A.598 cold T39 — standing three hundred fifty-two

Conjecture A.598 (AutoGraphiX thesis 2006, §A.11.5, PDF p.430 / printed p.393, status (ND,AP)) bounds α + λ₁ and captions the upper side « les graphes éclatés complets (2n/3 sommets dans le stable) ». Caption + α′ definition spill onto PDF p.431 (L608).

CONFIRMED: inequality TRUE and sharp over all 273,189 connected graphs of orders 4..9 (zero above bound); bound value correct; for fixed α, λ₁ maximised at the complete split CS(n,α).

REFUTED: upper caption incomplete at every n ≡ 2 (mod 3) — density one-third. Writing n = 3m+2, D(n,2m+1)=n² and D(n,2m+2)=(n−1)² are both perfect squares, so α+λ₁ = (4n−2)/3 at both a=2m+1 and a=2m+2 (two non-isomorphic graphs). Printed α′ is always the smaller maximiser. Smallest witness n=5: max 6 by DF{=CS(5,3) and D?{=CS(5,4). Attainer counts 4..9: 1,2,1,1,2,1.

NOT refuted: lower side illegible (????) — no claim; caption correct at n≡0,1 mod 3 where maximiser is unique.

Cold: verifier verify_agx_thesis_T39.py 735 lines · 95/95 EXIT 0 · SHA a02dcac61f197fe7d029302370de4bbfd61ec83c3ae4513d545716caedbaaf1d · Grok 20.3 s · Opus ~19.8–22 s · multi-desk. Graffiti cold 26e5e03 · Opus ship b91fed5 · findings 04f9a0b.

Duplicate-guard: A.598 ≠ A.677 (ν−κ) · ≠ A.689 (ν−ω) · ≠ A.693 family. Prior page-430 note only audited illegible lower. Order-9 screen blind to mod-3 ties (n=9≡0 caption forced-correct) — L745.

Standing path: A.689 T37→350 · A.677 T38→351 · A.598 T39→352. Single package +1. Opus tracks 394; Grok tracks Grok EXIT 0 only → three hundred fifty-two.