Mathematical Axiomatization of Non-Perturbative Continuum Spacetime via Rough Operator Algebra and Algebraic Reconstruction of Microstructure: Towards the Algebraic Resolution of Closed Timelike Curves (CTC)

The pervasive challenge in modern theoretical physics is not merely the unification of fundamental forces, but the mathematically rigorous evaluation of whether a continuous spacetime background can persist at the quantized microscopic scale. This paper presents a comprehensive mathematical axiomatization and algebraic reconstruction of spacetime, formalizedthrough a 14-framework, 3-stage architecture. By introducing the Universal Rough Operator Algebra (UROA) and Hyper-Resonant Quantum Continuum Field Theory (HR-QCFT), we provide a non-perturbative mechanism to resolve the ultraviolet (UV) divergences inherent in classical Quantum Field Theory, eliminating the reliance on ad-hoc renormalization schemes. Furthermore, this foundation demonstrates that the macroscopic spacetime continuum is an algebraic projection, providing a mathematically viable pathway to the Yang-Mills mass gap and establishing a rigorous framework for causality-preserving topological phase control of Closed Timelike Curves (CTCs).