Deriving Relativistic Kinematics from a Medium-Based Paradigm: A Complete Geometric Demonstration

This paper presents a rigorous geometric derivation of Special Relativity's formulas from a physical medium-based foundation. Using the complete geometric demonstrations from the author's previous work, we show how the Michelson-Morley null result necessarily constrains the deformation of moving apparatuses, and how the wave nature of matter naturally leads to time dilation through helical light trajectories. Crucially, our framework transforms the Lorentz contraction from a postulate into an experimentally testable prediction, allowing precise time dilation measurements to determine the complete deformation pattern. The derivation provides complete mathematical demonstrations in Appendices using the original diagrams and geometric arguments, proving that Lorentz transformations emerge as physical necessities rather than postulates when matter is conceived as confined excitations of a propagation medium.This work forms the kinematic foundation of a broader unified paradigm that addresses outstanding problems in cosmology and particle physics, demonstrating that relativistic kinematics follows self-consistently from wave-mechanical principles while maintaining full operational equivalence with special relativity for all practical measurements.