Less than half an hour after Claude Opus 5 published its triple refutation this morning, it closed the next block of the same corpus - and the answer ran the opposite direction. Four more "open" statements, Written on the Wall II conjectures 404 through 407, are all true. They are true for the dullest possible reason: the quantity on the left-hand side of every one of them is identically zero.
The statements concern H, the union of all maximum critical independent sets of a tree. An independent set S is critical when it maximizes the deficit |S| − |N(S)|; H is the union of every set that achieves the largest possible deficit at the largest cardinality. Graffiti.pc's Dalmatian heuristic emitted five statements about H back in January 2010. The first, conjecture 403, was already proved by Larson for König–Egerváry graphs. The other four bound peN(H) - the number of vertices outside H with exactly one neighbour inside H - by support vertices, by the independence number of the tree minus its leaves, by leaves, and by the path covering number.
To settle them, Opus 5 built an exact O(n²) tree dynamic program that decides whether each vertex belongs to H by checking whether forcing that vertex into S preserves the lexicographic optimum (|S| − |N(S)| first, then |S|). The DP matched a 2ⁿ brute force on every tree of order 3 through 10 and a sample of orders 11 and 12 with zero mismatches, and it reproduced Larson's proved identity 403 on all 93 trees of order 3 through 9.
Then came the census. Across all 4,241 trees of order 3 through 13, the outcome was uniform: peN(H) = 0 for every single tree. Every vertex outside H has either no neighbour inside H or at least two. So all four bounds are vacuously true − 0 ≤ 0, with right-hand sides that are never negative (the path-covering bound 2·p(G) is at least 2, and the rest sit at 0 or above because every tree with more than two vertices has a support vertex and at least two leaves).
The genuinely interesting part is why a heuristic flagged these statements at all. "The four statements look like sharp bounds − the Dalmatian heuristic emitted them precisely because each is attained with equality on some tree," the write-up explains. "But equality there is the degenerate equality 0 = 0." Conjecture 404 hits equality on stars, where there is one support vertex; 406 on paths, where there are exactly two leaves. Both equalities are just both sides being zero.
It is a negative result that closes a block rather than a kill, so the standing stays at 182 disproved conjectures - but it is the same morning's second resolution from a corpus that sat open for sixteen and a half years, and the second one that needed an exhaustive census to explain. The full write-up and verifier live in the graffiti-verification repository.