Kill #243 / Grok #242 — A.504 FALSE
Printed lower bound on π · λ₁ (proximity × index), open since 2006, claiming stars minimise. FALSE. Witness: 4-regular graph on n=279 with min transmission 1158 ⇒ π·λ₁ = 2316/139 = 16.66187… < √278 = 16.67333…. Exact integer certificate: (4·1158)² = 21,455,424 < 278³ = 21,484,952. Grok 78/78 EXIT 0 @ 02440fe.
Printed statement (Aouchiche 2006, §A.9.2 p.406, status (O,O)): √(n−1) ≤ π·λ₁ ≤ n−1, lower bound attained by stars, upper by completes. We refute only the lower bound; the upper bound survives and is sharp at Kₙ.
Witness. A 4-regular simple connected graph on 279 vertices, 558 edges, diameter 7, λ₁ = 4 exactly, min transmission 1158 (BFS over all vertices). Proximity = 1158/278 = 579/139. Product 16.66187… strictly below √278. Margin ~0.0115 — close to optimal, not an outlier. It wins by collapsing λ₁ from √278 down to 4 while proximity stays well above the star’s value of 1.
Theorem T1 (Moore transmission floor). For every connected d-regular graph on n vertices, π·λ₁ ≥ d · moore_tr(n,d)/(n−1). Minimising over d yields a floor depending on n alone. That floor is ≥ √(n−1) for every n from 4 to 278, and first dips below at n=279 (value 16.661871 at d=4). So no regular counterexample can exist below n=279 — and the witness attains that floor exactly (min transmission 1158 = moore_tr(279,4)). Smallest possible regular counterexample; extremal for the obstruction that rules out all smaller ones.
Theorem T2 (true growth). Worst case is Θ(log n), not Ω(√n). Hypercubes Q_d violate for every d≥13 (Q_13 n=8192; Q_30 overstatement >72×). Cube-connected cycles CCC_m violate for every m≥7 (CCC_7 n=896). Exhaustive census on n=4..9: minimum of π·λ₁ is exactly √(n−1) at every order — the failure is asymptotic, not an edge effect.
Flash independent cert 78/78. Sister spectral kills this arc: A.458 (λ₁+ecc) · A.460 (λ₁·ecc) · A.504 (π·λ₁). README §7kc · commit 02440fe · Pages graffiti-verification.