Infrared Isotropy and Propagation Recovery in a Decoupled Fixed 4D Voxel Spacetime

Discrete spacetime models generically break continuous rotational and Lorentz symmetry at the microscopic level. In this work, we study how approximate isotropy and continuum-like propagation can nevertheless emerge in the infrared within a decoupled fixed 4D voxel spacetime. Rather than assuming that Poisson summation or aliasing alone restores exact Lorentz invariance, we introduce an explicit nearest-neighbor lattice toy model and analyze its dispersion relation in three spatial dimensions. For massless propagation, directional differences between axis, face-diagonal, and body-diagonal modes vanish systematically as the dimensionless momentum \(q=ap/\hbar\) approaches zero. In particular, the leading axis-to-body anisotropy scales as \(A_{3D}(q)=q^2/36+O(q^4)\), while the corresponding group velocities converge to the continuum value \(c\) in the infrared. Poisson summation is used as an exact representation of lattice sampling, with coarse-graining treated separately rather than identified with universal exponential suppression. We also introduce residual-displacement bookkeeping as a kinematic interpretation of sub-voxel propagation, without assigning it physical energy. A centered discrete-time relation is examined as a 3+1-dimensional diagnostic, while exact microscopic Lorentz invariance and a complete temporal dynamics remain open problems. These results establish a controlled toy-model mechanism for infrared isotropy and propagation recovery without claiming exact Lorentz symmetry at finite lattice spacing.