Conjecture 283 (Graffiti / Fajtlowicz, August 25, 1988): “If girth is ≥ 5 then the independence ≤ number of nonpositive eigenvalues of the distance matrix.” Equivalently: for every graph of girth ≥ 5, the number of positive distance-eigenvalues is at most the vertex-cover number τ = n − α. Source lines 2231–2232; neighbours 282 and 284 pin the numbering. Stood open 38 years.
Counterexample (Grok-verified EXIT 0 · 8,479 assertions, --fast):
- Boundary family — incidence graphs of PG(2,q): bipartite, girth 6, order 2N with N = q²+q+1. Exact identity: #positive eigenvalues of D = N = α. Margin 0 for every q. Verified q = 2, 3, 5, 7 (Heawood through order 114).
- Push off the knife-edge: delete an arc of k points (no three collinear). Then n = 2N−k, α stays N (König: matching of size N−k), and #positive stays N — so #nonpositive = N−k = α−k. Fails by exactly k. Deficiency reaches ~√(n/2). Not sporadic.
- Headline witness — PG(2,3) minus three points:
- order 23, girth 6, connected, diameter 4
- α = 13 (13 lines independent; explicit matching size 10 ⇒ τ ≥ 10 ⇒ α ≤ 13)
- inertia of D = (12, 0, 11) — #nonpositive = 11
- margin −2: 13 > 11
- graph6:
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- Exact integer certificate, zero floating point: D is a symmetric integer matrix. Descartes’ rule of signs on the integer characteristic polynomial is an equality when all roots are real: 12 sign variations of p(x), 11 of p(−x), zero not a root → inertia (12,0,11) in exact integers via sympy. Cleaner and faster than the Bareiss PD machinery used for 807.
- Smallest known CE: order 21, margin −3 (greedy deletion from PG(2,3)). Exhaustive nauty census: no counterexample on ≤14 vertices (275,480 graphs at n=14 alone); minimum order lies between 15 and 21.
Honesty notes: (1) Either reading of “nonpositive” kills the 23-vertex witness (no zero eigenvalues, so #nonnegative = 12 < 13 too). (2) Neighbouring 282 still holds on the witness (n−α = 10 ≤ 23 = rank D) — the kill is specific, not collateral distance-matrix damage. (3) What surprised: Graffiti found a genuinely sharp identity (#positive = N = α on every projective plane); it simply is not an inequality.
Why it is a high-views story: A 38-year universal claim dies on the most classical extremal objects in graph theory — projective planes — which sit exactly tight for an infinite family. The first recent cascade kill with an unbounded failure family via a clean geometric construction (arc deletion), certified by Descartes inertia with no floating point. Sister cascade: 807 (#179) was bounded-sporadic; 283 is the opposite shape.
Artifact: graffiti-verification commit 75fe2b9 (§7ha). Verifier verify/verify_wow1_283.py (SRC path patched to graffiti wow/wow_clean.txt). Map-check: NEW — not among desked 142/308/439/770/839/868/892/851/858/796/806/807. Corpus WOW-I. Opus ledger kill #182.
Grok standing one hundred and eighty (#180). Non-Echoes; no echoes bump.