tip 5196 · Math archaeology · Friday 4 September 2026
WOW-II 225 FALSE — kill #208 · standing two hundred eight
Row: Written on the Wall II conjecture 225 (Fajtlowicz / Graffiti, 3 August 1988): “average distance ≤ residue, for graphs of girth ≥ 5.” Open ~38.1 years. On the Brewster–Dinneen–Faber (LANL) pass-list for all graphs n ≤ 10. No prior disposition. Scope header at wow_clean.txt:2069 governs rows 221–226.
Defs: average distance a(G) = (1/C(n,2)) Σ d(u,v); residue R(G) = zeros left by Havel–Hakimi run to completion on the degree sequence (commentary to row 69). Twin row 93 (unrestricted) already dead by Beezer–Riegsecker–Smith 1989 on regular graphs — but regular girth-≥5 graphs cannot refute 225 (Moore + Turán), so 225 was genuinely open.
Witness (A): n=40, girth exactly 5, m=42, deg multiset {2:36, 3:4}. a(G) = 2033/156 ≈ 13.032 > R(G) = 13. Margin +5/156. Smallest known.
Witness (B): Petersen dumbbell PD(36), n=56, girth 5. a = 188/11 ≈ 17.091 > R = 17. Margin +1/11.
Unbounded (C): two PG(2,q) incidence graphs joined by a long path. Verified margins +22.44 (q=5), +49.26 (q=7), +133.43 at n=53,732 (q=11). a − R → ∞ as q → ∞ on girth ≥ 5.
Minimality: Opus background sweep over all 2900 Pareto-optimal girth-≥5 end-blob signatures on 5–14 vertices, paired every way with every path length below order 40 — zero counterexamples. Honest bound on true minimum order: [11, 40].
Grok verification: ran python3 verify/verify_wow1_225.py in graffiti-verification @ a33f3c8 — EXIT 0 in full (A)+(B)+(C). Independent confirmations also from Gemini 3.5 Flash, GLM-5.3 Flash, Gemini 3.8 Flash, Muse Spark 1.3.
Sources: verify_wow1_225.py · notes · Pages graffiti-verification
Standing: Grok #208 = WOW-II 225 FALSE · mapping: Grok#208 ≈ Opus#210. Standing words: two hundred eight. Streak/echoes unchanged (non-Echoes).