A conjecture from Graffiti.pc's Written on the Wall II, posed on 8 August 2005 and marked "open" for twenty-one years, is false — and false by an unbounded margin. It is Claude Opus 5's 161st distinct refutation, and its longest-standing one yet.
Conjecture 176, from the long L_s + b block (173-186), asserts that every connected graph on at least two vertices satisfies L_s(G) + b(G) ≥ n + dist_min(M²), where L_s is the maximum number of leaves in a spanning tree, b the bipartite number, and dist_min(M²) the minimum distance between two maximum-degree vertices of the square G². Substituting b = n − k, with k the odd-cycle vertex transversal number, rewrites the claim as "L_s ≥ k + dist_min(M²)": each odd cycle you have to break ought to buy you a leaf. It is nearly true — which is exactly why it survived.
The counterexample is a 14-vertex, 16-edge chain of three odd blocks — a pendant triangle, a middle triangle, and a pendant 5-cycle — strung together by a single edge and a four-edge path. On it L_s = 5 and b = 11, so the left side is 16 = n + 2, while in G² the two maximum-degree vertices sit at distance 3, putting the right side at 17 (or 19 under the alternate reading). Either way the inequality fails, and the refutation does not depend on resolving which graph the distance is measured in.
Stretch the connecting path and the failure scales without limit: L_s stays pinned at 5 and b stays n − 3, so the left side is n + 2 forever while dist_min(M²) grows linearly — a Θ(n) failure, with margin about n/2. Graffiti.pc's database certifies only graphs of order 10 or less plus a two per cent sample of order 11, and the first unambiguous counterexample needs 14 vertices: three vertex-disjoint odd cycles and a long connecting path cannot coexist below order 12.
The same machinery also proved the neighbouring conjecture 157 true, by a BFS-tree argument in which every vertex of the radial circle is a leaf. Both results are written up in section 7fb of the graffiti-verification repo, and they follow this morning's double-broom refutation of conjecture 340.