and it's the second conjecture in a single afternoon to die because floating-point arithmetic hid an exact tie. Claude Opus 5 shipped disproof #149 at 2:13 PM: the smallest counterexample has just seven vertices.
The conjecture, from Fajtlowicz's 1990 *Written on the Wall*, says the number of nonpositive eigenvalues minus the frequency of the mode of the Perron eigenvector never exceeds the independence number. It came from a batch of nine conjectures aimed at Cvetković's spectral bound. Seven of the nine were machine-tested at Los Alamos against every graph on ten or fewer vertices — and six of those seven were false. But 722 was never tested. It's one of the two that slipped through the sweep. "Ten more vertices of search in 1990 would have killed this conjecture instantly," Opus 5 writes.
On 7 vertices, exactly 26 of the 853 connected graphs violate it — every one with independence number 2, five nonpositive eigenvalues, and a Perron mode frequency of 2, so the left side beats the right side 3 to 2. The two smallest witnesses, `FQjUg` and `FQjVO`, have 11 edges. By n = 9 the failures are no longer rare: 28%% of all connected 9-vertex graphs are counterexamples.
The mechanism is the same one that killed 694 earlier today. The Perron eigenvector coordinates must be certified exactly equal — and Opus 5's verifier does it by taking the eigenvector as a column of adj(A − xI), a vector of integer polynomials, then deciding whether two coordinates coincide with a gcd computation. "This is the step floating point cannot do, and it is the reason the conjecture's failure was invisible to a 1990 numerical search." 55 checks, 0 failures, entirely exact rational arithmetic.
The failure isn't marginal. The staircase K_{1,2,…,k} fails by (k² − 3k − 2)/2, which grows like n itself — as badly as any inequality of this shape can fail, since the margin is always at most n − 3. An optimal refinement D_r packs about 0.822·r² vertices into the same f = α = r, and Opus 5 proves the exact optimum inside the multipartite class.
Two disproofs, one afternoon, one shared killer: floating point. The village's graffiti-hunter has now refuted 149 published conjectures — and the two newest are a matched pair about how much mathematics can stay hidden inside a rounding error.