ECM Phase-Frequency Formalism: From Zero-Dimensional Latent States to Observable Redshift

This paper presents the formal derivation of the Extended Classical Mechanics (ECM) phase–frequency cascade, establishing a unified framework in which phase shift, frequency modulation, and energy transaction are treated as manifestations of a single primordial quantity. Beginning from a zero-dimensional latent state in which dimensional constraints are absent, we derive the transition from primordial frequency f0 to Planck-scale emergence fP, and subsequently to the observable redshift mechanism fsource = fobserved + Δfsource. Central to the formalism is the Formal Equivalence Principle, which establishes a bidirectional mapping x° ↔ Δφ between the energetic phase shift x°—the dynamic governor of frequency manifestation—and the geometric phase angle Δφ—the angular bridge of intra-cycle measurement. We derive the phase–frequency relation Δf = f(x°/360°) with explicit boundary conditions (x° = 0° ⇒ Δf = 0; x° = 360° ⇒ Δf = f), and formalize the distinction between trans-cycle energetic displacement and intra-cycle geometric rotation under Axiom 4 (Phase Duality). The energetic phase shift is shown to accumulate linearly without modulo constraint, driving cosmic dissolution at maximum universal radius. This work provides the axiomatic and mathematical foundation for ECM cosmology and establishes testable predictions regarding the redshift–distance relation and the ultimate fate of observable mass-energy.