Continuous Manifestation from Cyclic Phase Completion in Extended Classical Mechanics

This supplementary derivation note extends and clarifies the kinetic-energy formalism previously developed in A Comparative Framework for Extended Classical Mechanics' Frequency-Governed Kinetic Energy 10.2139/ssrn.5380919 and Preprints.org 10.20944/preprints202508.1031.v1 and in Emergence of Classical Kinetic Energy from Extended Classical Mechanics 10.13140/RG.2.2.14412.88963. This derivation formalizes a specific ECM interpretation in which observable frequency and kinetic manifestation arise from repeated latent phase-completion cycles. The analysis distinguishes between the latent apparent-mass domain, represented by Mᵃᵖᵖ ≡ −ΔPEᴇᴄᴍ, and the manifested energetic domain represented by ΔMᴍ and KEᴇᴄᴍ. The central result is that although latent phase evolution proceeds through discrete cyclic completion from 0° to 360°, the manifested frequency term hf remains continuous because each completed phase cycle contributes incrementally to observable frequency. The derivation further distinguishes between incremental phase evolution (dφ/dt) and wave-phase velocity (v₍phase₎), thereby establishing formal consistency with de Broglie wave mechanics, where manifested matter remains subluminal while associated phase propagation may be luminal or superluminal. This establishes a mathematically coherent bridge between cyclic phase mechanics, continuous energetic manifestation, and standard wave formalism within ECM.