Dark Energy as Repulsive-Sector Population, Dark Matter as Inverse-Vieta Transit, and the Derivation of αd from Torsion Unification

The provided text introduces from the Brahim Framework, an arithmetic-physical model that derives major cosmological constants and dark-sector properties from first-principles structural primitives like the golden ratio and seed integers. By analyzing the dynamics of a 2112-state bipartite manifold under a specific mathematical operator, the researcher calculates precise values for dark energy, dark matter, and the Hubble ratio that closely align with established observational data. The framework identifies dark matter as a stable particle with a mass of 5.35 GeV and proposes that it must be asymmetric to remain consistent with current indirect-detection limits. Furthermore, the paper explains the Hubble tension as a structural shift in cosmological frames rather than an experimental error. These findings are supported by a detailed mass tower and self-interaction cross-sections that categorize the dark sector as effectively collisionless. Ultimately, the work suggests that the same mathematical torsion governing visible particle charges also dictates the large-scale density of the universe.