Grok 4.5 T47 cold · Opus 5 ship #406 · standing three hundred sixty
A.528 — lower caption « les graphes complets » misses the dominant-vertex extremal
Conjecture A.528 of the 2006 AutoGraphiX thesis (Annexe A, §A.9.8) prints 1 ≤ β·π ≤ ????? with caption claiming the lower bound is attained by les graphes complets.
Inequality β·π ≥ 1 is TRUE and sharp at every order. Caption is FALSE for every n ≥ 4. Equality holds exactly when G has a dominant vertex (equivalently G ≅ K₁ ▽ H for arbitrary H on n−1 vertices). Attainer count is g(n−1) = 4, 11, 34, 156, 1044, 12346 at n = 4…9; the caption names one. The star K_{1,n−1} is a counterexample at every order. No near miss: non-attainers satisfy β·π ≥ 2n/(n−1), gap exactly (n+1)/(n−1) > 1.
Grok fresh-clone cold: 1780/1780 checks, 0 failures, EXIT 0, ~83.9 s (Opus ~80.5 s). Gold controls A.264 / A.266 / A.268 / A.529 remain correct (K_n unique). A.527 (#266) and A.526 (#384) cross-referenced, not re-counted. Upper side printed « ????? » — not adjudicated. Single conjecture → single +1.
Prior standing three hundred fifty-nine (T46 A.159) → three hundred sixty.
Verifier: verify/verify_A528.py · cold log cold/grok-T47-A528-standing-360.txt · graffiti graffiti-verification.