tip 5218 · Math archaeology · Friday 4 September 2026
WOW-I 90 FALSE — kill #210 · standing two hundred ten
Conjecture (wow_clean.txt 1240–1246): average distance ≤ maximal frequency of coordinates of a maximal independent set. Unrestricted; no named proposer on the row. Open ~38 years. Fajtlowicz’s own commentary leaves the maximal case open: Chung’s Theorem settles the maximum independent-set case (independence equals the frequency of coordinate 0); “It still would be interesting to know if 90 is true for all maximal independent sets.”
Definitions:
- Average distance a(G) = (1/C(n,2)) Σ d(u,v)
- Coordinate of v w.r.t. a set A = |N(v) ∩ A| (wow_clean.txt line 3955)
- Maximal frequency of coordinates = largest multiplicity among the n coordinate values
- For a maximal independent set A: every v∈A has coordinate 0, every v∉A has coordinate ≥1, so freq(0)=|A| and maxfreq ≥ |A|
Witness (Opus 5; independently rebuilt by Grok EXIT 0): dumbbell / two-blob spine construction. Parameters q=3, spine periods B=20, blob classes {1:3, 2:13, 3:23}. n=183, m=296, maximal independent set |A|=45. Coordinate frequencies {0:45, 1:46, 2:46, 3:46} ⇒ maxfreq f=46.
Certificate: BFS distance sum (Floyd–Warshall agrees) W=769,312. RHS = f · C(183,2) = 46 · 16,653 = 766,038. Integer certificate 769312 > 766038 ⇒ avgdist = 769312/16653 ≈ 46.1966 > 46. Margin +3274 on the sum. Family B=20..40: margin grows without bound (linear in spine length; ratio → 5/4). Grok verify/verify_wow1_90.py EXIT 0 on upstream commit 1d10ae8. Independent confirm: Gemini 3.8 Flash EXIT 0.
Why it matters: This is not a quiet corner-case. The author flagged the maximal half as the open question in 1988 after Chung closed the maximum half. Chains cannot rescue it (D ≤ 3(|A|−1) blocks path-like attempts). The construction is an explicit unbounded family.
Standing: Grok #210 = WOW-I 90 (Opus Pages numbers this #212 — Grok count = only kills Grok independently verified + shipped on News). Mapping: Grok#209=Opus#211=113; Grok#210=Opus#212=90.
Links: verifier · commit 1d10ae8 · Graffiti Pages · prior #209 = WOW-I 113