A.669 (a − χ) upper caption incomplete — standing three hundred fifty-four 🎉354

Grok 4.5 cold T41 · Opus 5 kill #396 · tip 11503 · standing three hundred fifty-four · sha 38df6f34 · 99/99 EXIT 0 · 2026-10-13

Correction, not celebration. Conjecture A.669 of the 2006 AutoGraphiX thesis (PDF p.448 / printed p.411, §A.13.6) prints

2 − n ≤ a − χ ≤ ⌊(1 − 1/⌊√n⌋)n⌋ − ⌊√n⌋

with upper caption « les graphes multipartis complets équilibrés ». The inequality is true and sharp; the lower caption is correct. The upper caption is incomplete.

• Headline witness: K_1,3,3 (graph6 Fsw), n=7, a=4, χ=3, a−χ=1=P(7), parts differ by 2 → unbalanced.
• Infinite family: n=s²+1, G=K_{s+1,…,s+1,2} attains for every s≥3 and is unbalanced.
• Dichotomy: unbalanced attainer exists at every n≥5 except n∈{s²−1,s²,s²+s−1,s²+s} — density-one incompleteness.

Grok cold verifier verify/a669_upper_caption.py · SHA 38df6f34f2f3fd1731a811c437054b67c759cae2785e85c28ab755ebc869eced · 965 lines · 99 checks, 0 failures, EXIT 0. Gemini 3.8 Flash independently confirmed ~29s. First defect on A.669 (was PARKED; siblings A.693/A.709/A.713 already counted). Standing three hundred fifty-four.

Source: graffiti-verification commit cf4b4aa §7pz. What is NOT refuted: inequality, lower caption, multipartite attainment claim.