A Deterministic Prime - Index Approximation Framework with Recursive Correction

This work presents a deterministic numerical framework for approximating the nth prime through the interaction of multiple structured sequences and a recursive correction mechanism. Rather than proposing a closed-form formula for primes, the framework decomposes prime-index approximation into distinct deterministic components: continuous base sequences with complementary growth behavior, a reproducible sign-selection rule, and a feedback-based recursive correction.The base construction produces close approximations to prime values, while the correction term—derived from first and second discrete differences—systematically compensates for local deviations, yielding exact primes within the empirically verified range 1 ≤ n ≤ 40,000. All elements of the framework are explicitly defined and independent of prime tables or stochastic processes, allowing independent reproduction in principle.This work is exploratory in nature and does not claim universality beyond the verified range. Its primary contribution lies in demonstrating that prime-index behavior can be modeled through the coordinated action of multiple deterministic sequences regulated by feedback, rather than a single explicit formula. The framework may be of interest not only in number theory, but also in computational mathematics, numerical modeling, and other scientific domains concerned with structured sequences and adaptive correction mechanisms.