  We construct a faithful and conservative functor from prisms to
$C_{p^\infty}$-Tambara functors; in appropriate situations, this gives an
algebraic description of $\underline\pi_0{\mathrm{TC}^-}$. We also present two
integral variants using the generalized $n$-series of Devalapurkar-Misterka.
The construction is based on the "twisted $I$-adic" or "$(p^\bullet)_q$"
filtration, and is closely related to $q$-divided powers. To verify the axioms,
we introduce a new technique for constructing Tambara functors, inspired by
transversal prisms. We apply this to give a conceptual construction of
Molokov's de Rham-Witt comparison map, and generalize it to a triangle
sandwiching prismatic theory between theories built from Witt vectors and
adjunction of $p$-power roots.
