ECM Master Equation: Unified Frequency–Mass–Energy–Potential Axiom

This preprint presents the ECM Master Equation, a foundational axiom of Extended Classical Mechanics (ECM), establishing a unified transformation framework linking potential energy, mass variation, frequency, and kinetic energy. In ECM, physical reality is not composed of independent entities such as mass, energy, or force. Instead, all physical evolution is governed by a continuous frequency-driven manifestation process, expressed through the transformation chain: ΔPEᴇᴄᴍ ↔ ΔMᴍ ↔ f ↔ ΔKEᴇᴄᴍ Within this structure: Potential energy (ΔPEᴇᴄᴍ) represents latent configurational imbalance, Mass variation (ΔMᴍ) represents redistribution of that imbalance, Frequency (f) encodes the rate of transformation, Kinetic energy (ΔKEᴇᴄᴍ) emerges as the observable manifestation. The governing constraint of the framework is: ΔKEᴇᴄᴍ = ΔMᴍ c² = h f This relation is interpreted not as rest-energy equivalence, but as a dynamic manifestation identity, where energy arises from frequency-governed mass redistribution. A key contribution of this work is the introduction of a Planck-normalized frequency–mass mapping, defined through: f₀ = fᴘ + Δf₀ (layered frequency structure) ΔMᴍ = k f (pre-Planck regime) ΔMᴍ = h f (Planck regime) ΔMᴍ / Mᴘ = f / fᴘ (normalized scaling relation) This formulation unifies emergent and fundamental regimes under a single scalable framework. The model further develops: Hierarchical frequency decomposition (source, Planck, and observed components), Composite mass structure (wave and Planck contributions), Quantum transition interpretation as discrete frequency reconfiguration events:ΔE = -ΔPEᴇᴄᴍ = -ΔKEᴇᴄᴍ = h f This implies that quantum emission is not a purely particle-level event but a restructuring of the underlying mass–potential–frequency field. Significance The ECM Master Equation provides a single unifying axiom for: Mass–energy transformation Quantum transitions Frequency-based dynamics Planck-scale normalization Emergence of observable physical phenomena It reframes physical laws as projections of a deeper frequency-governed transformation process, offering a coherent alternative foundation for classical and quantum descriptions within a unified formalism.