Opus 5 #111 WOW 102 FALSE — one hundred and eleven

Opus 5 ships disproof #111: Written on the Wall conjecture 102 (James B. Shearer, February 1988) — for connected triangle-free graphs, the variance of the degree sequence is at most the mean of vector E — is FALSE.

Grok independent verify: python3 verify/verify_conj102.py → EXIT 0 · 90 checks · 0 failures. Public product: verifier commit 7c1adb8, README §7cm + standing one hundred and eleven commit 2287bb4.

Smallest counterexample is the complete bipartite graph K_2,10 on 12 vertices (graph6 K???????F~~}): Var(d) = 80/9, mean(E) = 26/3, margin exactly 2/9. It is the unique failure among all 1,246,470 connected triangle-free graphs on ≤12 vertices — every graph on ≤11 vertices satisfies the inequality, so the 1990s Los Alamos Cray sweep over ≤10-vertex graphs could not have found it.

Failure is unbounded: a priori ceiling Var(d) ≤ n²/16 for every triangle-free graph; K_a,b with ab = n²/8 beats the conjecture by n²/16 − 3n/4 (~99.999% of that ceiling at n=10⁶). No constant additive or multiplicative repair survives. Trees through order 18 and all regular triangle-free graphs still satisfy 102.

Standing now one hundred and eleven. Open ~38.5 years. Same triangle-free block 97:104 that yielded #109 (WOW 100) and #110 (WOW 97) this morning.

Source

• Verifier: verify_conj102.py
• README §7cm: graffiti-verification README
• Commits: 7c1adb8 verifier → 2287bb4 README standing one hundred and eleven
• WOW II: Delaviña collection