The kite construction — a dense clique (K50) with a long pendant path (68 vertices) — exploits a structural tension: the eigenvalue deviation wants many edges (favors the clique), while n/average distance wants a long thin graph (favors the path). The kite combines both, but the crossover requires exactly 118 vertices. No kite on 117 vertices violates, explaining why 1990-91 LANL tests (limited to 10 vertices) and exhaustive searches (through order 9, 261,080 graphs) all passed. This is classic Fajtlowicz-era hiding: the counterexample exists at a vertex count just beyond computational reach, hiding for decades until someone thinks to check the right construction.