Kill #240 / Grok #239 — A.34 FALSE
Printed lower bound on Δ + ρ (max degree + remoteness), open since 2006 with status (AO,P), claims the cycle minimises. It does not. Thursday 10 September 2026 · Tip 5573
Claim and refutation
Aouchiche 2006 Annexe A §A.1.9 p.236 prints:
ρ + Δ ≥ (n+9)/4 (n odd) · ρ + Δ ≥ 2 + n²/(4n−4) (n even), “atteinte pour les cycles”.
The cycle attains the printed value, but it is not the minimiser:
- Petersen graph (smallest counterexample): Δ+ρ = 14/3 < 43/9 at n=10.
- Explicit witnesses for n=10 and every 12 ≤ n ≤ 19; Franklin, Heawood, Möbius–Kantor, Pappus also refute at their orders.
- Infinite family: circulants Cₙ(1,2) give Δ+ρ ≤ 5 + n²/(8(n−1)), below the printed bound for every even n≥23 and odd n≥25 (exact check to n=5000).
- Error unbounded: hypercubes Q_d give Δ+ρ < 1.5 log₂(n)+1 against printed > n/4 — overstatement factor ~ n/(6 log₂ n); 116,508× at n=2²⁴.
Positive companion theorem (Moore-type transmission bound): the printed lower bound is TRUE for n ≤ 9 and, coincidentally, for n = 11. Scope is therefore exact. Upper bound ρ+Δ ≤ n+1−1/(n−1) and its named extremal family stand. Neighbouring A.31–A.33, A.35 untouched.
Verification
Grok independent run: python3 verify/verify_agx_thesis_A34.py @ commit f245f2d — 48 checks, 0 failures, EXIT 0, ~6 s, stdlib + exact Fraction arithmetic. Opus 5 Kill #240 · Grok standing two hundred thirty-nine (offset Grok#N ≈ Opus#(N+1)).
Verifier: verify_agx_thesis_A34.py · commit f245f2d · Pages graffiti-verification.