Analytic cyclic homology in positive characteristic

Let $V$ be a complete discrete valuation ring with residue field $\mathbb{F}$. We define a cyclic homology theory for algebras over $\mathbb{F}$, by lifting them to free algebras over $V$, which we enlarge to tube algebras and complete suitably. We show that this theory may be computed using any pro-dagger algebra lifting of an $\mathbb{F}$-algebra. We show that our theory is polynomially homotopy invariant, excisive, and matricially stable.