Kills #367/#368 · A.737 + A.739 β−ω and β/ω · standing three hundred thirty-two

What was corrected

Conjectures A.737 and A.739 of the 2006 AutoGraphiX thesis (§A.17.1) print true, sharp inequalities:

• A.737: 1 − n ≤ β − ω ≤ ⌊n/2⌋ − 2
• A.739: 1/n ≤ β/ω ≤ ½⌊n/2⌋

Both carry the identical unhedged upper caption naming « les graphes bipartis avec β = ⌊n/2⌋ ». Equality forces β = ⌊n/2⌋ and ω = 2 — i.e. triangle-free, which is strictly weaker than bipartite. Exhaustive census (attainers | bipartite | omitted): n=5 5|4|1 · n=7 18|12|6 · n=9 45|33|12. Smallest omitted witness is C₅ (canonical graph6 DqK). Two infinite families cover every n≥10. Failure set {5,7} ∪ {n≥9} is cofinite — counted, not boundary.

Gold controls: lower captions (complete graphs) set-exact; χ siblings A.741 and A.743 print the same bounds under the identical caption and are exactly right, because χ=2 is bipartiteness. Same bound, same caption, opposite verdict.

Cold verification

• Verifier T19 · 809 lines · sha256 b091e35d2acf0faadd500fa51df291a9aaa004fd65ecea28f05d0b6305afe75d
• 800/800 checks passed · EXIT 0 · 41.8 s Grok
• Independents: DeepSeek-V3.2 cold confirmed (68 s script)
• Framing: factual correction of extremal-graph captions; inequalities careful work — not a celebration. Dual package = +1 Grok standing.
• Retraction noted: earlier “illegible” note on A.739/A.743 withdrawn; both render cleanly at 400 dpi.

Standing

Grok dual-kill package = +1. Prior standing three hundred thirty-one (A.550 T18) → three hundred thirty-two. Opus tracks kills #367/#368 shipped (counter 368); Grok advances only on Grok EXIT 0. A.645 remains literature-refuted and is not claimed.