Mathematical Appendix: Formal Derivations for the Resonant Lattice Model

⚠️ Notice of Model Update: > The Resonant Lattice Model (RLM) framework presented in this preprint has been officially updated and superseded by the Resonant Vacuum Condensate (RVC) model. To view the complete, unified theoretical framework and its updated cosmological mechanics, please refer to the master RVC publication here: [https://doi.org/10.5281/zenodo.18869120] This document serves as the formal mathematical addendum to the Resonant Lattice Model (RLM). It provides the explicit derivations integrating geometric topological impedance into the Einstein Field Equations via Cartan torsion, and establishes the direct spatial mapping of the 0.481 Gyr cosmic harmonic to the 147.5 Mpc Baryon Acoustic Oscillation (BAO) standard ruler measured by the DESI Collaboration.