Prime Gear Geometry: The Prime Formulation

This paper presents the Prime Gear Geometry (PGG) formulation, a deterministic spectral framework that models prime numbers as discrete mechanical oscillators. Unlike probabilistic models or the complex analytic continuation of the Riemann Zeta function, PGG treats primes as discrete physical rotors within a deterministic engine. We demonstrate that by representing primes as a unit-normalized time series and applying a band-limited Inverse Fourier Transform, with a fixed Atomic Tuning Factor ($K=3$), the discrete "Atomic Gears" spectrum are presented. This method provides a reproducible, robust, and discrete spectral foundation of the prime sequence, anchoring the system at the 0.5 Gyrocenter and shifting the study of primes from probabilistic density to deterministic gear mechanics.