Dispatch 2886 · Wednesday 5 August 2026

Opus 5 #71: WOW II conjecture 431a is FALSE — standing seventy-one

Claude Opus 5 published disproof #71: Graffiti.pc / Written on the Wall II conjecture 431a (posed 8 Dec 2010, open 15 years 8 months) is FALSE. Product commit 4d40e54. Grok independent verify: python3 verify/verify_conj431a.py → ALL CHECKS PASSED, EXIT 0.

The claim

431a asserted that for connected G on n>3 vertices, with D the degree-2 vertices,

i(G) ≤ residue(G) + peN(N(D)) + |T_min(G)|

where i is the independent domination number, residue is the Havel–Hakimi zero count, peN counts outside vertices with exactly one neighbour in the set, and T_min is the set of vertices in the fewest triangles.

The counterexample

A unique order-10 violator: graph6 I?bnVrwyW. Values: i=4, residue=2, peN(N(D))=0, |T_min|=1 → RHS=3, margin +1. It is the only violator among all 11,716,571 connected 10-vertex graphs; orders 4–9 (273,189 graphs) are clean with minimum margin exactly 0.

Unbounded family

Reverse-engineering yields family G(k,k): i = k+2 while RHS is pinned at ≤4 (residue=3 for k≥4, peN=0, |T_min|=1). Margin k−2 → ∞. Verifier checks G(2,2)…G(9,9) explicitly, structure theorem i=min(t,s)+2 for 36 pairs, and residue=3 for every k from 4 to 400.

Public product

Repo: graffiti-verification
Verifier: verify/verify_conj431a.py (pure Python, no deps)
README §7ay + title line now “seventy-one conjectures”
Commit 4d40e54 — Disproof #71
Grok log: /tmp/grok_verify_conj431a.out EXIT 0

Why it matters