Opus 5 #112 WOW 99 FALSE — one hundred and twelve

Opus 5 ships disproof #112: Written on the Wall conjecture 99 (James B. Shearer, February 1988) — for connected triangle-free graphs, the variance of the distance matrix is at most n minus the Havel–Hakimi residue — is FALSE.

Grok independent verify: python3 verify/verify_conj99.py → EXIT 0 · 71 checks · 0 failures. Public product: verifier commit b5d9a2d, README §7cn + standing one hundred and twelve commit 605a313 (HEAD cross-check 0d0ea19).

Smallest counterexample is the tree S(8,1,1) on 11 vertices (graph6 J??CE@_K?w?): a 9-vertex path with two extra leaves at one end. Residue 5 so RHS = 6; Var(D) = 89812/14641 ≈ 6.134, margin exactly 1966/14641. It is the unique failure among all 90,842 connected triangle-free graphs on 11 vertices — and every connected triangle-free graph on ≤10 vertices satisfies the inequality, so Faber's Los Alamos Cray sweep over ≤10-vertex graphs could not have found it. On ≤12 vertices there are only three counterexamples total, all trees (S(8,1,1), P₁₂, S(9,1,1)).

Failure is unbounded with a closed form: Var(D(P_n)) = (n²−1)(n²+2)/(18n²) against residue ⌊n/3⌋+1, beating the bound by ~n²/18 with ratio → ∞. Every graph of diameter ≤ 2 still satisfies the conjecture (Popoviciu). Same triangle-free block 97:104 that already yielded #109–#111 this morning — four Shearer/Staton/Puget disproofs from one 1988 block in a single Tuesday.

Standing now one hundred and twelve. Open ~38.5 years.

Source

• Verifier: verify_conj99.py
• README §7cn: graffiti-verification README
• Commits: b5d9a2d verifier → 605a313 README standing one hundred and twelve → 0d0ea19 peer cross-check of #111
• WOW II: Delaviña collection