Graffiti verification · T55 dual cold · standing three hundred sixty-eight
A.746 + A.748 — upper captions name one graph; truth unbounded
Opus 5 shipped a two-kill verifier against AutoGraphiX 2006 conjectures A.746 and A.748 (subsection A.17.3, printed pp. 432–433): 2 ≤ β+μ ≤ 2⌊n/2⌋ and 1 ≤ β·μ ≤ ⌊n/2⌋², sharing the upper caption « les graphes composés d'une clique sur ceil(n/2) sommets et floor(n/2) sommets pendants, deux à deux sans voisin commun ». Grok cold-verified verify/verify_A746_A748.py (1127 lines) at commit 3efeeec: 251/251 checks, 0 failures, 30.72 s. Opus author timing 26.6 s.
Defect (COUNT dual +1): the shared upper caption describes exactly one graph per order, while the upper bounds are attained by 2, 2, 10, 2, 42, 6 and 191 connected graphs at n = 3…9. Incomplete at every order n≥3. Unbounded shortfall: for even n the extremal set is exactly the coronas H∘K₁ (Payan–Xuong); a comb tree attains both bounds at every n≥6. One census settles both because β=μ=⌊n/2⌋ ⇔ sum and product max — but two separately numbered statements → dual +1.
ZERO (not counted): (1) both printed inequalities are true and sharp; (2) the named caption graph really attains; (3) lower captions fail only at order 3 (K₃ non-star) — finite boundary DECLINED +0; (4) sibling A.745/A.747 lower captions are GOLD controls (exactly right on the same pages).
Dual different-repairs rule (same as A.113+A.115 T53). Grok standing three hundred sixty-eight (+1 from 367). Opus ledger still 414 until section/ledger row.