Emergent Special Relativity and Gravity from a Discrete Substrate: The U-Cell Model

We present the U-cell model, in which matter fields are coupled to and carried by a discrete, non-relativistic elastic substrate, and Special Relativity emerges as the dynamically stable infrared fixed point of the renormalization group flow. We prove that a unique covariant coupling drives all field velocities exponentially toward the substrate wave speed c0 with a positive Lyapunov function — global, self-consistent, and quantum-stable. Gravity emerges from the longitudinal substrate flow via the Gullstrand-Painlevé metric; Einstein's field equations and the exact prefactor 8piG/c⁴ are derived from substrate compressibility. The Lorentz contraction of U-cells screens lattice granularity from high-energy probes, giving cTZ ~ 10^-49 for LHC electrons — consistent with all bounds. The model makes a single falsifiable prediction: a Lorentz-violating asymmetry cTZ ~ (vᵢn/c) (Eₖin/ELambda) ² in photoelectron emission, with annual modulation ~9% at a predicted galactic phase, accessible to a precision experiment with slow electrons.