Force-Based Phase-Frequency Continuum and Axiomatic Lorentz Transformation Alternatives in Extended Classical Mechanics

Extended Classical Mechanics (ECM) formulates a force-based, phase-frequency continuum as an alternative mathematical framework for describing large-scale cosmic motion, galactic dynamics, and velocity-dependent phenomena without invoking metric spacetime curvature. The framework treats space and time as fixed, abstract dimensional extensions and models large-scale phenomena through vacuum phase-kernel field interactions, energetic phase displacement, frequency transformation, and the manifestation of Negative Apparent Mass. Within this formulation, the energetic phase measure x° is causally related to source-frequency displacement and to the effective mass of an interacting system. Rather than introducing an additional phase-modification function into the gravitational force equation, ECM incorporates the phase-dependent contribution through the effective gravitating mass Mᵉᶠᶠ. Galactic rotation, large-scale cosmic motion, propagation redshift, and velocity-dependent transformations are therefore treated as manifestations of a common phase-frequency dynamical framework operating in fixed flat space. The present repository edition combines two linked ECM formulations: (1) the Force-Based Phase-Frequency Continuum for galactic dynamics, large-scale cosmic motion, and propagation redshift; and (2) the Axiomatic Formulation of Lorentz Transformation Alternatives, which reinterprets velocity-dependent relativistic scaling through dynamic mass parameters, energetic phase displacement, and absolute clock-rate variation within a fixed coordinate architecture.