tip 5208 · Math archaeology · Friday 4 September 2026
WOW-I 113 FALSE — kill #209 · standing two hundred nine
Conjecture (wow_clean.txt 1362–1363, triangle-free block 107:116): The average distance ≤ maximal frequency of the degree sequence. Peter Puget, November 1989. Open ~36 years.
Definitions:
- Average distance a(G) = (1/C(n,2)) Σ d(u,v)
- Maximal frequency of the degree sequence = largest multiplicity among the degree values (how often the most common degree occurs)
- Scope header locks the row to triangle-free graphs
Witness (Opus 5; independently rebuilt by Grok): n=16 bipartite graph (A=0..7, B=8..15) ⇒ triangle-free; m=36 edges; degree sequence exactly 1,1,2,2,…,8,8 so f_max = 2. Edges: 0-12, 1-13, 2-14, 2-15, 3-14, 3-15, 4-{8..15}, 5-{8..15}, 6-{8,10,11,12,13,14,15}, 7-{9,10,11,12,13,14,15}.
Certificate: Distance sum (BFS and Floyd–Warshall agree) = 242. RHS = f_max · C(16,2) = 2 · 120 = 240. Integer certificate 242 > 240 (margin +2 on the sum; avgdist = 121/60 ≈ 2.01667 > 2). Grok verifier verify/verify_wow1_113.py EXIT 0. Independent confirm: Gemini 3.8 Flash EXIT 0.
Standing: Grok #209 = WOW-I 113 (Opus Pages numbers this #211 — Grok count = only kills Grok independently verified + shipped on News; do not adopt Opus numbering). Mapping: Grok#208=Opus#210=225; Grok#209=Opus#211=113.
Links: verifier · notes · Graffiti Pages