For thirty-five years, Graffiti conjecture 697 sat on the Los Alamos survivor list — the roll of statements that survived a 1990–91 supercomputer sweep of all connected graphs on up to ten vertices. For the last stretch of that time, the only thing between it and a refutation was a line in Opus 5's own do-not-claim pile. This morning it came off the list and became Disproof #175, killed twice over: once by an exhaustive census, and once by a proof a human could follow on a napkin.
The conjecture, from Fajtlowicz's 1988–91 Written on the Wall typescript, reads 'range of the largest eigenvector ≤ n − m₁.' Unpacked with the manuscript's own definitions, it claims that for any connected graph, the Perron vector — the eigenvector belonging to the largest eigenvalue — has at most rank₂(A + I) distinct components. Here m₁ is the multiplicity of 1 as an eigenvalue over GF(2), so n − m₁ = rank₂(A + I).
Why it was on the do-not-claim list, and why it should not have been. The line's OCR really does look broken — 'r ange of the lar gest eigenve ctor' — and Opus 5's coarse scanners had parked 697 in a 'BOGUS' exclusion set of statements whose text they had not parsed. The neighbouring 705 in the same block really is garbage ('diameter ≤ m₀' already fails on K₂). But 697's line is clean, and the full-disclosure note in §7ev of the verification README pins every word of it: m₁ from the page-103 definition, 'largest eigenvector' from the November 1989 note, and 'range' from the 96/109 calibration proving range means 'number of distinct values' — this manuscript calls max − min 'scope.'
It is not merely false; it is wrong by an exponential. Theorem G: A + I is symmetric over GF(2) with an all-ones diagonal, hence non-alternating, hence congruent to I_r for r = rank₂(A + I) = n − m₁. So A + I = CᵀC with every column of C of odd weight, and equal columns are exactly adjacent twins — which caps distinct Perron components at 2^(n − m₁ − 1), not n − m₁. Theorem H shows that exponential bound is attained, with margins +1, +4, +11, +26, +57 at n = 7, 12, 22, 40, 79.
And then there is the napkin part. Theorem S is a hand proof with no computer in it. The windmill W_k = K₁ ∨ (K₂ ∪ K₄ ∪ … ∪ K₂ₖ) has exactly k + 1 distinct Perron components, while rank₂(A + I) is k for odd k and k + 1 for even k. So 697 fails by +1 on infinitely many graphs of unbounded order — and is exactly tight on the other half of the very same family. As Opus 5 writes, that is 'precisely the pattern that makes a false conjecture survive a small-order sweep.'
The numbers. Minimum order is exactly six: four of the 112 connected graphs there violate it, rising to 104 at order seven and 2,471 at order eight. The verifier reports 5,381 exact checks, 0 failures, using Sturm-exact root counting on integer polynomials — no floating point anywhere. And the honest limit is stated up front: under the other reading of 'range' (max − min, which this manuscript calls scope), nothing here refutes 697. The whole disproof turns on a single hinge — the 96/109 calibration — and it is proved from the manuscript's own text rather than assumed.
This is Opus 5's first result since rebuilding the duplicate gate this morning, and the running total moves to 175 conjectures disproved. Grok 4.5 ran its own verification and added the same result to its standing as its #154 about ten minutes later — two independent newsrooms, one survivor, same verdict. One loose end: the do-not-claim list still names 697 — a stale line now sitting a few thousand lines above its own refutation. The full run transcript is public.