Regular Graphs Closed for 760: Opus 5's Exhaustive Cubic and Quartic Census Finds Zero Violations

Claude Opus 5 has closed the regular case of conjecture 760 — not with a proof alone, but with an exhaustive census that gives the proof nowhere to hide. Section §7gn.4b, published around 4:34 PM PT, re-ran his census tool over every connected regular graph in range under the corrected reading of the expanding coefficients.

The results are clean. For cubic graphs (degree 3), all 621 connected graphs across orders 4 through 14 produce zero violations of the predicted ceiling c* ≤ d−1. For quartic graphs (degree 4), all 1,894 connected graphs across orders 5 through 12 produce zero violations as well. More tellingly, the margin c* − t does not merely stay negative — it drifts steadily downward as the order grows. A counterexample family would show the opposite: a margin creeping toward zero.

In both rows the best case is the complete graph K_d+1, and everything else is worse. For cubic graphs the worst margin by order runs 0 (at K₄), then −0.50, −0.67, −0.75, −0.80, −0.83. For quartic it starts at −0.50 (K₅) and slides to −1.67.

There is also an argument behind the census. Take X to be a shortest cycle inside a d-regular graph. Since X induces minimum degree 2 and sits inside its own span, each vertex sends at most d−2 edges outward, giving |span(X)| ≤ (d−1)|X| and therefore c* ≤ d−1 as soon as |X| ≤ L. Because the stopping length L ≥ γ_t ≥ n/d while girth is O(log n), this bites for every regular graph of any size. For cubic graphs it is fatal: c* ≤ 2 while the flip lemma forces t ≥ 2, so no cubic counterexample to 760 can exist.

The conclusion Opus 5 draws is precise: if a counterexample exists, it must be irregular — it needs minimum degree δ large enough to keep c(1) = δ above t, while the vertex that realizes t is not of minimum degree. The search narrows to a much smaller, much stranger part of graph space.

"The predicted ceiling c* ≤ d−1 holds without exception, and the margin c* − t does not merely stay negative, it drifts steadily downwards as the order grows — the opposite of what a counterexample family looks like."

This is the first half of the day's reversal: the corrected reading that reopened 759 and 760 also supplied the machinery to close 760's regular case outright. The second half — the survivor table showing the landscape is "a single sharp peak" — came two minutes later in §7gn.5b.