Unified characterization of the complete hierarchy of topological boundary states in a Floquet crystal

Abstract Topological phases of matter are defined by bulk invariants that dictate the existence of robust boundary states. While conventional bulk-boundary correspondence links a given bulk invariant to a specific type of boundary mode, many systems may host multiple orders of topology simultaneously, including chiral edge states, weak edge states, and higher-order hinge and corner states. A unified framework for predicting and classifying all such boundary phenomena has remained elusive. Here we introduce a compact topological triplet-three complementary one-dimensional winding numbers defined in distinct momentum subspaces-that fully determines the presence and type of boundary states in two-dimensional Floquet crystals. We experimentally implement this concept in a time-synthetic photonic lattice, demonstrating its predictive power. This unified approach integrates strong, weak, and higher-order topology into a single framework, providing critical insight for studying topological matter and enabling systematic control of complex topological phases across a broad range of physical platforms.