Abstract
Ramanujan primes encode a persistent form of prime abundance: after the nth threshold, every interval of the form (x/2,x] contains at least n primes. Existing work establishes their existence, global bounds, and leading asymptotic equivalence to the 2n th prime, yet the exact discrete mechanism and the next asymptotic correction are usually treated separately. This work provides a general framework for the analysis of stability and thresholds. It uses a prime window counting process as an event-driven step function and describes the latest deficit as well as the finite sieve process with a known upper bound guaranteeing its completeness. Asymptotic inversion of the prime-counting expansion formula provides the second-order approximation of the threshold with respect to a constant window value c from the interval (0, 1). In the classical case, the approximation shows that the difference between the 2n-th prime number and 2n converges to log 2. Exact computation of the first 100,000 Ramanujan primes confirms the predicted scale; at n=100,000, the second-order approximation has relative error 3.37×10^4. The framework separates rigorous asymptotics, certified finite computation, and historical terminology, providing a reproducible basis for further explicit estimates.