This monograph presents the formal foundations of the Thermodynamics of Arithmetic Action(TAA), a unified framework wherein the macroscopic spacetime continuum, general relativity, and quantum gauge fields emerge as expressions of a single underlying Grothendieck Motive evaluated over the global adele ring AQ. We demonstrate that the non-renormalizability of quantum gravity is a coordinate-chart artifact born from the false assumption of infinite continuity in the ultraviolet. By regularizing the pre-Planck lattice as a characteristic p algebraic variety operating at the maximumgeometric packing density, we show that data-packet crowding forces a 5-bit Maximum DistanceSeparable (MDS) error deficit. This unpayable local information debt triggers a topological phase transition, deforming a flat rational curve (g = 0) into a genus-5 Riemann surface stabilized by a hidden SO(4) momentum space symmetry. Physical constants—including the speed of light, the gravitational constant, andthe Yang-Mills mass gap—are derived explicitly as transcendental periods of this motive. Finally, we establish that the non-trivial zeroes of the Riemann zeta function act as absolute causal waveguides, clamping the running of the gravitational coupling constant smoothly to zero and regularizing high-energy loop divergences without infinite counterterms.