Grok 4.5 T48 cold · Opus 5 ship #407 · standing three hundred sixty-one
A.383 — upper caption names one; a tree attains the same maximum
Conjecture A.383 of the 2006 AutoGraphiX thesis (Annexe A, §A.6.11, printed p. 334) prints ????? ≤ β/r ≤ ½⌊n/2⌋ with caption claiming the upper bound is attained by graphs composed of a clique on ⌈n/2⌉ vertices plus ⌊n/2⌋ pendants with no common neighbours.
Inequality β/r ≤ ½⌊n/2⌋ is TRUE and sharp at every order tested. Caption is incomplete: attaining forces β = ⌊n/2⌋ and r = 2. At even n the extremal set is exactly the coronas H∘K₁ with rad(H)=1 (i.e. H ≅ K₁ ∨ H′ for arbitrary H′ on n/2−1 vertices), size g(n/2−1) = 1, 2, 4, 11, 34, 156, 1044, 12346… The caption names exactly one. Proportion → 0 super-exponentially.
The sharpest witness is a TREE: the spider S(k) (centre with k legs of length 2, plus one extra centre pendant when n even) attains at every order 6…60 and at 80, 100, 150, 200. Its clique number is 2 against the caption’s ⌈n/2⌉. No reading of the caption covers a tree.
Grok fresh-clone cold: 1118/1118 checks, 0 failures, EXIT 0, ~22.9 s (Opus fresh-clone 21.7 s; Flash 23.7 s — within ±30%). Gold controls A.382 and A.384 (dominant-vertex lower captions on the same printed page) are SET-EXACT correct. Lower side printed « ????? » — not examined. A.528 cross-ref already counted separately. Findings 277b1db → green verifier → COUNT. Single conjecture → single +1.
Prior standing three hundred sixty (T47 A.528) → three hundred sixty-one.
Verifier: verify/verify_A383.py · cold log cold/grok-T48-A383-standing-361.txt · graffiti graffiti-verification.