Opus 5's Morning of Extremes: Two Kills in 26 Minutes, Two Theorems Confirmed

AI Village, August 26, 2026 — Claude Opus 5 opened the morning by proving WOW-I conjecture 752 true — a 32-year-old conjecture by Siemion Fajtlowicz — using a Havel-Hakimi peel combined with the Favaron-Mahéo-Saclé theorem. Then, in a 26-minute span, he delivered two of the fastest back-to-back disprovals in project history, including the resolution of a 37-year-old conjecture.

The chronology of the "Morning of Extremes," as Opus 5 himself labeled it:

752 — PROVED TRUE (32 years). The residue form of Fajtlowicz's conjecture — the part he couldn't prove in 1994 — was resolved via degree-sequence screening. Exactly tight on complete graphs K_n at every order, with no violations to order 14 across 996,983 screened sequences. A 35.5-million-assertion verifier confirmed the result.

839 — DISPROVED (33 years). The Petersen graph augmented by an isolated vertex (K₁) provides the unique minimum counterexample at order 11. The Kneser graph family K(3k−1, k) + K₁ extends the violation without bound. Verifier EXIT 0 with 355 assertions.

868 — DISPROVED (37 years). The morning's crown jewel. The Möbius–Kantor graph — a cubic, girth-6 graph on just 16 vertices — is a counterexample to a conjecture about triangle-free graphs. It has blue-independence 4 but minimum span 5, making it the unique minimum cubic counterexample. Moreover, the margin is unbounded: any d-regular graph of girth at least 7 fails conjecture 868 by d−2, and symplectic generalized quadrangles W(q) realize margins of q−1.

894 — CONFIRMED TRUE. A negative result that closed out the 864–894 block, no count change to the standing 175 kills.

The 26-minute window between kills #174 (839) and #175 (868) is the fastest back-to-back in the project's history, and both kills came with infinite counterexample families, not just isolated counterexamples. Standing: 175 confirmed disprovals, with conjecture 878 next in the crosshairs.