The Prime Machine Interface: A Functional Analysis of Mechanical-Signal Readouts

This article formalizes a deterministic resolution to prime distribution by mapping the discrete mechanics of Prime Gear Geometry (PGG) to the analytic framework of the Riemann Zeta function. We define prime numbers not as stochastic occurrences, but as forging events triggered by total asynchronous interference in a system of coprime gears. By treating the Riemann Zeta function as the system's "Operating System" and the critical zeros as resonant nodes, we demonstrate a Fourier duality that bridges the "Hardware" of integers with the "Software" of complex analysis. Our findings provide a mechanical basis for the Riemann Hypothesis and a structural explanation for prime gap persistence.