Chronos Spatial Stability: Dispersion Relations for a Coupled Time Field and Density System

This paper extends the Chronos time-field framework by incorporating spatial dependence and Fourier-mode analysis into the coupled time field Θ(t, x) and density field ρ(t, x). Starting from a reaction–diffusion-type equation for ρ and a diffusion–relaxation equation for Θ, the system is linearized around a homogeneous equilibrium, yielding a 2×2 mode matrix M(k) for each wavenumber k. The dispersion relation λ±(k) derived from M(k) determines whether spatial perturbations grow or decay. In the symmetric parameter regime (Dρ = DΘ, κ = σ), the determinant simplifies to (κ + Dk²)² − CG, which is minimized at k = 0. This implies that global stability across all spatial modes is governed by the same Chronos threshold χ < 1 obtained in the homogeneous model. The Chronos constant χ thus controls stability universally, independent of wavelength. A verification protocol is provided to ensure that any researcher or automated reasoning system can independently confirm the linearization, mode matrix, dispersion relation, and Chronos threshold.