Spatiotemporal Topological Order B, From Modular Tensor Categories to Persistent Homology Pairings: Matrix Isomorphism and Monoidal Equivalence

Within the unified framework of spatiotemporal topological order, this paper establishes a rigorous categorical equivalence between the modular tensor categories(MTC) describing spatial topological order and the persistent homology pairingcategories (PH_pair) describing temporal multi-scale structures. We first provide a strict mathematical definition of the persistent homology pairing category,proving it is a finite semisimple braided monoidal category. By constructing aGrothendieck ring isomorphism, we match the fusion coefficients with the mergingcoefficients. Furthermore, we prove that both categories carry projective representations of SL(2,Z), and utilize the (persistent) Verlinde formula to construct anexplicit matrix isomorphism Ψ connecting the modular S-matrix and the persistent pairing matrix. Finally, we lift Ψ to a monoidal equivalence functor, provingit preserves fusion, braiding operations, and higher-order compatibility conditions(pentagon and hexagon equations). This paper verifies the realizability of the theoretical framework through concrete examples, providing a solid categorical foundation for spatiotemporal topological order.