This document presents the complete mathematical summary of the Dark Geometry (DG) theoretical framework, which unifies dark matter and dark energy as manifestations of the conformal mode of spacetime the Dark Boson. The model is built upon the Hertault Axiom and comes in three equivalent formulations: IDG (Informational Dark Geometry),QGU (Quantum Gravity Unication), and HDG (Holographic Dark Geometry). All parameters are derived from first principles, resulting in a zero free parameter model that simultaneously resolves the H0 and σ8 tensions.