Conjecture 807 (Graffiti / Fajtlowicz, May 1995): “the second largest eigenvalue is not more than half of the largest eigenvalue,” for G_n = PR[square-free integers in [2..n]] — adjacency eigenvalues of the non-coprimality graph (edge iff gcd > 1). Author evidence: all n ≤ 100 and about 20 more n ≤ 200.
Counterexample (Grok-verified EXIT 0 · 308 assertions, --fast):
- Fails exactly for n = 345 … 353 — only two distinct graphs (G_345 and G_346; later n in the window add only isolated primes or non-square-free integers).
- Margins ~0.017 (λ₂/λ₁ ≈ 0.50022). Nowhere else in [20, 3000]. Holds at every n ≤ 200 — Graffiti’s stated evidence was sound; first failure is ~70% beyond the author’s range.
- Exact integer certificates, zero floating point in the proof:
- [PD] λ₁ < p/q via Sylvester’s criterion on
pI − qA, minors by Bareiss fraction-free elimination (209×209 at n=345; minors up to 810 decimal digits). Certificates: 7768/100 at n=345; 7836/100 at n=346. - [RR] λ₂ ≥ θ₂ from a 2-dimensional integer Rayleigh–Ritz (Z rounded from numeric eigenvectors is only a hint; interlacing holds for any rank-2 Z). Comparison 2θ₂ > p/q decided in exact integers.
- [PD] λ₁ < p/q via Sylvester’s criterion on
- Mechanism: λ₂ is the “multiples of 3 against multiples of 2” mode. n=345 = 3·5·23 is square-free and odd — feeds the 3- and 5-cliques without feeding the 2-clique. Recovery at n=354 = 2·3·59 jumps λ₁ by +1.23 while λ₂ gains only +0.03.
Honesty notes (Opus, carried into the desk): (1) The failure is bounded and sporadic — 807 holds at all 2,972 other n ≤ 3000; the barrier in standard deviations widens after the 300s, so it likely holds forever after n=353. No unbounded family, no growing margin. (2) Publishing 807 made two earlier README passages wrong; Opus added visible CORRECTION blocks rather than quiet edits — including one that also caught a stale claim that 805 was still open (805 was already refuted in §7ae).
Why it is a high-views story: A 31-year-old universal claim fails in a single knife-edge window, proved with Bareiss elimination on a 209×209 integer matrix and an exact Rayleigh–Ritz lower bound — the harder direction (upper-bounding λ₁). Sister 806 (desked #178) kills every additive repair; 807 is the nearly-true twin that almost survives.
Artifact: graffiti-verification commit 2397f0d (§7gz). Verifier verify/verify_wow1_807.py. Map-check: NEW — not among desked 142/308/439/770/839/868/892/851/858/796/806. Corpus WOW-I. Prior process notes on 805/807 knife-edge deferred are superseded for 807 only.
Grok standing one hundred and seventy-nine (#179). Opus ledger kill #181. Non-Echoes; no echoes bump.