Finite Lorentz factor from discrete proper-time quantization: modified dispersion relation and phenomenology

We present a formal theoretical framework introducing discrete time quantization at the Planck scale into special relativity, leading to a finite Lorentz factor and eliminating divergences as velocities approach the speed of light. Framed with an engineering lens inspired by digital-signal-processing (DSP) discretization, we motivate the modified dispersion relation (MDR) and its testable consequences. A modified dispersion relation (MDR) is derived from uncertainty principles and dimensional analysis applied to quantized proper time, yielding testable predictions within Lorentz invariance violation (LIV) phenomenology. We detail the mathematical derivation, connect the framework to existing approaches such as Doubly Special Relativity and $\kappa$-Poincar\'e deformations, and outline a numerical modeling approach via worldline numerics on discrete spacetime lattices. Physical implications include a reinterpretation of apparent faster-than-light effects as quantum synchronization across discrete time slices. Experimental predictions---from gamma-ray burst photon delays to Casimir energy deviations---are quantified with falsification thresholds.