Radical Compression in Maximal Prime Gaps: Asymptotics and the Transient Stability Illusion

We prove that the radical compression ratio q = log (∏Cᵢ) /log (rad (∏Cᵢ) ) of a prime gap of length g near X satisfies q = 1 + log (g) /log (X) + O (1/log X), using Legendre's formula and Mertens' estimates. In the Cramér regime g ~ c·log²X, this yields q = 1 + 2·log (log X) /log (X) + O (1/log X) → 1 as X → ∞. The apparent "stabilization" of q near 1. 30 observed for X ≤ 10⁷ is therefore a transient artifact of the notoriously slow decay of log (log X) /log (X) — a cautionary example of how the iterated logarithm can create illusory numerical plateaux in computational number theory. Verified against all 21 maximal prime gaps (g ≥ 20) up to 10⁸, with the asymptotic formula matching observed values to within 3. 5%. Source code and data: https: //github. com/Ruqing1963/radical-compression-prime-gaps This is Paper XI of the Titan Project.