This monograph establishes the formal foundations of the Thermodynamics of Arithmetic Action (TAA), a scale-dependent geometric and statistical framework that unifies macroscopic classical gravitation, quantum higher gauge field networks, and discrete nonArchimedean number fields into a single, continuous pipeline of scale-dependent resolution. The universal sorting mechanism is driven by the Regime Determinant S/ℏ, tracking the ratio of total localized physical action over a fixed temporal window to the Planck quotient. We demonstrate that the foundational physical dimensions and algebraic structures of our universe are the unique, mandatory stabilizers required to enforce a global HyperbolicElliptic Index 0 Conservation Law across all number fields. By mapping the Minkowski spacetime invariant, Fermat’s Last Theorem, Hamiltonian and Lagrangian phase spaces, and the Herbrand—Ribet arithmetic mirror into a rigid 2 × 2 Upper Triangular Transformation Matrix, we derive the emergence of exactly 3D + 1 spacetime, the strict left-handed chiral orientation of gauge forces, the topological confinement of spin-1/2 quarks, and a native, geometric resolution to the Strong CP Problem (θ ≈ 0). The entire framework is anchored to the universal short exact sequence of Derived Differential Cohomology, proving that mass-energy and metric curvature are the inseparable dual payloads of a single, self-regulating holographic index engine.