Background-Independent Holonomy Regularizations, Non-Local Scale Tracking, and the Ultraviolet Finiteness of Covariant Quantum Gravity

Theoretical Research Manuscript / Quantum Gravity Unification FrameworkThis paper constructs a self-contained, non-perturbative mathematical framework unifying General Relativity with Quantum Mechanics by establishing the strict ultraviolet (UV) finiteness of the covariant gravitational functional integral. We map the abstract regularized tracking properties of generalized trace-map recurrences onto the peer-recognized structures of background-independent Ashtekar-Barbero connection variables, spin network states, and the discrete spectrum of the loop quantum volume operator. By evaluating the Hamiltonian constraint under a non-local, scale-adaptive tracking filter, we prove that high-frequency quantum curvature fluctuations are exponentially dampened outside the stable Planck-scale coherence domain, eliminating classical coordinate singularities and yielding an anomaly-free, unitary unification of quantum spacetime. Pipeline Disclosure: Core conceptual formulation—substituting the generalized trace-map parameters and synchopeshing potential profiles with the classical frameworks of background-independent Ashtekar-Barbero connection variables, Thiemann regularizations, and loop quantum volume gaps—was fully designed and authorized by the author. Initial structural outline and holonomy matrix structures organized via Grok (xAI); canonical geometric analysis validation, gauge-covariant kernel validation, and production-ready LaTeX typesetting finalized via Gemini (Google).