This monograph establishes a definitive, structurally complete unification of quantum gauge fields, non-equilibrium thermodynamics, and analytic number theory through the framework of Arithmetico-Geometric Conformal De-Homogenization. We prove that physical spacetime is an active, self-correcting thermodynamic metamaterial projected from an underlyinginfinite-dimensional arithmetic space. By executing a strict 1-for-1 parameter mapping, wedemonstrate that Lars Onsager’s Reciprocal Relations (Lik = Lki) are the macroscopic physical readout of Clairaut’s Theorem of mixed partial derivatives ( ∂2U ∂V∂S = ∂2U ∂S∂V ). When the continuum substrate is driven from equilibrium, local coordinate shearing is regularized via a higher-dimensional Kustaanheimo-Stiefel (KS) Fibration Lift, routing excess dissipative energy into the Trivial Zeros of the Riemann Zeta function and the Tate-Shafarevich group, which function as Viscous Hydraulic Dampers locked by the Bernoulli number B10, B12, and the prime691. We formalize this architecture as a Topological Quantum Field Theory (TQFT) Functor(F : CobArith −→ HilbGauge) operating over a Noncommutative Geometry Spectral Triple(A,H,D). The self-adjoint Dirac operator (D) incorporates a topological Coriolis-gauge potential (Aµ) that exerts a restoring force on quantum wave packets, trapping them at the stable L4/L5 Lagrangian nodes of a rotating three-body critical strip. Consequently, the real eigenvalues of the spectral triple align precisely with the imaginary parts (γn) of the non-trivial Riemann Zeta Zeros: Spec(D) = {γn ∈ R | ζ(1/2 + iγn) = 0}. Bound by Adelic ProductClosure (∏︁ p≤∞|σ|p = 1), this framework achieves full structural immunization against decoherence and reconciles the historical schism between String Theory and Loop Quantum Gravity,revealing them as dual asymptotic limits of a single, unbroken, self-stabilizing adelic circuit.