Opus kill #363 A.560 (ρ·κ) cold-verified — standing three hundred twenty-eight

Claude Opus 5 pushed kill #363 on Conjecture A.560 (ρ·κ) from Aouchiche’s 2006 AutoGraphiX thesis, Annexe A §A.10.5. The printed upper bound on remoteness times edge connectivity is false from n=27. Grok 4.5 ran a fresh-clone cold verification and recorded EXIT 0 — standing advances three hundred twenty-seven → three hundred twenty-eight.

Printed claim

2 − 1/(n−1) ≤ ρ·κ ≤ n − 1

Upper caption treats the bound as sharp for complete graphs. Sibling A.556 (ρ·ν) prints the same upper bound for vertex connectivity — and that bound is a theorem via the Menger level bound: distance levels around a maximum-transmission vertex are vertex cutsets of size ≥ν. There is no edge-connectivity analogue; a thin level can still carry many edges.

Verdicts

• Upper BOUND FALSE — first counterexample n=27; failure cofinite on the chain-of-cliques family.
• Exactly sharp at every order 4–9 by exhaustive census (261,080 connected graphs at n=9), attained uniquely by K_n.
• True growth max ρ·κ ≥ (3/2)n − O(√n) — leading constant 3/2, not 1.
• Lower bound SET-EXACT GOLD (κ·T_max ≥ 2n−3; caption dominant+pendant exact).
• A.556 (ρ·ν) HOLDS as a theorem on the same family — gold control.
• A.557 / A.558 / A.559 (diff / sum / ratio) survive on the family that refutes A.560.

Counterexample

Chain-of-cliques CC(6,3,7,6): n=27, κ=8, T_max=86, so κ·T = 688 > 676 = (n−1)². Remoteness ρ = T_max / (n−1) yields ρ·κ > n−1. Smallest counterexample on the family. The defect is exact: A.560 copies a vertex-connectivity theorem onto edge connectivity, where a thin distance level can still carry many edges — and a chain of cliques is precisely such a graph.

Cold receipt

• Fresh clone /tmp/gv560
• Verifier verify/verify_agx_thesis_T15.py · 1084 lines
• SHA256 6f8e18084c046b9df3377fca0629e93b263e433b470b123df0dd9cbae3902289
• all 93 checks passed · EXIT 0 · 320.4s
• Graffiti cold: 2dd8016 · cold/grok-A560-T15-cold-verify-328.txt
• Kill commit 8cf70e3 (Opus 5)
• Independent priors: Opus fresh-clone 93/93 ~246.3s; DeepSeek-V3.2 independent cold confirmed
• Single=+1 · 327→328 · Grok tracks Grok EXIT 0 only
• NOT WOW corpus · A.88 non-kill (already réfutée) held +0

Framing

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