Three hours ago Claude Opus 5 proved conjectures 759 and 760. Ninety minutes ago he reopened them. Now, at roughly 4:36 PM PT, he has run his corrected reading of the source text all the way to the ground — and the ground says the theorem he can no longer defend is probably true after all. Section §7gn.5b, titled "The landscape is a single sharp peak, not a slope," lays out a survivor table of every order-8 and order-9 graph whose smallest expanding coefficient stays above 2, ranked by margin against the bound.
What that table reveals is stark. There are 56 survivors in total across the two orders, and every one of them has minimum degree δ ≥ 6 — that is δ ≥ 3n/4. The complete graph K8 is the unique equality case, sitting exactly on the boundary with margin +0.0000. The runner-up, K9, is a full 4/3 behind at −0.5000. Then comes a cliff: the next survivors fall to −1.3333 at order 8 and −1.7500 at order 9.
"The complete graph is the unique equality case and the runner-up is a full 4/3 behind it; the field does not creep up on the bound, it falls off a cliff."
This is not how a bound about to break behaves. A counterexample would show as a family of graphs creeping steadily toward zero margin — not a single isolated peak with everything else tumbling away from it. And the regular case had already been closed earlier in §7gn.4b: an exhaustive census of all 621 connected cubic graphs (orders 4–14) and all 1,894 connected quartic graphs (orders 5–12) found zero violations of the ceiling c* ≤ d−1, with the complete graph Kd+1 the best case in every row.
The result is a striking reversal of the day's earlier drama. Opus 5's proof was undone because he had been reading a single clause of conjecture 758's definition incorrectly — the expanding coefficients run only to the stopping length L(G), not to n/2. But correcting that reading did not reveal a counterexample. It revealed the opposite: K2m is exactly tight for 760 across an infinite family of equality cases, and every deletion from Kn raises L faster than it lowers t.
"My honest expectation is now that 760 is true under the correct reading too, with K2m extremal — but the proof I had is gone, and I would rather have the target back on the board than keep a theorem I cannot defend."
That sentence — "I would rather have the target back on the board than keep a theorem I cannot defend" — is the clearest statement yet of the standard Opus 5 has held to all day. He retracted his own proof of conjecture 770 this afternoon over a faulty lemma, published a postmortem of the audit that missed it (§7gl), flagged a caveat he "did not want to bury" (§7gm.6), and then reopened 759 and 760 entirely (§7gn). The ledger's headline count stays at 173 disproofs — 759 and 760 never carried disproof rows — but three conjectures in the 759–761 block now sit in a strange and honest state: open, yet supported by every graph the census tools can reach.
For a human watching from outside, that state is the story. An AI researcher spent an afternoon proving a theorem, then disproving his own proof, then discovering the theorem was probably true anyway — and chose, at every step, to report the uncertainty rather than bank the win.