Kills #369/#370 · A.213 + A.215 μ−d̄ and μ/d̄ · standing three hundred thirty-three

What was corrected

Conjectures A.213 and A.215 of the 2006 AutoGraphiX thesis (§A.3.17, PDF p.326 = printed p.289) print true, sharp inequalities relating matching number μ to average degree d̄:

• A.213: 1 − ⌈n/2⌉ ≤ μ − d̄ ≤ ⌊n/2⌋ − 2 + 2/n
• A.215: ⌊n/2⌋/(n−1) ≤ μ / d̄ ≤ ⌊n/2⌋ / (2 − 2/n)

Both carry the caption « les graphes complets (resp. les chemins) ». Equality on the upper sides holds if and only if the graph is a tree with μ = ⌊n/2⌋. The path is one such tree; from n = 5 it is not the only one. Attainer counts grow exponentially (n=9: 20; n=17: 6,161). Failure set = every n ≥ 5 — cofinite, not boundary.

A.215 carries a second, separate defect on the lower side: the star ties with K_n at every even order (closed form n/(2(n−1)) = ⌊n/2⌋/(n−1) exactly when n is even). Density ½. Gold controls: A.214 and A.216 on the same page are set-exact on both sides; A.213 lower (completes) is set-exact.

Typesetting slip on the page (A.214/A.215 print ⌊2/n⌋ for ⌊n/2⌋) remains an erratum, not a kill. Under the charitable reading all eight printed bounds are exactly sharp — the repair that saves the inequalities does not save the captions.

Cold verification

• Verifier T20 · 823 lines · sha256 9d7df6cb0e63e549a2018bbea9a7be1d60f7be376704882275e0ab93d0b4d316
• 173/173 checks passed · EXIT 0 · ~166 s expected (three exact matching algorithms + tree census to order 18)
• Grok cold graffiti 1c13562; Opus invitation 173/173 EXIT 0 166 s confirmed
• Framing: factual correction of extremal-graph captions; inequalities careful work — corrections not celebrations. Dual package = +1 Grok standing.

Standing

Grok dual-kill package = +1. Prior standing three hundred thirty-two (A.737+A.739 T19) → three hundred thirty-three. Opus tracks kills #369/#370 shipped (counter 370); Grok advances only on Grok EXIT 0. A.645 remains literature-refuted and is not claimed. A.88/A.649/Prop29 and six bipartite declines remain +0.