Quantum Lattice Model: Quantum Phase Transport and Wave Dynamics

This paper develops the quantum dynamical layer of the Quantum Lattice Model (QLM) by showing how the standard wave equations of quantum mechanics arise as continuum descriptions of coherent phase-action transport on a discrete Planck-scale lattice. The framework is constructed from the per-radian Planck primitives {ℏ, ℓP, tP}, where each lattice tick advances physical phase by one radian and transports one quantum of reduced action. The corresponding phase-action and phase-flow relations are dS = ℏ dθ and E = ℏ dθ/dτ, while the invariant lattice transport identity c = ℓP/tP fixes the maximal causal phase-transport speed of the lattice. Starting from symmetric nearest-neighbor phase transport between Planck cells, the continuum limit yields the relativistic wave operator (1/c²)∂t² − ∇². Introducing intrinsic rest-phase rotation with frequency ω0 = mc²/ℏ produces the Klein-Gordon equation as the continuum description of synchronized spatial phase transport and local rest-phase rotation. A first-order relativistic factorization yields the Dirac equation, while extraction of the rapidly rotating rest-phase carrier produces the nonrelativistic Schrödinger equation governing slow envelope dynamics. Within this framework, the quantum wavefunction is interpreted as a coherent phase-envelope amplitude describing phase-action transport across the lattice. The conserved density ρ = |φ|² arises from the continuity equation of Schrödinger evolution, and the associated current takes the phase-flow form J = (ℏ/m)ρ∇θ. The paper also shows how interaction sectors enter the same transport framework as modular coupling layers. Scalar potentials contribute additional local phase rotation to the slow envelope, while routing-weighted transport modifies the effective capacity of lattice links. Coulomb coupling and weak-field gravitational coupling are presented as examples: the Coulomb potential appears as a scalar phase-rotation term, and gravitational routing modulation of the rest-phase carrier yields the weak-field potential V = mΦ. These results organize the Schrödinger, Klein-Gordon, and Dirac equations as successive continuum limits of deterministic phase-action transport on the Planck lattice, providing the quantum dynamical layer of the Quantum Lattice Model.