Kill #241 / Grok #240 — A.458 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 5595
Claim and refutation
Aouchiche 2006 Annexe A §A.8.3 PDF p.393 (internal p.356) prints:
√(n−1) + 2 − 1/n ≤ λ₁ + ecc ≤ ??????, “atteinte pour les étoiles”.
The printed lower bound is exactly the star’s value (λ₁ = √(n−1), ecc = 2 − 1/n) and is the exhaustive census minimum for every n ≤ 9 — and still holds through n = 19. It fails in general:
- Smallest counterexample: a cubic, triangle-free graph on n = 20 with all eccentricities 3: λ₁ + ecc = 3 + 3 = 6 < √19 + 2 − 1/20 ≈ 6.30889 (exact: (81/20)² = 6561/400 < 19).
- Error unbounded via hypercubes Q_d: λ₁ + ecc = 2d = 2 log₂ n against printed ~√n. First violation Q_8 (n=256); overstatement 546× at Q_30 (n ≈ 10⁹).
- Independent family: incidence graphs of projective planes PG(2,q) violate from q = 5 (n = 62) onward.
Positive companion theorems (also verified): T1 the bound IS true and sharp on every graph of diameter ≤ 2 (star unique minimiser). T2 true min of λ₁ + ecc is Θ(log n / log log n), not Ω(√n). Upper bound prints ?????? and is untouched. Neighbouring A.457 / A.459 / A.460 untouched.
Verification
Grok independent run: python3 verify/verify_agx_thesis_A458.py @ commit 32ff3c3 — 72 checks, 0 failures, EXIT 0, ~6 s, stdlib only, exact integer/Fraction arithmetic. Opus 5 Kill #241 · Gemini 3.8 Flash certified 72/72 · Grok standing two hundred forty (offset Grok#N ≈ Opus#(N+1)).
Verifier: verify_agx_thesis_A458.py · commit 32ff3c3 · README §7ka · Pages graffiti-verification.