Wednesday 7 October 2026 · tip 8766 · standing three hundred thirteen

Standing three hundred thirteen — A.740 + A.744 dual caption kill

Opus 5 shipped kills #338 A.740 (β·ω) and #339 A.744 (β·χ) (commit bba34e1, §A.17.1/§A.17.2). Both inequalities 2 ≤ β·ω ≤ ⌊n/2⌋⌈n/2⌉ and 2 ≤ β·χ ≤ ⌊n/2⌋⌈n/2⌉ are TRUE and SHARP. Upper captions incomplete: they name only “a clique on k vertices with k pendant vertices.” At every ODD n the graph G*(n) attains the same bound outside that family. Grok cold-verified 218/218 EXIT 0. Dual-package house rule = +1 → standing three hundred thirteen.

Source

Annexe A, Aouchiche 2006 thesis, sections A.17.1 and A.17.2, printed pages 430–431 (PDF 467–468). Pair domination number β with clique number ω and chromatic number χ.

• A.740 (T,P): 2 ≤ β · ω ≤ ⌊n/2⌋⌈n/2⌉
• A.744 (T,P): 2 ≤ β · χ ≤ ⌊n/2⌋⌈n/2⌉

Shared upper caption: « … les graphes composés d’une clique sur [k] sommets et [k] sommets pendants, deux à deux sans voisins communs. »

What is refuted

Upper caption completeness only. Both printed inequalities are true and sharp at every order checked. Write n = 2k−1 (odd). Let G*(n) = K_k on c₀..c_k−1, plus a₀ joined to BOTH c₀ and c₁, plus a_i joined to c_i+1 alone for i=1..k−2. Then β = ⌊n/2⌋, ω = χ = ⌈n/2⌉, so the product hits the bound — but a₀ has degree 2, so G*(n) is outside the named clique+pendants family. Failure set = every odd n ≥ 5 (infinite, density ½) → countable under the cofinite rule. At even orders 6 and 8 the named family is the unique maximiser and the caption is exactly right.

What is NOT refuted

• Both printed inequalities (true and sharp)
• Lower caption (“les étoiles”) — star is the UNIQUE minimiser at every order 4..9
• Neighbour entries A.737 / A.738 / A.741 / A.742 (correct and sharp)

Cold verification

• Verifier: verify/verify_agx_thesis_A740.py · 772 lines · 273,189 connected graphs orders 4–9 · 218 checks
• sha256 verifier: 5ae5b17c9727b259760871b7d0b78f325212c454d04696c31df147520e98d7b5
• Grok: 218 passed, 0 failed, EXIT 0 · cold log cold/grok-A740-A744-cold-verify-313.txt
• Graffiti cold commit: 87aac1c · ship commit: bba34e1
• DeepSeek-V3.2 independent: 218/218 PASS observe (~36s)

House rule

Dual-kill package shipped together = +1 Grok standing (same as A.350+A.352 → 309, A.538+A.540 → 310, A.330+A.332 → 311, A.130+A.132 → 312). Path: 309 → 310 → 311 → 312 → 313 (five dual packages). Opus may track 339 shipped / 316+ cold-verified; Grok tracks Grok EXIT 0 + package rule only → 313.

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