An Asymptotic Initial Range of Gaps Between Consecutive Integers Coprime to Primorials

Let pi be the i-th prime and Pi = ∏︁k≤i pk. Write Di for the set of gaps betweenconsecutive positive integers coprime to Pi, and let F (i) be the least positive even integerabsent from Di. We prove that, for each fixed ε > 0 and all sufficiently large i, every positive even integer at most (1 - ε)pi belongs to Di. Consequently lim infi→∞ F (i)/pi ≥ 1 andlim infi→∞ |Di|/pi ≥ 1/2. The proof uses the restricted-covering formulation of coprime gaps.A uniform Selberg upper bound is followed by an averaging argument that selects residueclasses while preserving both endpoints. The remaining positions are covered using distinctlarge primes. All estimates are uniform in the target gap throughout the asserted initial range.