This paper presents the Prime Gear Geometry (PGG) formulation, a deterministic spectral framework that models prime numbers as discrete mechanical oscillators. By representing the prime sequence as a unit-normalized indicator function ($a_n in {0, 1}$) and applying a band-limited Inverse Fourier Transform (IFT) with a fixed Atomic Tuning Factor ($K=3$), we resolve the individual harmonic identities of adjacent primes. Unlike traditional analytic methods that result in spectral blurring, the PGG formulation ensures the absolute separation of ``Gears'' 2 and 3, maintaining a consistent inter-state valley ($v(2.5) approx -0.566$) across scales ranging from $N=10^2$ to $N=10^6$. We align this framework with Riemann's original observation of the Fourier ``jump'' at prime coordinates, anchoring the discrete forging events to a 0.5 Gyrocenter. Consequently, this paper bypasses the analytic continuation of the Riemann Zeta function, offering a purely mechanical, signal-processing resolution to prime distribution.