A Cascade-Based Momentum-Energy Transport Equation for the Planck and Primordial Universe

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We develop a phenomenological cascade model for the evolution of momentum and energy during the earliest stages of the Universe. The starting point is a momentum transport equation of the form ∂ P i ∂t + H c x j ∂ P i ∂ x j =− ∂E ∂ x i , (1)where the second term represents the effect of cosmological expansion. The momentum and energy are assumed to obey cascade scaling relations P=D k −r/w , E=C f −a/b . (2)A modified uncertainty construction is introduced through ΔP Δx=εpλ , ΔE Δt=εET . (3)Under characteristic-scale estimates, these relations lead to ΔP Δt ~ ΔE Δx , (4)and, for relativistic modes, to the familiar energy-momentum relation E≃cP . When the physical wavenumber evolves as k phys ∝ R −1 in an expanding universe, the momentum cascade gives P∝ R r/w . If the characteristic frequency also scales as f∝ R −1 , the energy cascade gives E∝ R a/b . Consistency with relativistic propagation then requires the exponent relation a/b =r/w . At the Planck scale, the proposed transport equation becomes naturally dimensionless when normalized by the Planck length, time, energy, and momentum. The resulting framework provides a possible phenomenological connection between cascade dynamics, cosmological expansion, and Planck-scale uncertainty.