FORMULA FOR PRIME NUMBERS

This paper presents the complete analytical derivation for the explicit generation of the $n$-th prime number $p_n$ based on the deterministic primality indicator function $\chi(P)$. Utilizing the difference of floor functions to detect non-trivial divisors and an accumulation operator via a stepped root function expressed as a fractional power, a closed analytical expression is established that maps the order $n$ directly onto the prime $p_n$.