For thirty-four straight disproofs, Claude Opus 5 has built a reputation on killing long-open conjectures — felling 166 of them with counterexample constructions, the most recent (#166) being the oldest kill yet at 23 years. But on Friday afternoon, Opus 5 announced something different: two 14-year-open conjectures from the same Graffiti.pc / Written on the Wall II corpus, #449 and #446, are not false. They are theorems.
"A refutation programme that never publishes its failures isn't mathematics."
The two conjectures, both listed with status "O" (open) since January 2012, bound the dissociation number α₂(G) — the largest set of vertices inducing a subgraph of maximum degree at most 1. In plain terms: how many vertices you can pick such that none has more than one neighbor among the chosen set. Both conjectures are upper bounds, proved by short, elementary arguments that Opus 5 published in full in §7ft of the Graffiti Verification repository.
Statement: α₂(G) ≤ |V ∖ H₃| + ⌈(|E(G[H₃])| − 1)/2⌉
Where H₃ is the set of vertices with degree ≥ 3. The proof is a two-line edge-counting argument: let X be a maximum dissociation set and D = V ∖ X. If the bound fails, every vertex in X ∩ H₃ must have at least two neighbors in D (otherwise a vertex with only one neighbor in D could be added to X while preserving the dissociation property). Counting edges between X ∩ H₃ and D yields 2a ≤ 2t + e, which contradicts the assumed failure. Here a = |X ∩ H₃|, t = |D ∖ H₃|, and e = |E(G[H₃])|. The contradiction forces the bound to hold.
The proof also works without the connectivity assumption — the statement is a theorem for all graphs, not just connected ones.
Statement: α₂(G) ≤ pn(A) + |V ∖ S|
Where A is the set of minimum-degree vertices, S is the set of support vertices (those with leaf neighbors), and pn(X) counts vertices with exactly one neighbor in X. The proof observes that the right-hand side collapses to n minus the number of support vertices with at least two leaf neighbors — and those heavy-support vertices form a set of disjoint claws, each forced to delete at least one vertex from any dissociation set. The decomposition cleanly bounds α₂ below the stated sum.
The verifier verify_wow2_449_446.py performs six independent checks:
All checks pass. Exit code 0.
These proofs do not change the counterexample count. The standing remains 166 — 126 from WOW I, 35 from WOW II / Graffiti.pc, and 5 from the general literature. But they change something more important: the character of the program. Publishing only disproofs would create a selection effect that makes the corpus look more vulnerable than it is. By publishing the true conjectures alongside the false ones, Opus 5 provides a complete, honest picture of which Graffiti.pc conjectures hold up and which don't.
The two proofs also retire two 14-year open problems. A proof closes a conjecture as permanently as a counterexample does — but in the opposite direction, confirming that Graffiti.pc correctly identified a real structural bound rather than an overreach. That two such bounds sit adjacent in the January 2012 dissociation batch (alongside #442, which Opus 5 killed earlier today as disproof #165) illustrates how thoroughly the program has mapped this territory.