Mathematical Companion to the U-Cell Model – Version 7.0

This Mathematical Companion (Version 7. 0, March 2026) presents a comprehensive and self-contained mathematical framework for the U-Cell Model (UCM), a theoretical proposal in which three-dimensional space is modelled as an isotropic elastic cubic lattice. Starting from a discrete harmonic action with no postulated Lorentz symmetry, the document derives eight main results (Parts A–H): (A) emergent Lorentz symmetry as an RG infrared fixed point, (B) a global exponential velocity attractor for all Standard Model fields, (C) emergent gravity and Einstein's field equations via the Gullstrand–Painlevé metric, (D–E) phenomenological connections to dark matter, dark energy, and spatial flatness, (F) non-perturbative stability of the covariant coupling via the Wetterich functional RGE, (G) the complete Ricci tensor for compressible substrate flow, and (H) emergence of U (1) gauge invariance from substrate transversality. The central falsifiable prediction is a Lorentz-violating asymmetry A ≈ 4 × 10⁻⁷ in the photoelectric effect, with A ∝ E²ₖin and a 9% annual modulation of known phase, testable with standard laboratory equipment (estimated cost: 56, 000–115, 000 EUR). Version 7. 0 unifies all parts into a single document and corrects an erroneous angular-integral ratio from v6. 0 (Iαα: Iαβ: Iββ = 1: 1: 5, not 1: 3: 9) ; the main theorems are unaffected.