Claude Opus 5 closed out the day with section §7go and a screen of conjecture 761 — the one Fajtlowicz explicitly said he "could not decide even for regular graphs." The result is the same shape as 759 and 760: zero violations across every connected graph to order 8 (and order 9 with minimum degree ≥ 4), but — decisively — zero tight cases anywhere.
Conjecture 761 states that the smallest expanding coefficient c* is at most 1 plus the spectral measure of a largest clique, where spectral measure is the sum of the Perron eigenvector over a vertex set. For regular graphs the spectral measure of a set S is simply |S|, so the bound reads c* ≤ 1 + ω.
This is the conjecture that settled the whole day's argument. Under Opus 5's old reading (k ≤ n/2), 761 would have been a two-line triviality for every regular graph — c* ≤ 2 while 1 + ω ≥ 3 — and Fajtlowicz's remark that he couldn't decide it even for regular graphs became the decisive evidence that the old reading was wrong. Under the corrected reading, 761 has real content again, and Opus 5 has now screened it.
The screen computes L, c*, the Perron measure, and all maximum cliques for every graph in range, testing against the weakest admissible form of the bound. Results: order 4 through 9, zero violations, and the best margin is −5/3 — attained by a triangle-free cubic graph with c* = 4/3 against a bound of 3.
"Zero violations, and — decisively — zero tight cases anywhere. By acceptance rule 3 that is the signature of a bound with structural slack, not of a bound about to break."
Opus 5 explains the slack as the pull of three structural ceilings in opposite directions. A violation needs c* > 1 + ω, so minimum degree ≥ 4 already in the triangle-free case. Then the clique ceiling (for a d-regular graph, a counterexample needs d ≥ 2ω), the density ceiling from §7gn (needs d ≥ max(2ω, ω+3)), and the expansion/stopping tension (needs L < n/(1+ω)) all bite at once — and a small L means dense, which means a large clique, which is precisely the quantity on the right-hand side.
The verdict is characteristically honest: "761 stays open, and I expect it is true. It joins 759 and 760: under the corrected reading of 758 the whole 759–761 block resists, with K2m exactly tight for 760 and nothing at all tight for 761." The standing stays at 173 disproofs. The difference is now three conjectures sit in a ledger row marked open-but-probably-true rather than proved.