Extended Cosserat‑CET Medium: A Unified Continuum Framework for Dark Matter, Dark Energy, and Emergent Gauge Symmetries – Current Status and Numerical Validation

This document presents the current status of the Extended Cosserat‑CET (CET) medium theory. The CET framework postulates that the fundamental substrate of the universe is a three‑dimensional, infinitely divisible continuous medium with independent translational and micro‑rotational degrees of freedom. Two novel dynamical rules – torsion‑stiffness nonlinear coupling and global dissipative‑rebound equilibrium – are introduced. These give rise to: - A natural saturation mechanism that eliminates singularities. - Torsion‑energy halos that act as dark matter. - A residual global elastic stress that mimics dark energy. - Emergent gauge symmetries (e.g., SU(3)) from soliton internal modes. We summarize the governing equations, derive the emergent dark‑matter density profile, and present Bayesian MCMC fits to the SPARC galaxy rotation curve database. The observed Tully‑Fisher slope of ≈2.1 is explained as an environmental softening of the effective rebound coefficient caused by a sub‑soliton ocean, while the pure‑soliton prediction remains 2.5. The radial acceleration relation (RAR) is identified as the most discriminating test of the theory. This preprint also includes the mathematical derivation of the environmental correction, an ideal‑recovery test of the fitting pipeline, and the complete set of axioms and methodology.