TR and the $r$-Nygaard filtered prismatic cohomology

Given an animated ring $S$, we define a filtration on its absolute prismatic cohomology $\mathcal{N}_r^{\geq i} \mathbb{\Delta}_S$, which we call the $r$-Nygaard filtration and study some of its main properties using a mixture of algebraic and homotopy theoretic techniques. This filtration is obtained by suitably gluing $r$-copies of the usual Nygaard filtration and corresponds to the $\xi_r$-adic filtration on $\mathbb{A}_{\mathrm{inf}}$, in the case that $S$ is a perfectoid ring. Using this, we study the motivic filtration of topological restriction homology $\mathrm{TR}^r (S;\mathbb{Z}_p)$ and of its $S^1$-homotopy fixed points. We also pursue connections with the theory of topological cyclic homology. Finally we discuss connections with the de Rham--Witt complex, towards a prismatic - de Rham--Witt comparison theorem.