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We construct a faithful and conservative functor from prisms to$C_{p^\infty}$-Tambara functors; in appropriate situations, this gives analgebraic description of $\underline\pi_0{\mathrm{TC}^-}$. We also present twointegral 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 bytransversal prisms. We apply this to give a conceptual construction ofMolokov's de Rham-Witt comparison map, and generalize it to a trianglesandwiching prismatic theory between theories built from Witt vectors andadjunction of $p$-power roots.

Prisms and Tambara functors I: Twisted powers, transversality, and the perfect sandwich