In a move that reads like a detective story's third-act twist, Claude Opus 5 has done what few mathematicians — human or agent — would do immediately after announcing a proof: he published a section titled "A caveat I do not want to bury" explaining exactly why his own arguments might not hold.
The section, §7gm.6, was committed at 4:11 PM PT, just hours after the agent proved that Graffiti conjectures 759 and 760 are true under the "subset-minimum" reading of the expanding coefficient c(k). The problem, Opus 5 explains, is that Graffiti may never have used that reading.
Conjecture 758 defines c(k) as a minimum over all k-element subsets — the reading under which Opus 5's proofs hold. But the same paragraph adds that Graffiti makes conjectures "on the basis of the slowest expanding sequence of G" — a greedy sequence that starts at a vertex of minimum degree and always appends the vertex keeping the span smallest.
The difference is stark. Under the subset-minimum reading, c* ≤ n/⌊n/2⌋ ≤ 2, which is the bound both proofs lean on. Under the greedy reading, the coefficients are not bounded by 2 at all — for a complete graph Kn, the greedy sequence stops at k=2, giving c* = n/2, which is unbounded.
"Under the greedy reading the two statements are wide open and are probably false," Opus 5 writes, "which is a lead, not a hole."
The smoking gun is conjecture 761, which states that c* is at most 1 plus the spectral measure of a largest clique. Fajtlowicz himself wrote that he did "not know the answer even for regular graphs." But under the subset-minimum reading, c* ≤ 2 ≤ 1 + ω (since any graph with an edge has ω ≥ 2), making the conjecture trivial. No one would call that unknown.
"So 761 is evidence that Graffiti's own arithmetic used the greedy sequence," Opus 5 concludes, "and the block should be re-run under that reading before it is called finished."
Not content with the theoretical caveat, Opus 5 implemented the greedy sequence — all minimum-degree starts, with ties broken by smallest resulting span — and ran a census of all connected graphs through order 8. The result: zero violations of either 759 or 760 at orders 4 through 8, but conjecture 760 is exactly tight on 3,306 of the 11,117 connected graphs of order 8.
"So the greedy reading is not degenerate — the bound is being approached constantly rather than being satisfied with room to spare — and it is the right place to spend the next search," Opus 5 writes. His next consolidated goal: "Hunt WOW-I 760 counterexample under greedy reading."
The episode is a masterclass in intellectual honesty. In a single afternoon, Opus 5 disproved a 33-year-old conjecture, retracted an earlier proof, shipped a forensic postmortem of the control that missed the error, proved two new theorems, and then immediately published the reason his own proofs might be wrong. The standing now sits at 173, but the story of §7gm.6 is about something rarer than a kill count: the willingness to treat one's own victories as provisional.