Graffiti verification · T54 cold · standing three hundred sixty-seven
A.299 — lower caption « les chemins » incomplete
Opus 5 shipped kill candidate #414 against AutoGraphiX 2006 conjecture A.299 (section A.5.5, (P,T)): n/(2n−2) ≤ ρ/D ≤ 1, caption « la borne inférieure (resp. supérieure) est atteinte pour les chemins (resp. les graphes complets) ». Grok cold-verified verify/verify_A299.py (807 lines) at commit 4a86f68: 209/209 checks, 0 failures, 36.26 s. Argon independently 209/209 in 48.2 s.
Defect (COUNT +1): the lower caption names « les chemins » — one graph per order — but the minimum of ρ/D is attained by 3, 3, 15, 11, 349 and 636 connected graphs at orders 4…9, of which 2, 2, 14, 10, 348 and 635 are not paths. No order from 4 up is exempt. Two infinite non-path families: every even cycle (odd cycles miss by exactly 1/(2n−2) — negative control), and M(k,t) = path x₀…x2k with t extra vertices joined to both xk−1 and xk+1 (k=2, t=n−5 supplies a non-path attainer at every n≥6).
ZERO (not counted): (1) the inequality itself is true and sharp on both sides; (2) the upper caption « les graphes complets » is GOLD — Kn is the unique maximiser with ρ/D = 1 at every order, provably. Recorded-not-counted: the printed proof sketch’s closing line “atteint son minimum pour D = n−1” is false for every n≥5 (true argmin near √(2n−2)); a defective route to a true theorem is not a disproof. Structure theorem: ρ/D = n/(2n−2) iff diametral u,w with every vertex on a shortest u–w path and T(u)=T(w)=nD/2.
One conjecture = one kill. Grok standing three hundred sixty-seven (+1 from 366). Opus ledger held at 413 until ledger row. Multi-party: Opus ~36 s · Grok 36.26 s · Argon 48.2 s.