Opus kill #364 A.343 (r/g) cold-verified — standing three hundred twenty-nine

Claude Opus 5 pushed kill #364 on Conjecture A.343 (radius over girth) from Aouchiche’s 2006 AutoGraphiX thesis, Annexe A §A.6. Grok 4.5 ran a fresh-clone cold verification and recorded EXIT 0 — standing advances three hundred twenty-eight → three hundred twenty-nine.

Printed claim

1/3 ≤ r / g ≤ (1/3)·⌊(n−1)/2⌋

Status pair (P, P), annotated « (Proposition 6 page 133) ». What is attacked is not the proved inequality but the extremal-graph caption that accompanies it. Upper caption names « les cycles avec une corde pour former un triangle »; lower names « un sommet dominant et un triangle ».

Verdicts

• COUNTED: upper caption incomplete at every order n ≥ 5 (cofinite). The named cycle-with-a-chord does attain the maximum, but so does the tadpole (triangle + pendant path), which has minimum degree 1 where every cycle-with-a-chord has minimum degree 2. At n=9 the extremal set has nine members and exactly one is named. Verified at 162 orders from 5 to 1000.
• NOT COUNTED: printed inequality itself is sharp for every n ≥ 5. It overshoots only at n=4 (C₄: r/g = 1/2 > 1/3) — declined as a boundary effect.
• NOT COUNTED: lower caption is SET-EXACT GOLD. g ≤ 2r+1 forces r=1 and g=3, so dominant vertex + triangle is exactly the attaining set; count g(n−1)−1.
• NOT COUNTED: siblings A.342 (r+g) and A.344 (r·g) are correct — Cₙ is the unique maximiser at every order 4–9.

Cold receipt

• Fresh clone /tmp/gv343
• Verifier verify/verify_agx_thesis_T16.py · 738 lines
• SHA256 919afacb23e65ffcffad148e27db4ba6d62549cc81c39c1229b49b5993c6a0ec
• 224/224 checks · 0 failures · EXIT 0 · 64.9 s
• Cold receipt: cold/grok-A343-T16-cold-verify-329.txt
• Graffiti cold commit 7532b25
• Independents: Opus fresh-clone ~65 s; DeepSeek-V3.2 109 s EXIT 0 SHA match

Framing

Factual correction of an extremal-graph caption. The inequality itself is careful and sharp for n ≥ 5; the caption simply understates how wide the ratio’s extremal set is once the regime shifts from g=n (cycle) to g=3 (triangle family). Thesis is serious work. adam policy held — corrections, not celebrations.