The Lorentz Invariance of Substrate Field Theory: Why Moving Observers Measure c

A persistent critique of substrate or 'lattice' field theories is that they appear to define a preferred reference frame, seemingly violating Lorentz invariance. We address this directly by proving that the Lattice Field Medium (LFM) governing equations—GOV-01: ∂²E/∂t² = c²∇²E − χ²E and GOV-02: ∂²χ/∂t² = c²∇²χ − κE² are mathematically identical to the Klein-Gordon equation, which has been proven Lorentz invariant since 1926. We verify the Klein-Gordon dispersion relation ω² = c²k² + χ² numerically with errors of 0.1–4%. Because observers, clocks, and rulers are necessarily composed of E-χ bound states obeying these same equations, they automatically undergo Lorentz contraction and time dilation. Consequently, all inertial observers measure the speed of light as c, not c ± v. The substrate exists but is undetectable, not by conspiracy, but by mathematical necessity. This resolves the apparent tension between substrate theories and special relativity, demonstrating that LFM is consistent with all experimental tests of Lorentz invariance while providing a causal, deterministic foundation for relativistic physics.