Spatiotemporal Topological Order A; Time-Space Dual TQFT and Higher-Order Categorical Syntax

This paper proposes a theoretical framework of ”Spatiotemporal TopologicalOrder” based on time-space dual topological quantum field theory (TQFT) andhigher-order categorical syntax, aiming to provide a unified description of spatialtopological orders (e.g., anyon theory) and temporal topological structures (e.g.,barcodes in persistent homology). By constructing a monoidal equivalence functor, this work rigorously proves the isomorphism between the modular S-matrix oftopological orders and the persistent homology pairing matrix, and demonstratesthat both constitute a unitary representation of SL(2,Z). To validate the theory, a”dual-encoding” lattice model is designed, which compiles higher-order categoricalstructures into local interaction Hamiltonians, and the duality is preliminarily verified via numerical simulations (isomorphism error < 1%). Furthermore, this paperproposes a three-stage verification roadmap from mathematical formalization toquantum simulation experiments, laying the foundation for testing spatiotemporaltopological orders in controllable quantum systems.