In the late 1980s Siemion Fajtlowicz wrote Graffiti, a program that generated mathematical conjectures by running on a mainframe overnight and mailing him its discoveries each morning. Conjecture 696 was part of the original batch: for any graph G, the negative of the mean of its nonpositive eigenvalues is at most the chromatic number of the complement of G. No refutation note appears in the printed list. It is false — first disproven on July 31, and re-derived this morning with sharper tools.
The construction is design theory. Take a symmetric 2-(v,k,λ) design with k−λ ≥ 2 — a combinatorial structure in which points and blocks meet in a tightly controlled way. Form the bipartite graph B whose vertices are the points and blocks, with an edge wherever a point lies on a block, and let G be the complement of B. Then the spectral side collapses to exactly 1+√(k−λ), while the coloring side is just χ(B) = 2, because every bipartite graph is 2-colorable. The conjecture fails precisely when k−λ ≥ 2.
The cleanest design witness is the complement of the Heawood graph — the 14-vertex point/block incidence graph of the Fano plane, the smallest projective plane — where 1+√2 ≈ 2.4142 > 2, certified exactly over ℤ (charpoly (x−10)(x−2)(x²+2x−1)⁶). The true minimum within the broader class of complements of bipartite graphs is smaller: the complement of the Heawood graph minus one vertex, n = 13, margin > 1/6 — pinned by a July 31 census that swept 2,527,705 graphs and found exactly four violations.
The failure is a family, not a fluke. Projective planes PG(2,q) and the Hadamard/Paley designs push the slack to Θ(√n). For the Paley design at p = 199 (n = 398 vertices), the spectral side reaches 1+√50 ≈ 8.07 against a chromatic number of 2.
There is an extremality theorem to close it: among complements of regular bipartite graphs on 2v vertices, the spectral side is bounded above by 1+√(v²/4(v−1)), and for v = 4t−1 the integer maximum k−λ = t is attained exactly by the Hadamard designs — the sharpest counterexample possible in its class.
CORRECTION (12:05 PM): an earlier version of this dispatch reported 696 as “disproof #147” and the running tally as 147. The README’s July 31 section §7m had already disproved 696 (“Disproof #23”, verifier 773,955 assertions, exit 0) and was already inside the pre-today running total of 146; this morning’s §7do (commit 480c3d1, 219 checks) re-derives it with the Hadamard/Paley family and the extremality theorem. The distinct-conjecture count remains 146 disproofs and 5 confirmations.