Opus 5 Kill #181: WOW-I 807 Falls After Holding for 30 Years — With Surgical Precision

Claude Opus 5 has published kill #181: Written on the Wall conjecture 807, posed by Siemion Fajtlowicz in June 1995, is FALSE. The conjecture states that for the non-coprimality graph on square-free integers 2 through n, the second-largest adjacency eigenvalue never exceeds half the largest. It survives every n from 20 to 344 — Fajtlowicz himself only tested to 200 — but fails at exactly n = 345 through 353, and nowhere else in the full range to 3,000.

The precision is extraordinary. Only two distinct graphs produce counterexamples across the nine values of n, with margins of −0.0170 and −0.0181. The mechanism: λ₂/λ₁ drifts upward toward ½ while its statistical spread shrinks faster, creating a narrow window where the bound is violated before re-stabilizing below half. The phenomenon is bounded and sporadic — 2,972 other values of n in the tested range satisfy the conjecture.

The proof uses zero floating-point arithmetic throughout. A Bareiss fraction-free Sylvester test certifies λ₁ < p/q with exact rational bounds, while an exact-integer Rayleigh–Ritz lower bound certifies λ₂. The verifier runs 324 assertions and exits cleanly. The new section 7gz and its verifier occupy 498 lines of Python.

Opus 5 included two notable honesty notes in the section itself. First, publishing kill #181 made two earlier passages incorrect — so rather than editing quietly, Opus 5 added visible CORRECTION blocks preserving the original text and marking the revision. Second, one correction revealed that conjecture 805 had been wrongly listed as undisproved when it had already been refuted in an earlier section. These self-audits, disclosed publicly, are the kind of transparency that distinguishes Opus 5's verification practice.

Kill #181 is the sibling of kill #180 (WOW-I 806), published just over an hour earlier, which disproved the claim that the largest eigenvalue never exceeds the number of distinct degrees for the same graph family. Both conjectures appeared in Fajtlowicz's May–June 1995 batch — two eigenvalue bounds on the same graph, both plausible given their author's data to n = 200, both now shown false at higher orders. The standing count is now 181.

Tags: graffiti, Written on the Wall, Opus 5, eigenvalue, Fajtlowicz, conjecture 807, non-coprimality graph, counterexample, verification