A.566 FALSE — standing two hundred forty-five
Aouchiche 2006 Annexe A p.422 · upper bound on β + ρ · β = domination number · ρ = remoteness = max transmission/(n−1) · status was “(T, AO)” open upper since 2006 · Grok EXIT 0 · 57/57 · commit 1fd1c7b · verifier verify_agx_thesis_A566.py · sha256 07a92f88…acee6
Printed bound (three cases by n mod 3): β+ρ ≤ (5n+6)/6 − 2/(n−1) [n≡0], (5n+4)/6 [n≡1], (5n+8)/6 − 6/(n−1) [n≡2], attained (says the author) by path / path+1 pendant / path+2 pendants. Those three are the s=0 members of one growing caterpillar family W(n,t) with t pendants on spine vertices 2..t+1.
FALSE for every n≡1 mod 3 with n≥16, every n≡0 mod 3 with n≥21, every n≡2 mod 3 with n≥26 — hence for every n≥26. Excess identity: V(n, t0+3s) = printed + s(n−c−9s)/(n−1) ~ n/36 → ∞. Smallest counterexample n=16: W(16,3) gives 71/5 = 14.2 vs printed 14 (unique maximiser). n=13 is an exact tie (23/2). Separate structural failure at n=5 (P5 gives 9/2 > 4; named family needs n≥8). Lower bound 2 true sharp on K_n. Corrected bound: printed + max_s s(n−c−9s)/(n−1).
Exhaustive free trees n≤16; tree-maximum lemma (edge deletion never decreases β+ρ). Flash cold 57/57; scope confirmed (a)(b)(c). Opus Kill #246 = Grok #245. Standing two hundred forty-five.