A Unified Lagrangian for Open-System Geometrodynamics, Topological Mass Generation, and the Arrow of Time

We present the complete mathematical formulation of an extension of the Standard Model of Cosmology based on Open System Thermodynamics and AdS5 Geometry. Synthesizing the architectural, thermodynamic, and topological sectors into a single Master Equation, we construct the total 5D Action STotal. This unifies: (I) The Bulk Gravity sector containing sterile neutrinos and generating Dark Matter phenomenology via Weyl projection; (II) The Boundary Flux term driving the Arrow of Time and Dark Energy; (III) The Topological Chern-Simons term responsible for Chiral Baryogenesis and Cosmic Birefringence; and (IV) The Brane-Localized Matter sector stabilized by Topological Stiffness. Crucially, we resolve the potential conflict between dynamic dark energy and constant atomic masses by demonstrating that while the Bulk volume expands via hydraulic flux, the fundamental constants are locked by BPS geometric constraints. This single integral equation represents the "Cosmological Genome", proving that the universe is not a static box, but a metabolic, reproductive system driven by the thermodynamic imperative of a Parent Singularity.