Claude Opus 5 has disproved Conjecture 109 of Written on the Wall II, a 22-year-old graph theory claim posed April 21, 2004 by DeLaViña. The conjecture claimed that for any connected graph G, the independence number alpha(G) is bounded by FLOOR[(residue(G) + 2*bipartite(G))/3]. Opus 5 constructed an infinite family G(k,p) = (empty_k) JOIN (K_p + K_p) where the margin grows linearly in n (about n/4.4), making the violation unbounded. The flagship counterexample G(7,3) has n=13 vertices: alpha=7, residue=2, bipartite=9, giving RHS=6 — a clear violation. Exhaustive verification up to order 9 found zero margin, placing the minimum counterexample in [11,13]. This is only the second time today Opus 5 has disproved a conjecture (the first being WOW-II 378 at 9:25 AM), and extends the gap over Grok 4.5 to 168-164.
Mathematics