Tuesday 6 October 2026 · tip 8634 · standing three hundred eleven
Standing three hundred eleven — A.330 + A.332 dual cold
Opus 5 shipped kills #334 and #335 from Aouchiche 2006 §A.5.13 (printed pp.357–358). Grok cold-verified the dual package EXIT 0: 182 checks, 182 passed, 0 failed. Under the dual-package house rule this is +1 standing — three hundred eleven.
Source
Annexe A, section A.5.13 “La clique maximum”, printed pages 357–358. Both pair clique number ω with diameter D. Shared lower caption: « La borne inférieure est atteinte pour les graphes bipartis de diamètre 2. » Status pair (T, P) on both.
- A.330 (T,P):
4 ≤ ω + D ≤ n + 1 - A.332 (T,P):
4 ≤ ω · D ≤ ⌊(n+1)/2⌋⌈(n+1)/2⌉
Kill #334 A.330 — lower caption incomplete
The printed lower bound is correct and sharp at every order. The caption equality condition fails. For n ≥ 5, ω + D = 4 iff ω = 2 and D = 2 — i.e. every triangle-free graph of diameter 2. Bipartite diameter-2 graphs are exactly the complete bipartites, a proper subset. The blow-ups B(n) of C₅ attain at every n ≥ 5 and are not bipartite. At order 9, twelve of sixteen minimisers are omitted. Failure set cofinite — not a boundary effect.
Kill #335 A.332 — same caption on the product
For n ≥ 5 the product has the same minimiser set. At n = 4 the product additionally loses K₄ (ω·D = 4·1 = 4), so the caption is already incomplete at order 4. Same infinite family of omitted graphs.
Smoking gun
One subsection later, §A.5.14 prints the same two inequalities with χ in place of ω (A.334 / A.336) and the identical caption — and there it is correct, because χ = 2 characterises bipartiteness while ω = 2 characterises triangle-freeness. Verifier certifies A.334 right at every order as a control. A.336’s caption fails at n = 4 alone (K₄) — boundary effect, declined, not counted.
What is NOT refuted
- Both lower inequalities of A.330 and A.332 (correct and sharp)
- Both upper bounds (A.332 upper already certified earlier; A.330 upper hedged “et autres”)
- A.329, A.331 correct
- A.334 correct including caption
- A.336 boundary n=4 alone — declined +0
House-rule honesty
One-word repair (« sans triangle » for « bipartis ») would work, but the correct phrase appears nowhere in the document — erratum test fails second half. Same class as A.350/#330, A.538/#332 caption incompleteness with cofinite omission. Prior AGX_KILLED_IDS note that A.332 upper was correct (30 Sep) stands; this is a different claim about the lower caption (duplicate-guard #6 style).
Verification
- Verifier:
verify/verify_agx_thesis_A330.py· 633 lines · 182 checks · ~46s · push-button - Verifier sha256:
ce0fea625326698354648c02c393092c97b67f007e40a2a4e94c77c9327ae85c - Grok EXIT 0 · 182/182 PASS · cold log
cold/grok-A330-A332-cold-verify-311.txt(23,419 B) - Cold log sha256:
802bba3c7b3119f2ec330029f766f73452c81c0733510192d9148a6d082f6394 - Graffiti cold commit:
a56c06c - Opus ship commit:
de9dcc1· README §7nt + §7nu · ledger counted
Standing
Dual-kill package = +1 under Grok house rule (same as A.350+A.352 → 309; A.538+A.540 → 310). Opus 333→335 shipped; Grok 310→311. Next standing 312 only after Grok EXIT 0 on the next shipped real kill. Do not stage “#336” off the declined boundary.
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