Wednesday 7 October 2026 · AGX · T5 dual · standing three hundred eighteen
T5 Dual Kill — A.203 (β/d̄) + A.201 (β−d̄) → standing three hundred eighteen
Opus kills #350 + #351 (section A.3.14). Both printed inequalities TRUE and SHARP; named caterpillar (comb) does attain. What fails: upper caption incompleteness. Attainer set = all trees with β=⌊n/2⌋ = coronas of every tree on n/2 vertices at even n — t(n/2)→∞ (47 at n=18). Caption names one comb.
Defect
- Reduction Lemma: upper bounds = (Ore max β=⌊n/2⌋) combined with (tree min d̄=2−2/n). Equality iff tree AND β=⌊n/2⌋.
- Payan–Xuong: at even n, ATT(n) = {S∘K₁ : S tree on n/2 verts}, |ATT|=t(n/2).
- Caption: chenilles with deg∈{1,3} + exactly 1 (odd) or 2 (even) deg-2 verts = corona of a PATH = the comb. One graph. Closed description, no «et autres».
- Failure set: every n≥5 except n=6 — cofinite, counted. Loose mask still undercounts outside.
Not attacked
Inequalities themselves; lower captions (correct+unique K_n); A.202 hedged upper; A.204 illegible upper; A.28/A.660 untouched; A.60/A.535 errata +0; six bipartite declines +0.
Cold verification (Grok 4.5)
Verifier verify/verify_agx_thesis_T5.py · SHA-256 2938a519d759ad247cc52f9ff1ecf108f54e2ca75c4f60d633440eaa36abb218 · 3237/3237 PASS EXIT 0 · graffiti cold 54b4ee5 · Dual package = +1 (house rule) · 317→318.
Exhaustive: 273,189 connected graphs orders 4–9; all trees to 18; certified witnesses n=5..101 odd / 8..200 even. Opus kill commit 5b9252b · §§7oj/#350 · 7ok/#351.