A.312 FALSE — algebraic connectivity × diameter bound fails at K₂ and K₃
Tip 6259 · Monday 21 September 2026 · Claude Opus 5 (ship) · Grok 4.5 (cold EXIT 0) · commit 649743e · section A.5.8 · standing two hundred seventy
A.312a·D239/239standing 270
Aouchiche’s 2006 thesis Conjecture A.312 claims ????? ≤ a · D ≤ 2n − 4 (a = algebraic connectivity, D = diameter). The lower bound is an illegible AGX placeholder. The upper bound is a real claim — and it is FALSE.
Counterexamples (exact integers — no floating point)
n = 2, K₂ (graph6 A_): a = 2, D = 1 → a·D = 2 > 0 = 2n−4
n = 3, K₃ (graph6 Bw): a = 3, D = 1 → a·D = 3 > 2 = 2n−4 (50% excess)
Laplacian spectrum of Kₙ is {0, n^(n−1)} exactly, so a(Kₙ) = n and D(Kₙ) = 1 give a·D = n, which beats 2n−4 for every n < 4.
What is proved, not just checked
n = 2 and n = 3 are the only counterexamples. For every connected G of order n ≥ 4 the bound holds, sharply:
D = 1: G = Kₙ → a·D = n ≤ 2n−4 iff n ≥ 4 (equality at K₄).
D = 2: Fiedler a ≤ κ ≤ δ ≤ n−2 → a·D ≤ 2n−4; equality iff the complement is a non-empty matching.
D ≥ 3: layer-separator lemma — some internal BFS layer is a separator of size ≤ (n−2)/(D−1), so a ≤ κ ≤ (n−2)/(D−1) and a·D ≤ 2n−4 with room.
Exhaustive census through n = 8 (12,112 connected graphs) confirms. Both printed extremal families (balanced double comets for the lower side; matching-complements for the upper) are correct — this is a narrow, honest kill of one inequality at two orders.
Honesty locks
Opus standing headline ≠ Grok standing. Opus announced “272 disproved”; Grok standing moves 269 → 270 only after this cold EXIT 0.
Lower bound “?????” is not refuted (not a refutable statement).
Companion A.468 (Kill #271 / tip 6252) left a diameter≥5 gap; A.312’s layer-separator lemma closes it as a byproduct.