A Phase-Geometric Framework with Fixed Planck Length, Mathematical Subdivided Coordinates, and Cyclic Cosmic Evolution: Extended Classical Mechanics

Abstract

Extended Classical Mechanics (ECM) presents a phase-, frequency-, energy-, and propagation-based framework for describing physical emergence from an initial zero-length potential-energy state. The formulation retains the Planck length ℓₚ = 1.616255 × 10⁻³⁵ m as a fixed physical reference scale, while introducing the mathematical subdivided Planck-length coordinate ℓₚ₍ₓ°₎ = (x° / 360°)ℓₚ, which provides a mathematical representation of phase progression without treating the physical Planck length as phase-dependent. At the initial x° = 0° state, ECM represents the system as an unextended potential-energy state from which finite phase progression can emerge. The first finite phase increment—the Genesis Pixel—is associated with x° = 1°, for which λₚₕₐₛₑ₍₁°₎ = ℓₚ / 360° ≈ 0.00278 ℓₚ. The corresponding proposed threshold condition is Eₚₒₜₑₙₜᵢₐₗ₍₁°₎ ≥ h fₚ₍₁°₎. The origin-frequency decomposition is represented by f₀₍ₓ°₎ = fₚ + Δf₀₍ₓ°₎, fₚ = c / ℓₚ, with the pre-Planck transformation component expressed through the mathematical phase coordinate. During the pre-Planck phase domain 0° < x° < 360°, the phase wavelength is represented by λₚₕₐₛₑ₍ₓ°₎ = ℓₚ₍ₓ°₎ < ℓₚ. The associated ECM propagation state is proposed to be superluminal and approaches the causal limit from above as x° → 360°−: lim₍ₓ°→₃₆₀°⁻₎ v₍ₓ°₎ = c⁺. At the exact 360° state, v₍₃₆₀°₎ = c, while the mathematical phase coordinate undergoes a cyclic reset: λₚₕₐₛₑ₍₃₆₀°₎ = ℓₚ₍₃₆₀°₎ = 0. Thus, the limiting approach to the fixed Planck scale is explicitly distinguished from the subsequent mathematical reset. ECM further formulates the effective mass through Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ) = Mᴍ − Mᵃᵖᵖ, with the zero-effective-mass condition Mᴍ = Mᵃᵖᵖ. This provides an effective-mass interpretation of the balance associated with a zero-gravity state. In this context, Chernin's zero-gravity radius Rzɢ is proposed as an observational bridge to the ECM maximum-radius concept through Rzɢ ≡ rₘₐₓ,ᴜₙᵢᵥ, subject to dynamical derivation and observational testing. The resulting framework connects phase progression, frequency decomposition, wavelength, propagation velocity, energy transformation, effective mass, and cosmological evolution within a common operational description. ECM further proposes a late-time transformation regime in which increasing negative apparent-mass dominance may be associated with progressive dissolution of manifested matter and, in the limiting cyclic interpretation, a return toward the zero-length potential state. The framework therefore provides a testable conceptual connection between sub-Planckian phase dynamics and macroscopic gravitational and cosmological behaviour while maintaining the Planck length as a fixed physical reference scale.