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Abstract

The St\v{r}eda formula establishes a fundamental connection between the topological invariants characterizing the bulk of topological matter and the presence of gapless edge modes. In this work, we extend the St\v{r}eda formula to periodically driven systems, providing a rigorous framework to elucidate the unconventional bulk-boundary correspondence of Floquet systems, while offering a link between Floquet winding numbers and tractable response functions. Using the Sambe representation of periodically driven systems, we analyze the response of the unbounded Floquet density of states to a magnetic perturbation. This Floquet-St\v{r}eda response is regularized through Ces\`aro summation, yielding a well-defined, quantized result within spectral gaps. The response features two physically distinct contributions: a quantized charge flow between edge and bulk, and an anomalous energy flow between the system and the drive, offering new insight into the nature of anomalous edge states. This result rigorously connects Floquet winding numbers to the orbital magnetization density of Floquet states and holds broadly, from clean to disordered and inhomogeneous systems. This is further supported by providing a real-space formulation of the Floquet-St\v{r}eda response, which introduces a local topological marker suited for periodically driven settings. In translationally-invariant systems, the framework yields a remarkably simple expression for Floquet winding numbers involving geometric properties of Floquet-Bloch bands. A concrete experimental protocol is proposed to extract the Floquet-St\v{r}eda response via particle-density measurements in systems coupled to engineered baths. Finally, by expressing the topological invariants through the magnetic response of the Floquet density of states, this approach opens a promising route toward the topological characterization of interacting driven phases.