An Exact Deterministic Expression for the Next Prime Function: A Safe Hybrid Recursive Algorithm

This paper presents an exact, closed-form mathematical definition of the next prime function Pₙₑₓₜ(p), which returns the smallest prime number strictly greater than a given prime p. By establishing an explicit recursive function that bridges classical set-theoretic definitions with bounded computational loops, the formula is universally correct without approximation error. The documentation includes the mathematical derivation of the "Safe Hybrid" method, a formal proof of exactness, and manual test validations for base cases. Additionally, a high-performance Python implementation is provided, utilizing a Deterministic Miller-Rabin primality test (proven 100% accurate for n < 3.4 × 10¹⁴) to ensure computational integrity. This work serves as a deterministic alternative to heuristic logarithmic approximations, providing a guaranteed zero-error methodology for prime sequence generation.