A Frobenius-Descent Formalism for the Motivic Filtration on K-Theory of Analytic Adic Stacks

This paper develops a novel framework for studying the algebraic K-theory of analytic adic stacks, a class of geometric objects central to modern p-adic and arithmetic geometry. We introduce a Frobenius-descent formalism rooted in the recent theory of prismatic cohomology developed by Bhatt and Scholze. This formalism allows us to construct a canonical and functorial filtration, which we term the motivic filtration, on the K-theory spectrum of certain well-behaved analytic adic stacks. Our primary result establishes the fundamental properties of this filtration, including its exhaustiveness and convergence under suitable finiteness conditions. We further demonstrate that this filtration is intrinsically connected to the arithmetic of the underlying space by proving its compatibility with the cyclotomic trace map to topological cyclic homology (TC). The graded pieces of our filtration are shown to be related to prismatic cohomology, thereby providing a new structural link between K-theory, which encodes information about vector bundles, and the deep arithmetic data captured by p-adic cohomology theories. As an application, we provide a sample computation for the adic affine line, illustrating how the filtration recovers and refines known structures. This work opens new avenues for investigating arithmetic invariants of p-adic spaces through the lens of stable homotopy theory.