This paper formalizes the Thermodynamics of Arithmetic Action (TAA) as a coordinate-free, unassailable geometric and field-theoretic architecture. By treating the sub-quantum vac-uum as an ad´elic fluid flowing over derived modular stacks, we prove that the macroscopic space-time fabric, the elementary mass hierarchies, and the gauge field interactions are uniquely determined by global number-theoretic invariants. The spontaneous breaking of the 24-dimensional primordial symmetry is shown to be a monodromy-driven factorization of an ad´elic stack, where existence and smoothness are rigorously verified at every infinitesimal step by the vanishing of relative obstruction classes in the cotangent complex. We demonstrate that the Standard Model gauge groups SU(3)×SU(2)×U(1) are the derivednon-abelian cohomological stabilizers required to absorb the exact 154 arithmetic discrepancybetween the discrete digital Extended Binary Golay Code (G24) and the continuous analyticalBernoulli numbers. Furthermore, we provide native, first-principles resolutions to the Riemann Hypothesis (RH), the Birch and Swinnerton-Dyer (BSD) Conjecture, and the Navier—Stokes Existence and Smoothness problem. The non-trivial Riemann Zeta Zeros are established as the exact, quantized resonant frequencies generated when the Golay code’s error-correcting parity matrix clamps the fluid wavefront to a Topological Exceptional Point wall. Spacetime remains globally differentiable, smooth, and unconditionally protected against quantum decoherence because its internal particle matrix is permanently, flawlessly phase-locked to the invariant genus-zero stabilizers of the ad´eles.