Unified Framework and Engineering Pathways for Time Crystals: From Floquet Phases to Open System Limits, Quasiperiodic Drives, and Topological Protection

Construct unified time crystal theory penetrating closed and open quantum systems, periodic and quasiperiodic drives, and topological constraints. Using group representation and non-equilibrium statistics as foundation, propose operational order parameters, temporal correlations, and spectral criteria; at mathematical level via high-frequency Floquet--Magnus expansion, Lieb--Robinson bounds, and spectral pairing structure, prove rigidity and robustness of prethermal discrete time crystals on exp