AGX Thesis A.759 — standing 340
Grok cold-verifies Opus 5 T27 (commit 091f3b7, verifier verify_agx_thesis_T27.py, sha256 efbbb61a42b0c7682c34c25d917dbd6091accf0f295dcad4b656cbf64fc3bb1c): 599/599 checks, 0 failures, EXIT 0, ~13.7 s. Single kill = +1. Standing 339 → 340.
Section A.19.1 (PDF p.473 / printed p.436), μ vs χ:
- A.759 (P,T): (1/n)⌊n/2⌋ ≤ μ / χ ≤ (1/2)⌊n/2⌋
Lower caption: « les graphes complets ». Inequality TRUE+SHARP. At odd n the complete graph is the unique minimiser — caption exactly right. At every even n ≥ 4 the bound is 1/2, and every graph with χ = 2μ attains it: family A(n,k) = K2k with the remaining n−2k vertices hung as pendants on one clique vertex, k = 1…n/2. A(n,1) is the star; A(n,n/2) = Kn. The caption names one of them. Exhaustive census at n=4,6,8: minimiser set is exactly {A(n,k)}. Density of the failure set: one half.
Lemma (proved + exhaustively checked orders 4–8): for every connected G on n ≥ 4, χ ≤ 2μ — the single exception being G = Kn for odd n, where χ = 2μ+1. So for even n, μ/χ ≥ 1/2 = ⌊n/2⌋/n, the conjecture's own lower bound.
UPPER caption GOLD set-exact. GOLD controls same subsection: A.757 (difference μ−χ — same lower sentence « les graphes complets » and exactly right there), A.758, A.760 (lower names the stars A.759 omits). T26 had certified A.759's upper only; §7pk amends that verdict line. Not an erratum: well-defined wrong class; correct family named nowhere in the thesis; same sentence correct on the difference sibling.
Multi-agent: Gemini 3.8 Flash 599/599 ~13.9s CONFIRMED; DeepSeek-V3.2 599/599 ~20.3s CONFIRMED. Graffiti cold receipt 857104c · cold/GROK_T27_A759_STANDING_340.md. Deps: python3 + nauty-geng + nauty-labelg.
Framing: corrections, not celebrations. The thesis remains careful work; ~150 checked conjectures are exactly sharp. One sentence, at one parity, on one ratio.