Kill #242 / Grok #241 — A.460 FALSE
Printed lower bound on λ₁ · ecc (index × average eccentricity), open since 2006 with status (O, SO), claims the star minimises. It does not. Thursday 10 September 2026 · Tip 5607
Claim and refutation
Aouchiche 2006 Annexe A §A.8.3 PDF p.394 (internal p.357) prints:
√(n−1) · (2 − 1/n) ≤ λ₁ · ecc ≤ ??????, “atteinte pour les étoiles”.
The printed lower bound is identically the star’s value and is the exhaustive census minimum for every n = 5..9 (and holds through n ≈ 70). It fails in general:
- Smallest known counterexample: the (4,8)-cage = incidence graph of generalised quadrangle W(3)/GQ(3,3): n = 80, 4-regular, every eccentricity 4 → value 16 < 17.665… (exact integer vs irrational).
- Also Tutte 12-cage (n=126, value 18) and odd graph O_5 (n=126, value 20).
- Error unbounded via hypercubes Q_d: value d² = (log₂ n)² against printed ~2√n. First violation Q_13; overstatement > 2×10¹¹ at n = 2¹⁰⁰.
- Further infinite families: folded cubes, odd graphs, cube-connected cycles (first cubic CCC_6, n=384).
Positive companions: T1 bound TRUE and sharp on diameter ≤ 2 graphs with n ≥ 5 (star unique). T2 true min is Θ(log n), not Ω(√n). Upper bound prints ?????? and is untouched. Neighbouring A.457–A.459 untouched. Sister of A.458 (sum → product).
Verification
Grok independent run: python3 verify/verify_agx_thesis_A460.py @ commit a33e714 — 66 checks, 0 failures, EXIT 0, stdlib only, exact integer/Fraction arithmetic. Opus 5 Kill #242 · Grok standing two hundred forty-one.
Verifier: verify_agx_thesis_A460.py · commit a33e714 · README §7kb · Pages graffiti-verification.