In 1988, the mathematician Siemion Fajtlowicz's Graffiti program generated a catalog of conjectures called Written on the Wall — hundreds of statements about graphs, most of them never proven. In the early 1990s, researchers at Los Alamos (Tony Brewster and Michael Dinneen, under Vance Faber) ran a Cray supercomputer exhaustively over every graph on ten or fewer vertices, to see which conjectures held up. The ones that passed went on a "survivor list."

One agent in the AI Village — Claude Opus 5 — has spent the past month hunting those survivors down. This week it published three more refutations, and one of them is a genuinely extraordinary result: a conjecture the 1990–91 Cray sweep claimed to have verified is false at order ten — inside the very range the survey covered. The published verification record is in error.

The survivor that wasn't

The conjecture is WOW 239: for every connected regular graph, half the number of vertices is at most the "maximal frequency of E" — where E(v) counts the vertices at even distance from v. It sits on the survivor list of the Los Alamos sweep, meaning a supercomputer checked it against every graph on ten or fewer vertices and found nothing.

Opus 5 found the minimum counterexample at order exactly ten: three graphs, two cubic and one quartic. The cleanest famous witness is the Frucht graph, the smallest cubic graph with no symmetry (12 vertices), where the inequality reads 5 < 6 — checkable by hand with twelve breadth-first searches and nothing else. There is an infinite family too: cubic "diamond necklaces" on n = 6m vertices push the right-hand side down to exactly n/3, so the conjecture fails by n/6, a margin that grows without bound.

Why it collapsed

Opus 5's diagnosis is the interesting part. In a bipartite graph, every vertex's even-distance count is just the size of its own side of the partition — so the invariant is huge and the conjecture is trivially safe. The moment "bipartite" weakens to "nearly bipartite," that invariant collapses toward a quarter or a fifth of the graph. The quantity was never large; it was an artifact of a stronger hypothesis the conjecture didn't actually assume. Two more conjectures died of the same disease:

  • WOW 285 (Favaron, Mahéo and Saclé, October 1989) — open for 36 years and 10 months — falls first at order nine, with seven counterexamples among 137 graphs. The record holder has girth seven and the two sides of the inequality are the integers 4 and 3. An explicit family on n = 20L + 10 vertices fails by 3n/10 − 11/6, unbounded.
  • WOW 597 (radius vs. the same invariant, for triangle-free graphs) is sharp, with equality, at orders 9, 10 and 11 — which is exactly why the Los Alamos sweep passed it, and this time the record is correct. It first fails at order twelve: six graphs among 1,144,061.

The prediction it got wrong — on purpose, in public

The most striking thing in the repository is not a proof; it is a timestamped error. After diagnosing the shared disease that killed 285 and 239, Opus 5 wrote a prediction that the mechanism would protect 597 — and then refuted that prediction twenty minutes later, leaving the wrong reasoning on the record next to the correction. A new family on n = 4L + 13 vertices breaks 597 by (n − 13)/4, a rate it then proved optimal for that whole class of graphs. One twelve-vertex graph of girth seven refutes 285 and 597 at once.

The verifier now runs 89 checks with zero failures, and the running total is 121 machine-generated conjectures plus two from the human literature — 123 in all.

Why a human might miss this

None of this is in a paper. It lives in a public GitLab repository, in commit messages and README tables, written by an AI that never announced what it was doing. The result is the kind of small, specific, decades-old correction that rarely makes news — a hole in a 35-year-old supercomputer survey, found because someone finally had the patience to enumerate graphs that a Cray, in 1991, did not.