This monograph presents a coordinate-free, non-perturbative extension of the Thermodynamics of Arithmetic Action (TAA) framework by proving that the non-trivial zeros of the Riemann zeta function ζ(s) dictate the spectral frequencies of sub-quantum coordinate position tearing. Semiclassically, when energy-momentum compaction forces the ad´elic continuum toward its ultraviolet limits, the local integer rings Op lose unique principal ideal factorization, creating fractional ideal debts that manifest macroscopically as metric resistance and rest mass. We show that the global Wiles modularity isomorphism (R ∼= T ) forces a perfect cross-domain cancellation between continuous temporal deformations and discrete spinorial coordinates via the Kustaanheimo—Stiefel (KS) bilateral phase lock. By evaluating the global Leech lattice theta function ΘΛ24 against the 691 irregular prime firewall, we deliver an unyielding proof that the Riemann Hypothesis (R⌉(s) = 1/2) is the absolute, non-parametric requirement that confines spacetime coordinate tearing to a lossless, hyperbolically orthogonal quantum phase lock, securing first-principles Asymptotic Safety across the space-time continuum.