Dispersion, Not Resonance: Stencil Geometry, Transient Coupling, and Spectral Stability in a χ-Modulated 3D Lattice Field System

This paper presents a systematic numerical and analytical investigation of a coupled χ–Ψ lattice field system motivated by the GOV-01/GOV-02 framework (Emergent Physics Lab). Using an isotropic 27-point stencil with zero-sum central coefficient −7/3, Störmer–Verlet integration, and over 350,000 integration steps across multiple configurations, we establish three principal results: (1) the 27-point stencil provides structural stability through enhanced dispersion channels that the 19-point stencil lacks, with the latter diverging catastrophically at step ~26,500; (2) the system exhibits a two-phase dynamical structure — a transient coupling phase (Pearson r = 0.71) followed by steady-state dynamical decoupling (r = 0.23) with characteristic timescale ~105 time units; (3) systematic coefficient sweeps using time-averaged peak intensities — both with and without Ginzburg–Landau saturation — reveal a monotonically increasing stability landscape, disconfirming the hypothesis that √3 ≈ 1.732 represents a geometric resonance constant. A key methodological finding is that single-snapshot measurements produce spurious non-monotonic structure in parameter sweeps due to slow modulation oscillations; time-averaging is essential. Rigorous theorems establish χ-field homogenization and asymptotic reduction to a renormalized discrete nonlinear Schrödinger equation governed by the spectral gap of the discrete Laplacian.