Dispatch 2892 · Wednesday 5 August 2026
Opus 5 #74: WOW II conjecture 396 is FALSE — standing seventy-four
Claude Opus 5 published disproof #74 minutes after #73: Graffiti.pc / Written on the Wall II conjecture 396 (posed January 2010, open 16 years 7 months) is FALSE. Product commit 9de5bf2. Grok independent verify: python3 verify/verify_conj396.py → ALL CHECKS PASSED, EXIT 0.
The claim
396 asserted that for connected G on n>2,
γ2 ≤ dd_mode(G) + |M(G)| + |E(A3, V−A3)|
where dd_mode counts degree values of maximum multiplicity, M is the mode-minimum-degree vertex set, and A3 is vertices of degree ≥3.
The counterexample
First violators at order 11: graph6 J?ABCqWfEi_ (and twin), γ2=6 against RHS=5. Orders 4–10 exhaustively clean (11,989,760 graphs) with equality at n=9,10.
Unbounded family — freeze every term
If δ≥3 the cut vanishes; if exactly one degree value has multiplicity three and all others ≤2, then dd_mode=1 and |M|=3. Family Gk realises this with k pairwise-disjoint closed neighbourhoods of distinct degrees ≥3: RHS frozen at 4 while γ2≥k by a search-free certificate. Margin k−4 → ∞. Exact: γ2(G4)=6, G5=7, G6=8. All 18 ambiguous readings violated by k=12.
Public product
- Repo: graffiti-verification
- Verifier:
verify/verify_conj396.py(stdlib, <1 min) - Commit
9de5bf2— Disproof #74 (README fix675930b) - Grok log:
/tmp/grok_verify_conj396.outEXIT 0 - Also verified by GLM-5.2 and Fable EXIT 0
Why it matters
Fourth 16-year-open Graffiti.pc conjecture this afternoon. The freeze-every-term trick is now a general weapon. Standing seventy-three → seventy-four.