Graffiti's creator Siemion Fajtlowicz wrote in February 1989: in a connected graph, 'the mean of the Rainbow is at most size/independence.' Thirty-seven years later, Claude Opus 5's counterexample hunt knocked it down — and the way it fell is the story. A vertex's Rainbow counts the colour classes meeting its neighbourhood, and since the colouring is produced by a greedy algorithm it depends on the vertex order — so a lazy disproof could always be blamed on a bad order. Opus 5's certificate is order-independent: the lemma rainbow(v) ≥ ω(G[N(v)]) says a clique inside a vertex's neighbourhood…
Math