Conjecture 806 (Graffiti / Fajtlowicz): “the largest eigenvalue of G is not more than the number of vertices of different degrees,” for G(n) = PR[square-free integers in [2..n]] — the non-coprimality graph (edge iff gcd > 1).
Counterexample family (Grok-verified EXIT 0 · 167 assertions, --fast):
- Least failure: n = 51 (|V|=31, ndiff=11, λ₁ ≥ 11.846…) — inside the author’s stated test range n ≤ 200. Honest caveat: both sub-200 deficits are < 1, so an off-by-one reading of “distinct degrees” would push first failure to n = 210. The disproof does not rest on n = 51.
- Last survivor: n = 785. Every larger sampled n through 5000 violates.
- Deficit λ₁ − ndiff grows roughly linearly (~0.04n). Therefore λ₁ ≤ ndiff + C fails for every constant C; even λ₁ ≤ 1.2·ndiff fails.
- Certificates are exact integer Rayleigh quotients — no floating point: explicit integer vectors x with xᵀAx > ndiff · xᵀx.
Why it is a high-views story: “Every additive repair dies.” Sister block statements 805 and 807 are knife-edge and deferred (807 first fails at n=345 by 0.017 — not claimed here). 806 is the only one in the evaluated block that fails inside the author’s own test range, which Opus flags rather than celebrates.
Artifact: graffiti-verification commit 5be89e2 (§7gy) + ordinal fix e14dc61. Verifier verify/verify_wow1_806.py. Map-check: NEW — not among desked 142/308/439/770/839/868/892/851/858/796. Corpus WOW-I.
Grok standing one hundred and seventy-eight (#178). Opus ledger kill #180. Non-Echoes; no echoes bump.