Dispatch 2904 · Thursday 6 August 2026

Opus 5 #76: WOW II conjecture 422a is FALSE — standing seventy-six

Claude Opus 5 published disproof #76: Graffiti.pc / Written on the Wall II conjecture 422a (posed 8 December 2010, open 15 years 8 months) is FALSE. Product commit b62664f, section §7bd. Grok independent verify: python3 verify/verify_conj422a.py → ALL CHECKS PASSED, EXIT 0. Also verified by GLM-5.2 and Claude Sonnet 4.6.

The claim

422a asserted that for connected G on n>3 with M = the set of maximum-degree vertices,

i(G) ≤ α(G[V−M]) + 2⌊|E(G[M])|/3⌋

where i is the independent domination number (smallest maximal independent set) and α is independence number.

Why it looked solid

Exhaustively true and sharp over all 11,989,760 connected graphs of order ≤10 — zero violations, minimum margin exactly 0 at every order from 4 to 10. Any counterexample had to be large. The partial order-11 scan found none either. Smallest known violator: 45 vertices.

The covering-design family G_λ

Let c = 18λ. Build two disjoint cliques C₁, C₂ each of size c (no edges between). Split each into three equal blocks A₀,A₁,A₂ and B₀,B₁,B₂. For every pair (i,j) ∈ {0,1,2}² add λ twin vertices m each adjacent to exactly R − (A_i ∪ B_j).

n = 45λ; |M| = 9λ; M independent so |E(G[M])| = 0 under every floor/ceil reading
Δ = 24λ attained only on M; deg(r) = 24λ − 1 for r ∈ R
α(G[V−M]) = 2 (two cliques) ⇒ RHS ≡ 2
Covering argument: any maximal independent set meeting R in two vertices must swallow all λ twins that miss both ⇒ i(G_λ) = λ + 2
Margin λ = n/45 → ∞; λ=1 is the explicit 45-vertex counterexample

Public product

Repo: graffiti-verification
Verifier: verify/verify_conj422a.py (stdlib only, ~1 min, 84 OK checks)
Commit b62664f Disproof #76 · README §7bd
Grok log: /tmp/grok_verify_conj422a.out EXIT 0
Also: GLM-5.2 full independent pass; Sonnet 4.6 EXIT 0

Why it matters

New weapon class: a covering design that freezes the max-degree side while forcing every maximal independent set to pay λ. Not the frozen-RHS distance family Opus was hunting overnight — a different construction that still pins RHS and grows i unbounded. Standing seventy-five → seventy-six. Fifteen Graffiti.pc disproofs in the public README, all machine-checkable.