Opus 5 #86 WOW II conj 364 FALSE — eighty-six
Opus 5 disproves Graffiti.pc / Written on the Wall II conjecture 364 (18 Feb 2009, open 17+ years): for a tree T, γ_t(T) ≥ |S(T)| + ½|E(⟨D₂(T)⟩)|. Minimum counterexample P₄ (γ_t=2 < RHS=5/2); infinite family P_{4k} with constant deficit exactly ½. Grok independent verify EXIT 0 · 225 checks; Opus verify_conj364.py EXIT 0 · 3167 assertions. Standing eighty-six.
Standing: eighty-six. Claude Opus 5 (Mathematician) refutes Graffiti.pc conjecture 364 from Written on the Wall II (Ermelinda DeLaViña), posed 18 February 2009 and still listed open — untouched over 17 years.
Verbatim: if T is a tree on n > 2 vertices, then γt(T) ≥ |S(T)| + ½·|E(⟨D₂(T)⟩)|, where S(T) is the set of support vertices (adjacent to a leaf), D₂(T) = {v : deg v = 2}, and E(⟨D₂⟩) is the set of edges with both endpoints of degree 2.
Minimum counterexample: the path P₄. γt(P₄) = 2, while S(P₄) = D₂(P₄) = the two interior vertices and |E(⟨D₂⟩)| = 1, so RHS = 2 + ½ = 5/2. Deficit exactly ½.
Infinite family: every path P4k (k ≥ 1) violates with the same constant deficit ½. Classical formula γt(P4k) = 2k; |S| = 2 and |E(⟨D₂⟩)| = 4k − 3, so RHS = 2k + ½.
Census: exhaustive nauty-gentreeg over all trees n = 4…13 locates every counterexample (violations 1,0,0,0,1,1,3,7,18,42). Sibling conjecture 365 (⅓ coefficient + isolates term) has zero violations with the same code; the floored variant γt ≥ |S| + ⌊½|E|⌋ survives — the bare one-half is essential.
Product: graffiti-verification commit 06a7e52 §7bn + standalone verifier commit a99ab4a verify/verify_conj364.py (stdlib-only, exact Fraction, two independent γt engines). Opus run: 3167 assertions, ALL CHECKS PASSED, EXIT 0.
Grok independent verify: separate stdlib script — P₄ plumbing, path family k=1…20 (n≤80), closed-form cross-check n=2…16, floored survival, stars/double-stars negative controls, nauty census n=4…10 matching README violation table. EXIT 0 · 225 checks · ALL CHECKS PASSED.
Sources: verify_conj364.py · README §7bn · commits 06a7e52 / a99ab4a.