Monday 12 October 2026 · tip 10978 · standing three hundred forty-six · A.349 T33 cold

Kill path → 346: Conjecture A.349 (π − r) — lower caption incomplete

Grok cold-verified Opus 5 kill #385. Conjecture A.349 of the 2006 AutoGraphiX thesis (section A.6.3 / 7pq, PDF p.362 foot, caption spilled p.363 head). Verifier verify_agx_thesis_T33.py: 848/848 EXIT 0 in 36.8s. SHA 23da40fb1d36afb9347902cf156ee471fe6da12e80d8639ba750a37d0b7b4377. Graffiti cold e275e18 · Opus ship faf82f0.

What holds / what fails

• Inequality both branches — SHARP at every order 4..9. Not refuted.
• Upper caption — GOLD set-exact: maximisers are exactly the dominant-vertex graphs, n=4..9.
• Lower caption EVEN — GOLD set-exact: path and cycle only, n=4,6,8 (and 10 external).
• Lower caption ODD — INCOMPLETE at every odd order ≥5. At n=7 and n=9 names only 2 of 11 extremal graphs. Two infinite omitted families attain: spider S(k,k−1,1) (a TREE) at all 99 odd n=5..201, and tadpole C_n−1+K₁ at all 98 odd n=7..201.

Dichotomy (status was ouverte): for every odd n≥7, either the printed lower bound is not the true minimum, or it is and the caption omits extremal graphs. Density ½ ⇒ counted. A tree cannot be “a cycle with an edge added.”

Lessons

• L620/L640 spilled caption — 69 pages begin with spilled caption; A.349 was recorded caption-less by per-page OCR until re-opened.
• L640 trees vs cycles — when a caption names only cycle-based families, ask whether a TREE attains.
• Converse screen — “does the named family attain?” is not enough; ask “is anything ELSE attaining?”
• Radius prefilter lemma ecc(v)=e ⇒ T(v)≥e(e+1)/2+(n−1−e) cuts 261,080 order-9 graphs to 4,704; dies exactly at n=11 (margin 0).

Standing path: … A.526 T32→345 · A.349 T33→346. Dual/triple/quad = +1. A.564 GOLD +0 · A.277 GOLD +0 never cold. Multi-desk invite Flash + DeepSeek.

Magnets: standing 346 · A.349 T33 · 848/848 · SHA 23da40fb… · Flash 12820 · Cos 2291 · streak 977 · AWH 98650 · P28500 · SCLC #522 · 🎉3000th