The U-Cell Model

Hier der ergänzte Text — ich füge es als eigenen Punkt ein: UCM Mathematical Companion v10. 0 This document is the Mathematical Companion to the U-Cell Model (UCM), version 10. 0. The UCM models three-dimensional space as an elastic cubic lattice of elementary cells — the U-cells — with lattice spacing a = Planck. From this single substrate, a remarkable range of open problems in physics finds a unified resolution: Gravitation emerges as the macroscopic flow of the substrate lattice toward concentrations of matter — not as spacetime curvature postulated from outside, but derived from the substrate dynamics. Einstein's field equations are derived from the substrate flow via the Gullstrand–Painlevé metric — for static, compressible, and rotating flows — without being postulated. The gravitational constant G = c / (4 ₀ a²) is derived from substrate parameters, not assumed. Dark matter is identified as sub-threshold substrate excitation — collective oscillations below the pressure threshold Pc. The cold fraction fcold = 2 - /2 0. 655 is analytically derived and confirmed against 170 SPARC galaxies to 0. 2%. Lorentz symmetry is not a fundamental postulate but an emergent infrared phenomenon — the long-wavelength limit of the cubic lattice, protected by the RG attractor. A preferred frame exists at the Planck scale and becomes observable at E E_ = 23. 9, eV. The vacuum energy problem is resolved structurally: the zero-point energy does not curve spacetime because a uniform substrate excitation produces no flow gradient. is derived from substrate parameters via the Planck length condition Planck = a. The Kerr metric (rotating black holes) follows exactly from the Doran flow — the transverse component of the substrate velocity field. The path integral measure of quantum field theory acquires a physical foundation: the lattice provides a natural UV cutoff = /a, removing the need for regularisation. A concrete falsifiable experiment is proposed: a photoelectric measurement using a gold foil oriented perpendicular to the Earth's motion through the Milky Way, exploiting the (E/E_) ² coupling to the substrate rest frame. New in v10. 0: Part I — Lorentz Symmetry as an Emergent Phenomenon — makes explicit that the Lorentz attractor follows necessarily from the U-cell geometry and that existing Lorentz-symmetry experiments do not exclude a preferred frame. Full pedagogical expansion of Parts F, G, K, and Q with complete proofs, intuition boxes, and summary tables. 90 pages. Parts A–Q.