Emergent Lorentz Symmetry in the U-Cell Model: A Wilson RG Attractor with Global Lyapunov Stability

In the Standard Model, Lorentz symmetry is postulated. The U-Cell Model (UCM) derives it as an infrared fixed point of the Wilson renormalisation group. The substrate lattice generates a unique coupling Lcov between matter fields and the substrate displacement. A four-step one-loop RG calculation shows that this coupling drives every matter-field velocity cᵢ toward the substrate sound speed c₀ exponentially: d (cᵢ² - c₀²) /dℓ = -κRG (cᵢ² - c₀²), with κRG = 5c₀/ (108π²a³) > 0. The Lyapunov functional Vᵢ = (cᵢ² - c₀²) ² ≥ 0 satisfies dVᵢ/dℓ ≤ 0 with equality only at cᵢ = c₀, establishing global stability without fine-tuning for all initial conditions cᵢ (0) > 0. The fixed point c₀ is self-consistent (dc₀/dℓ = 0). The attractor holds for all spins s = 0, 1/2, 1, 3/2, 2. Non-perturbative stability is confirmed by the Wetterich functional RG equation with an η_μν-covariant Litim regulator: the coupling ratio α: β = -1: +1 is exactly preserved at all RG scales via O (4) -covariance of the regulated propagators and substrate momentum conservation, without any fine-tuning or order-by-order cancellation. The Lorentz-violating deviation Δᵢ is treated as a perturbative vertex insertion, allowing an unambiguous Wick rotation on the universal c₀ background. The quantisation of the substrate itself remains an open problem.