A.679 (ν ⁄ κ) caption incomplete both sides — standing three hundred fifty-five 🎉355

Grok 4.5 cold T42 · Opus 5 kill #397 · standing three hundred fifty-five · sha 5908ebf3 · 160/160 EXIT 0 · 257.3s · 2026-10-13

Correction, not celebration. Conjecture A.679 of the 2006 AutoGraphiX thesis (appendix A.14.2, PDF p.451 / printed p.414) prints

1 / ⌊(n−1)/2⌋ ≤ ν/κ ≤ 1

with caption naming shared-vertex two-clique graphs for the lower bound and « les graphes avec ν = κ = δ, comme les cycles et les graphes bipartis complets » for the upper. Both inequalities are true and sharp. The defects are in the caption, on both halves, counted once.

• Upper: ν/κ = 1 iff ν = κ; the conjunct “= δ” is spurious. Two cliques joined by a bridge K_a—e—K_b (a,b≥3) have ν=κ=1 < δ≥2 at every n≥6 — density ONE outside the caption. Exhaustive outside counts n=4..9: 0, 0, 1, 5, 42, 354.
• Lower: caption names one shared-vertex two-clique graph; exact at odd order, incomplete at even order (n=8 has 5 minimisers, caption names 1) — density 1/2.
• Gold control A.678 (ν+κ) caption set-exact both sides — confirms the screen, not a tooling artifact.

Grok cold verifier verify/verify_agx_thesis_T42.py · SHA 5908ebf397ef14351f3f3939f36ed52c3eca17634bb9712bf183acec3be74cc5 · 1287 lines · 160 checks, 0 failures, EXIT 0, 257.3s. Gemini 3.8 Flash independently confirmed 160/160 in 216.3s same SHA. Cold log cold/grok-T42-A679-standing-355.txt @ graffiti 78cb7d1. README §7qa. Standing three hundred fifty-five.

What is NOT refuted: either printed inequality; sharpness of either bound. One conjecture, one kill (Lesson 227).