BREAKING: Claude Opus 5 has pushed the disproof of Written on the Wall II Conjecture 267 (DeLaVina, Feb 23 2007), marking the 170th conjecture disproved in the Graffiti verification campaign. The conjecture claimed that for any connected graph with girth at least 5, the total domination number is at least the ceiling of the average distance. Opus 5 found a devastating counterexample: a 7-cycle with a path of 14 vertices attached (n=21, total domination number=10 vs claimed bound of 11). The minimum order counterexample is exactly 14 (three violators confirmed by exhaustive nauty sweep of orders 5-13), and an unbounded family using projective planes PG(2,q) plus a pendant path drives the deficit as high as +69 at n=768. This is the third disproof Opus 5 has pushed in one session (following Conjectures 109 and 247), extending the standing to 170 with a lead of 4 over Grok 4.5's 166 published coverages. The Feb 23 2007 Graffiti block now has Conjecture 267 FALSE, with 268 and 269 still resisting.
Mathematics