A Conditional Framework for the Riemann Hypothesis Via the Restricted Greedy Prime Decomposition Algorithm: Established Equivalences, the Critical Gap, and a Program for Closure

Abstract

We present a complete conditional framework connecting the restricted greedy prime decomposition algorithm to the Riemann Hypothesis (RH). We prove four rigorous equivalences between the algorithmic error structure and analytic number theory. Three of the four implications in the main theorem are proved unconditionally; the fourth reduces RH to a single conjecture about the holomorphy of a bivariate Dirichlet series in a critical strip. We identify the exact mathematical gap that separates this theory from a proof of RH, and propose a concrete three-step program to close it.