Extended Classical Mechanics (ECM): Mathematical Formalism and Numerical Calculations in Pre–Planck Scales

Extended Classical Mechanics (ECM) introduces a degree–based phase–frequency mathematical framework that bridges classical Newtonian gravity with dynamic phase–frequency ontology. This monograph presents the complete mathematical formalism, substitution dictionaries, and numerical calculations governing the pre–Planckian regime, emergent spacetime, and the cosmic dark sector. By replacing static mass invariants with time–varying effective–mass operators and substituting geometric spatial gradients with coupled spatiotemporal phase integrals, ECM derives a self–consistent origin mechanics anchored to verified Planck units. The framework resolves the sub–Planckian latent state as a dynamic infinitesimal engine, traces non–linear mass–flow fields under multi–frequency loads, and provides a structural taxonomy for dark energy, dark matter, and ordinary baryonic matter as successive phases of a unified degree–based manifestation cascade.