Sharp Two-Sided Causal Cones for Lattice Klein--Gordon Dynamics

We determine matching constructive inner and operator-norm outer propagation bounds for a free nearest-neighbor lattice Klein--Gordon model. Its positive-frequency group velocity is at most one. A finite-range Hermitian Clifford lift squares to the scalar lattice symbol and has no additional massless nodes. For every spatial direction we construct positive-energy wave packets with macroscopic ballistic motion and position error \(O(a^{2/5})\), uniformly at fixed times. The exponent is optimal within the stated diffraction--discretization error balance. Complexifying the finite-range symbol yields an exact directional growth function and a Legendre-transform leakage rate; a direction-independent estimate suppresses propagation outside the relativistic cone. Combining both sides gives a threshold-arrival law with radial error \(O(a^{2/5})\) and an explicit early-time leakage estimate. The result is a finite-regulator free-field theorem under the stated packet and measurement conventions; it does not establish many-body microcausality or nonlinear dynamics in curved spacetime.