The U-Cell Model (UCM) models three-dimensional space as an isotropic elastic cubic lattice of microscopic units (U-cells). This Mathematical Companion (version 9.0) derives the following results from first principles. (A–K) Substrate lattice, emergent Lorentz symmetry (Wilson RG, exponential attractor), Gullstrand–Painlevé metric, Einstein's field equations, dark matter as sub-threshold excitations (fcold=2−π/2f_\mathrm{cold} = \sqrt{2-\pi/2} fcold=2−π/2, confirmed on 170 SPARC galaxies), falsifiable photoelectric experiment, non-perturbative stability (Wetterich FRGE), full Ricci tensor, Kerr metric via Doran flow. **(Q)** The UCM lattice generates, in the continuum limit, the spacetime required by the Feynman path integral. All three fundamental constants of physics are derived from three substrate parameters (ρ0,a,κc)(\rho_0, a, \kappa_c) (ρ0,a,κc) without free parameters: c=κc/ρ0,G=c212πρ0a2,ℏ=12π ρ0 a4 c.c = \sqrt{\kappa_c/\rho_0}, \quad G = \frac{c^2}{12\pi\rho_0 a^2}, \quad \hbar = 12\pi\,\rho_0\, a^4\, c.c=κc/ρ0,G=12πρ0a2c2,ℏ=12πρ0a4c. The Planck length equals the lattice spacing (ℓPlanck=a\ell_\mathrm{Planck} = a ℓPlanck=a) as a consequence, not a postulate. The vacuum energy problem is resolved: the zero-point energy does not curve spacetime because a uniform, isotropic energy distribution generates no substrate flow gradient, and Lcov\mathcal{L}_\mathrm{cov} Lcov is linear in ∂μua\partial_\mu u^a ∂μua. Scope: The UCM provides the foundation for QFT — the continuum, the path integral measure, ℏ\hbar ℏ, Lorentz symmetry, and the UV cutoff. The field content of the Standard Model is not derived in this version.