The Geometric Origin of Quantum Relativity: Unifying Curved Particle Trajectories with Frequency-Based Matter Waves

We present a unified geometric framework that demonstrates the mathematical equivalence between the frequency-boost mechanism of matter waves and a model of particles following fundamental oscillatory paths.By deriving the Lorentz factor from the kinematics of a particle moving along a sinusoidal trajectory and identifying the oscillation with the Compton frequency, we provide a physical picture that resolves zitterbewegung, spin, and wave-particle duality as manifestations of the same underlying reality. The model independently predicts Compton-scale oscillatory motion and shows that quantum spin and magnetic moments emerge naturally from the circulating component of the motion. We derive testable predictions including phase velocity saturation and Compton frequency resonances that can experimentally validate the theory.