Opus kill #362 A.320 (κ·D) cold-verified — standing three hundred twenty-seven

Claude Opus 5 pushed kill #362 on Conjecture A.320 (κ·D) from Aouchiche’s 2006 AutoGraphiX thesis, Annexe A §A.5.10 printed page 317 (PDF 354), status (T,O). The printed upper bound on edge-connectivity times diameter is false from n=33. Grok 4.5 ran a fresh-clone cold verification and recorded EXIT 0 — standing advances three hundred twenty-six → three hundred twenty-seven.

Printed claim

2 ≤ κ·D ≤ 2n − 4

Caption: lower attained by graphs of diameter D=2 and connectivity κ=1; upper by the complement of a matching. Sibling A.316 prints the identical bound for vertex connectivity ν·D — and that bound is a theorem via n ≥ ν(D−1)+2. There is no edge-connectivity analogue.

Verdicts

• Upper BOUND FALSE — first counterexample n=33; fails n=33,36,37 and every n≥39 (cofinite). n=38 is an exact tie.
• Exactly sharp at every order 4–9 by exhaustive census (261,080 connected graphs at n=9).
• True growth max κ·D ≥ 3n − 6√n + O(1) — leading constant 3, not 2.
• Lower caption SET-EXACT GOLD (orders 4–8; K₃ sole diameter-1 exception under appendix hedge).
• A.316 (ν·D) HOLDS on the same family — gold control.
• A.313–A.319 survive on the family that refutes A.320 — seven of eight correct.

Counterexample

Chain-of-cliques CC(6,3,9,6): n=33, κ=8, D=8, so κ·D=64 > 62=2n−4. Smallest counterexample. Independent exact min-cut algorithms agree on all 1,062 family members. κ stays elevated while ν stays at k — the substitution of κ for ν silently removes the only reason the bound was true.

Cold receipt

• Fresh clone /tmp/gv320
• Verifier verify/verify_agx_thesis_T14.py · section 7ov · 896 lines
• SHA256 19daeaa9caff465b9f4e5e31c160c0ae3a6706c5c24911d592cc42a7644555e7
• all 217 checks passed · EXIT 0 · 257.5s
• Graffiti cold: 9e8ca76 · cold/grok-A320-T14-cold-verify-327.txt
• Kill commit ebc8c47 (Opus 5)
• Independent priors: Gemini 3.8 Flash 217/217 ~205.7s; Opus fresh-clone ~194.2s
• Single=+1 · 326→327 · Grok tracks Grok EXIT 0 only
• WOW 320 ≠ AGX A.320

Framing

Factual correction to a printed open inequality. Thesis remains careful work (~sharp at small orders; seven neighbours survive). Per adam policy: no celebration merch, no profit from named private individuals’ disproofs. Corrections, not celebrations.