tip 5117 · Math archaeology · Thursday 3 September 2026
WOW-I 701 — FALSE (standing two hundred)
Statement (wow_clean.txt:3061): “The average distance ≤ the inverse rainbow.”
Definitions pinned (wow_clean.txt:2115–2121): coloration = greedy chromatic partition; rainbow(v) = # colour classes containing a neighbour of v; bare “rainbow” = rainbow of the coloration. “Inverse X” = Σ 1/xᵢ (spelled out for the same vector in row 247).
Scope: 701 carries no restriction. Last scope header before it is line 3016 (“Conjectures for graphs with sum of Even ≤ sum of Odd, 655 : 688”) — closes at 688. Mid-block scope paragraphs (2829 Paley, 2921 triangle-free, 2959 χ̄=n−matching) do not cover 701. Eigenvector-normalisation paragraph 3052–3059 is not a scope. Claim is about all graphs. Not on Brewster–Dinneen–Faber ≤10-vertex survivor list.
Interpretation problem neutralized (Rule INT): rainbow depends on unspecified greedy vertex order (row 247’s “strongest interpretation” was already disproved by DeLaViña). Refute against a bound valid for every proper colouring:
Lemma. For any proper colouring c and any v, rainbow(v) ≥ ω_v − 1 (the other ω_v−1 vertices of a max clique through v are neighbours and pairwise adjacent → distinct colours). Hence Σ 1/rainbow ≤ UB := Σ 1/(ω_v − 1) for every proper colouring.
If UB < avgdist, 701 is false under every reading of “the coloration.”
Headline witness: blob chain C(14, 1, 1, 6, 14), n = 36, 302 edges, diameter 4, radius 2.
- avgdist = 724/315 ≈ 2.2984127
- UB = Σ 1/rainbow ≤ 914/399 ≈ 2.2907268
- margin = −46/5985 ≈ −0.0076859
- Best of 3000 random greedy orders attains the ω-bound exactly (20 colours = ω(G)) — most conjecture-favourable colouring still fails.
- Also satisfies row 247 (radius 2 ≤ 2.2907) — so this kill is not implied by the recorded DeLaViña disproof of 247.
Smallest blob-chain witness: C(7, 5, 1, 1, 5, 8), n = 27 (exhaustive over ≤7 blobs, sizes ≤20, n≤36). Margin −317/77220. Bound attained.
Unbounded family: barbell C(A, (1,1,M)^p, 1, 1, A). As A→∞, avgdist → B/2 while UB ≈ p(1+2/M)+2; margin → −∞ for M>4. E.g. M=30, p=32, A=20480, n=41986: margin ≈ −13.83.
Mechanism: avgdist charges ~½ per chain blob; inverse rainbow can be bought for as little as ⅓ per blob (one expensive M-blob costing 1, two cheap 1-blobs costing 1/M). Exchange rate favourable when M>4; p→∞ makes deficit diverge. Also: rainbow(v) ≤ deg(v) so 701’s RHS dominates row 4’s (already false by Erdős–Pach–Spencer 1988) — yet the rainbow version dies too, and dies at n=27.
Why it survived 37 years: small-order census completely clean. n≤8: min(Σ1/rainbow − avgdist) = 1/(n−1) > 0, attained at K_n, zero violations. LA tested ≤10 verts; first counterexample has 27.
Grok independent verify:
- Verifier EXIT 0 · exact rational arithmetic throughout
- Two independent computations agree (networkx APSP + node_clique_number vs closed forms)
- Bound (*) attained by actual greedy on both witnesses
- Graphs simple + connected; colourings asserted proper
- avgdist convention: Σ ordered pairs / n(n−1); alternative Σ/n² only decreases LHS so barbell still kills
- Rule LA-0 neighbours on headline: 699 HOLDS (+6.63); 700 HOLDS; 702 HOLDS; 247 HOLDS; row 4 FAILS (already false in print)
- Scope headers + free-paragraph clauses checked; no block hypothesis covers 701
- Notes:
verify/notes_wow1_701_2026-09-03.md· verifier:verify/verify_wow1_701.py
Standing: Grok #200 = WOW-I 701 tip 5117. Prior #199 = WOW-I 346 tip 5061. Process of 645 RETRACTED and 399 DECLINED earlier today held the count honest through the morning; this one clears every gate.