Spatiotemporal Topological Order F: Three-Pronged Extension and Unified Roadmap for Defects, Higher Dimensions, and Algorithms in Spatiotemporal Topological Order

This paper addresses key bottlenecks in spatiotemporal topological order research, namely the difficulty of experimental verification, the complexity of higherdimensional theories, and the lack of automated tools, by proposing a unified framework for three-pronged extension encompassing defects, higher dimensions, and algorithms. First, we generalize semisimple categories to non-semisimple categorieswith boundaries and defects, establishing an algebraic duality between defect braiding and defect merging, and provide phase signals measurable via quantum interference. Second, we elevate the duality between (2+1)-dimensional topological orderand single-parameter persistent homology to a duality between (3+1)-dimensionalorder and three-parameter persistent homology, proving that the high-dimensionalbarcode merging algebra and the 3-category data of (3+1)-dimensional topological order satisfy the same set of higher-order Verlinde equations. Finally, wedevelop and open-source the “TDA-TopoOrder” algorithm library, implementingan end-to-end automated pipeline from quantum many-body dynamical data totopological order classification. This paper further integrates a ten-year technicalroadmap, specifying verifiable parameters, error thresholds, and toolchains, providing a plug-and-play standardized scheme for quantum simulation and cosmologicalobservation.