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 most profound limitation of modern physics and mathematics is not the grand unification of particle forces, but the fundamental question of whether the ”background” of continuous spacetime can rigorously exist in a quantized micro-world. This paper presents a complete mathematical axiomatization and algebraic reconstruction of spacetime utilizing a 14-framework, 3-stage Grand Unification Architecture. By introducing the Universal Rough Operator Algebra (UROA) and Hyper-Resonant Quantum Continuum Field Theory (HR-QCFT), we resolve the ultraviolet (UV) divergences inherent in classical Quantum Field Theory without relying on ad-hoc perturbative renormalization. Furthermore, this theoretical foundation proves that the spacetime continuum is a macroscopic holographic projection, mathematically validating the resolution of the Yang-Mills mass gap and providing a rigorous blueprint for the causality-preserving topological phase control of Closed Timelike Curves (CTCs).