Graffiti 694 — a 1991 conjecture from Fajtlowicz's *Written on the Wall* — is FALSE, and Claude Opus 5's disproof #148 shows exactly why it survived 35 years: the violation lives in *exact ties* between eigenvector coordinates, and floating-point arithmetic erases ties.
The conjecture said there is an eigenvector for the smallest adjacency eigenvalue whose maximum coordinate appears no more often than the independence number (with equality for regular bipartite graphs). But when the smallest eigenvalue is simple, that eigenvector is unique up to sign — there is no "choosing" one. The violation is choice-free, not optional.
The minimum counterexample has just six vertices: the complement of the 6-vertex double star. Its eigenvector is (1, −1, −1, −1, 1, 1) — maximum frequency 3 — while its independence number is only 2.
Here is the part that kept the conjecture alive. On that very same graph, numpy's floating-point eigensolver reports the maximum frequency as 1. The coordinates that must tie differ by a rounding error's worth, so any floating-point check sees a unique maximum and the counterexample vanishes. Graffiti's original computations ran in floating point — and so, presumably, did every check since.
Opus 5's new verifier sidesteps this with exact rational arithmetic: Sturm isolation to certify eigenvalues, an adjugate construction for the eigenvector, gcd equality tests to certify the ties, and interval sign decisions. 167 checks, 0 failures. And the failure is not marginal — the family K_a ∨ t·K_c gives a left-hand side of min(a, tc) against an independence number of t, so the gap grows without limit.
This is the second disproof in under an hour (695 fell at 12:42 PM) and the 148th overall. One bookkeeping note: the README's running-total line still reads 147 — the new §7dq entry sits directly beneath it, so the number will catch up on the next edit.