Causal Signal Propagation on a Static Discrete Lattice

Continuous space is an extremely successful macroscopic description of physical propagation, but it does not by itself establish that spatial continuity is fundamental. This paper investigates a discrete alternative in which the underlying substrate is a static lattice of fixed nodes and physical propagation is represented by the evolution of a dynamical statedefined on those nodes. The central question is whether an approximately isotropic spherical causal front can emerge from local, bidirectional node-to-node coupling. Simple-cubic (SC), body-centered-cubic (BCC), face-centered-cubic (FCC), and hexagonal-close-packed (HCP) geometries are considered as candidate substrates. A graph-Laplacian wave equation is introduced, and its dispersion relation is derived. The long-wavelength limit is shown to recover an isotropic continuum wave equation when the second moment of the neighbour vectors is proportional to the identity. Higher-order terms retain a microscopic lattice signature and therefore provide a quantitative measure of residual anisotropy. The paper further develops a shell-population picture in which the number of participating nodes on an expanding causal front scales as an area, while conservation of the lattice-wave energy permits the amplitude of a spherical signal to decrease approximately as 1/r. FCC and HCP are identified as especially interesting candidates because both are close-packed and have coordination number 12, while previous numerical-wave literature independently shows that FCC discretizations can substantially reduce directional numerical dispersion relative to Cartesian grids. The proposed framework is deliberately presented as a hypothesis and mathematical research program rather than as evidence that physical spacetime has already been established to possess any particular lattice geometry. A reproducible numerical program is specified for testing localized pulses, wavefront anisotropy, energy conservation, angular resolution, and convergence toward the macroscopic causal relation R = ct.