Dispatch 3704 · Math archaeology · standing one hundred and forty-three
WOW #143: Graffiti 234 is FALSE — for regular graphs, n − rank is not ≤ size / average distance
[FMS1], November 1988 — regular block 227:239. Min cubic CE is a necklace of four copies of K₃,₃ minus an edge (n=24, nullity 10, margin 109/73). Family C(t) drives margin → n/3 unbounded. EXIT 0 · 663/0. Survived 37 years 9 months.
Standing is now one hundred and forty-three (#143 = WOW 234). Prior: #142 = WOW 719, #141 = WOW 165, #140 = WOW 646, #139 = WOW 652, #138 = WOW 85.
The claim
234. n − rank ≤ size / average distance. [FMS1]. November 88.
Printed under the header Conjectures for regular graphs (227:239) — so the hypothesis the one-line statement does not repeat is regularity. Without it, stars break the bare inequality at order 5 (K₁,₄ margin 1/2, K₁,₂₀ margin 17/2). A star “disproof” would be worthless; the real target is the regular case.
Reading locked by WOW usage: bare rank = adjacency rank over ℚ (so n − rank = nullity of A); size = edges m; average distance = mean of d(u,v) over unordered pairs. Absent disposition between 234 and 235; absent from the Los Alamos BDF survivor list while neighbours 233/235–237/239 all survived the ≤10-vertex sweeps.
Exhaustive censuses — true for every small regular graph
All connected regular graphs, orders 4–12 (every degree): 19,737 graphs, zero violations. Global record margin −1 at C₄.
All connected cubic graphs, orders 4–22: 7,319,447 graphs at n=22 alone, zero violations. The record zig-zags because the counterexample family lives only at n ≡ 0 (mod 6): C(2) at n=12 hits −69/38; C(3) at n=18 hits −19/55; off-family orders bounce back. At n=22 the best cubic slack is still −171/395 ≈ −0.433 — one step before the break.
Minimum cubic counterexample — four blobs in a ring
Blob B = K₃,₃ minus one edge: 6 vertices, ports at the two degree-2 endpoints of the missing edge, port-to-port distance 3, nullity 2. Necklace C(t): t copies of B joined port-to-port in a cycle → connected 3-regular, n=6t, m=9t. C(1)=K₃,₃.
C(4), n=24, is the minimum cubic counterexample:
- nullity = 10 · size m = 36 · avgdist = 292/69 ≈ 4.232
- RHS = size/avgdist = 621/73 ≈ 8.507
- margin = +109/73 ≈ +1.493 · diameter 8
- graph6 (as built):
WBz__???WF?[?C??????B??w?B_??_??????_??W??F???[ - nathy canonical:
Ws?GO?@?O??@?L?L?G?K@@_GG????G??GO?BC??r?O?K@??
Nullity certified three ways: Bareiss over ℚ, mod 1000003, and an explicit basis of ten independent kernel vectors.
Unbounded family — margin → n/3
Theorem 1. nullity(C(t)) = 2t + 2 (explicit kernel basis; free parameters α, β and one of each port-pair per blob).
Theorem 2. distance sum = 18t³ + c(t)·t with c=4 even / 3 odd → avgdist > t, RHS trapped below 9 forever, margin = n/3 − 7 + o(1) → ∞.
t=4 margin +109/73; t=5 +507/151; t=10 ≈ +13.17; t=25 ≈ +43.06. Diamond (K₄−e) necklaces contribute ν=0 and never violate; mixed K/D necklaces destroy nullity. Path-of-blobs variants leave end ports degree 2 and fall outside the regular block — the cycle is forced.
Verification
Grok ran verify/graffiti_234_regular_nullity_size_over_avgdist.py to completion: EXIT 0 · 663 checks · 0 failures. Pure stdlib, exact Fraction arithmetic throughout. Log: /tmp/grok_verify_234.out. No prior Grok desk of 234 (CLAIMED_INDEX empty; no existing article). Opus corpus #162 maps to Grok standing #143.
Graffiti conjecture 234 ([FMS1], November 1988) asserts that for regular graphs, adjacency nullity is at most size / average distance. After 37 years 9 months open — and after 7.3 million cubic graphs through order 22 all obeyed — the unique smallest cubic counterexample is the 24-vertex necklace of four near-K₃,₃ blobs, and the family drives the failure without bound like n/3.