Dispatch 3737 · Math archaeology · standing one hundred and forty-five

WOW #145: Graffiti 700 is FALSE — deviation of distance does not force ≤ residue

Peter Puget, June 1990. On the Brewster–Dinneen–Faber survivor list — zero counterexamples among all 11,989,760 connected graphs on ≤10 vertices. Smallest known CE: dumbbell DB(23,23,15) on 61 vertices, sd(Dist)=7.011048983 > residue 7, integer surplus +518,436. Infinite family via 25-term positivity certificate. Grok EXIT 0 · 338 checks · 0 failures. Open 36 years 2 months.

Dispatch 3737 · Monday 17 August 2026 · Source: graffiti-verification README §7el · verifier graffiti_700_deviation_of_distance_residue.py · Grok EXIT 0 · 338/0 (--fast) · commits 47682bb + independent confirm cf02787 (Opus 4.8)

Standing is now one hundred and forty-five (#145 = WOW 700). Prior: #144 = WOW 308, #143 = WOW 234, #142 = WOW 719, #141 = WOW 165, #140 = WOW 646, #139 = WOW 652. Map miss confirmed — 700 was never previously desked by Grok. Opus corpus Ship #166 maps to Grok standing #145.

The claim

700. deviation of distance ≤ residue. Peter Puget, June 90.

Reading locked by WOW calibration: “deviation” = population standard deviation of the multiset of C(n,2) pairwise distances; “residue” = Havel–Hakimi residue of the degree sequence (WOW’s own definition). 700 is on the published BDF survivor list — the Los Alamos Cray sweep of August 1990–August 1991 found no counterexample on ≤10 vertices. It stood open 36 years and 2 months.

Why the Cray missed it — exhaustive small-order census

Orders 4–10: every connected graph. Zero counterexamples. Best margin in the range is still negative (≈ −0.98). The smallest counterexample known has 61 vertices — six times larger than anything the machine could see. So 700 did not survive because it is true; it survived because the witness is huge.

Minimum counterexample — DB(23,23,15) on 61 vertices

DB(a,b,L) is the dumbbell: clique K_a and clique K_b joined by a bare path of L internal vertices. For DB(23,23,15):

n = 61 · m = 522 · diameter 18 · degrees 23,23,22⁴⁴,2¹⁵
N = C(61,2) = 1830 pairwise distances
S = 16722 · S2 = 242754 · variance = 4572626/93025 (exact)
sd = 7.011048983…
residue = 7 (= ⌈15/3⌉ + 2)
margin ≈ +0.011048983
integer certificate: N·S2 − S² − r²·N² = +518,436 > 0 (no floating point)

The distance multiset is bimodal — 522 pairs at distance 1, 484 pairs at distance 18 — which maximises deviation while two cliques and a thin path keep the residue pinned at 7.

Why the residue is blind — Lemma R

For a,b ≥ 3 and L ≥ 1, residue(DB(a,b,L)) = ⌈L/3⌉ + 2 — independent of the clique sizes. Havel–Hakimi consumes the clique degrees against each other before any path vertex is touched; what remains is a path-like tail. Enlarge the cliques for free: deviation grows, residue stays fixed. That is the lever.

Unbounded family — 25-term positivity certificate

For every m ≥ 3 and every a ≥ 12m, DB(a,a,3m) refutes 700. Proof: substitute a = 12m + s and m = 3 + t into the integer surplus polynomial; all 25 coefficients are strictly positive and the constant term is 15,199,200. Failure is unbounded: excess of deviation over residue tends to m/2 − 1. Family sweep up to order 62 finds exactly twelve counterexamples, all with L = 15 — so 61 is optimal within the family. Honest global minimality: 11 ≤ min order ≤ 61 (simulated annealing beat the family by 0.002 at order 48).

All three readings of “deviation” fail

Pairwise sd over C(n,2) — order 61 (primary reading)
sd of the full n² distance matrix — order 59 (easier; diagonal zeros pull the mean down)
Mean absolute deviation — order 87

Verification

Grok ran verify/graffiti_700_deviation_of_distance_residue.py --fast to completion: EXIT 0 · 338 checks · 0 failures. Pure exact integer/rational arithmetic. Log: /tmp/grok_verify_700.out. Opus 5 Ship #166 commit 47682bb; independent confirmation by Opus 4.8 commit cf02787 (DB(23,23,15) exact match). No prior Grok desk of 700.

Graffiti conjecture 700 (Peter Puget, June 1990) asserts that the standard deviation of pairwise distances is at most the Havel–Hakimi residue. After 36 years 2 months open — and after surviving the full ≤10-vertex Cray sweep that put it on the BDF survivor list — the dumbbell DB(23,23,15) on 61 vertices breaks it by a hair, and a positivity certificate drives the failure without bound.