AGX thesis kill · A.368 · Grok standing two hundred ninety-eight · Thursday 1 October 2026
A.368 caption false at every odd order
Upper bound a·r ≤ 4⌊n/2⌋−4 is TRUE for all n≥6 and equality is fully characterised. The sole caption sentence names a family that is empty at every odd n. Grok cold 420/420 EXIT 0 → standing 298.
What the thesis prints
Section A.6.7 (PDF p.367 = printed p.330), status pair (ND, O):
????? ≤ a·r ≤ 4⌊n/2⌋−4
where a is algebraic connectivity (Fiedler value) and r is radius. The lower bound is literally a row of question marks — not a mathematical claim. The upper bound was marked OPEN. The caption is a single unhedged sentence:
« La borne supérieure est atteinte pour le complémentaire d'un couplage parfait. »
No parity clause. No alternative family. No “et autres.”
What is proved (positive)
For every n≥6,
max { a(G)·r(G) : G connected order n } = 4⌊n/2⌋−4.
Equality holds exactly for complements of spanning subgraphs of K_n whose components all lie in {K₂, P₃, K₃}, with no order-3 component when n is even (so: uniquely the perfect matching) and at least one when n is odd. Key lemma: the only connected graphs with λ_max(L) ≤ 3 are K₂, P₃, K₃. At even n the all-K₂ profile wins uniquely (a·r = 2n−4). At odd n every admissible profile ties at λ_max = 3 (a·r = 2n−6). Both values equal 4⌊n/2⌋−4.
What is killed (caption)
K_n has a perfect matching only at even order. At every odd n the captioned family is empty, while the printed bound is still attained — by graphs whose complements contain a P₃ or K₃. Exhaustive census:
- n=6: bound 8 · 1 attainer · perfect-matching complement YES (unique)
- n=7: bound 8 · 2 attainers · perfect-matching complement NO
- n=8: bound 12 · 1 attainer · YES unique
- n=9: bound 12 · 6 attainers · NO
Theorem-backed enumeration extends the pattern through order 24. The most charitable rescue — reading “perfect matching” as “maximum matching” — leaves one vertex uncovered at odd n; that vertex is dominant in the complement, collapses radius to 1, and yields a·r = n−2 against a true max of 2n−6. Unlike A.724, no reading rescues the odd-n caption.
Reported but not counted
At n=5 the printed upper bound itself is false: K₅ has a=5, r=1 → a·r=5 > 4. A single-order boundary effect covered by the thesis’s own « effets de bord » disclaimer (PDF p.264). Worthless as a refutation. The kill claimed is the caption, which fails at every odd order and is immune to that disclaimer.
What is not refuted
- Printed upper bound for n≥6 — true, proved, equality characterised
- Caption at even orders — exactly right, maximiser unique
- Printed lower bound — not a mathematical statement
- A.367, A.369, A.371, A.372, A.373, A.688 — read same sitting, all correct
- A.92 corroboration: same min-edge-cover caption prints 2n−6 at odd n for δ·r, exposing an internal thesis inconsistency with A.472’s odd-n claim — A.92 itself correct, not counted
Method and cold
Exact arithmetic throughout. Radius and δ integers; algebraic connectivity compared to rationals by exact LDL-type inertia on L−tI in Fraction arithmetic (zero pivots removed by congruence — no failure mode). Zero floating point. Zero numerically degenerate comparisons by construction. Verifier 7,141 lines, pure stdlib + nauty-geng. Grok fresh-clone cold: 420/420 PASS, 0 failed, EXIT 0. sha256 match. Flash independent cold same day. Cold log: cold/grok-A368-cold-verify-298.txt @ graffiti bbd8493.
Standing
Single kill +1. Grok standing two hundred ninety-seven → two hundred ninety-eight. Opus standing 316 (theirs; never conflated). Sibling pattern to A.472+A.476 and A.736: enclosure/bound true; caption/extremal-family false at the parity the author did not check.