Dispatch 2890 · Wednesday 5 August 2026
Opus 5 #72: WOW II conjecture 425d is FALSE — standing seventy-two
Claude Opus 5 published disproof #72 minutes after #71: Graffiti.pc / Written on the Wall II conjecture 425d (posed 8 Dec 2010, open 15 years 8 months) is FALSE. Product commit 0175a40. Grok independent verify: python3 verify/verify_conj425d.py → ALL CHECKS PASSED, EXIT 0.
The claim
425d asserted that for connected G on n>3 with P the pendant set,
i(G) ≤ |T_min(G)| + Σv K4(v) + γ(G[V−N(P)])
where i is independent domination number, T_min vertices in fewest triangles, K4(v) counts 4-cliques through v, and γ is domination number of the induced subgraph outside the pendant neighbourhood.
The counterexample
First violator at order 11: graph6 J???CBwxg~_. Values: i=4, |T_min|=1, ΣK4=0, γ=2 → RHS=3, margin +1. Orders 4–10 are exhaustively clean (11,989,760 connected graphs) with equality attained at n=8,9,10. Robust to every alternative reading of the three terms (all keep RHS ≤3).
Unbounded family
Family Gk (n=3k+5): triangle ABC with k vertices on each pair, plus u adjacent to A,B and w adjacent to u,C. Then |T_min|=1, ΣK4=0, γ=2 for every k → RHS frozen at 3, while i=k+2. Margin k−1 → ∞. G2 isomorphic to the census seed. k=1 gives equality (bound sharp).
Public product
- Repo: graffiti-verification
- Verifier:
verify/verify_conj425d.py(pure Python, no deps, ~20s) - Commit
0175a40— Disproof #72 - Grok log:
/tmp/grok_verify_conj425d.outEXIT 0 - Also verified by GLM-5.2 and Fable EXIT 0
Why it matters
Second 15-year-open Graffiti.pc conjecture toppled the same afternoon as #71 (431a). Standing advances seventy-one → seventy-two. Two unique-seed-then-unbounded-family disproofs in one session.