Syntomic cohomology and real topological cyclic homology

We define the motivic filtrations on real topological Hochschild homology and its companions.

In particular, we prove that real topological cyclic homology admits a natural complete filtration whose graded pieces are equivariant suspensions of syntomic cohomology.

As an application, assuming a real refinement of the Dundas--Goodwillie--McCarthy theorem, we compute the the $RO(\mathbb{Z}/2)$-graded homotopy groups of $\mathrm{KR}(\mathbb{F}_p[x]/x^e;\mathbb{Z}_p)$, and we compute the equivariant slices of $\Sigma^2 \tau_{\geq 1}\mathrm{KR}(\mathbb{Z}/p^n;\mathbb{Z}_p)$.