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On the equivalence of two theories of real cyclotomic spectra

We give a new formula for real topological cyclic homology that refines the fiber sequence formula discovered by Nikolaus and Scholze for topological cyclic homology to one involving genuine $C_2$-spectra. To accomplish this, we give a new definition of the $\infty$-category of real cyclotomic spectra that replaces the usage of genuinely equivariant dihedral spectra with the parametrized Tate construction. We then define an $\infty$-categorical version of Høgenhaven's O(2)-orthogonal cyclotomic spectra, construct a forgetful functor relating the two theories, and show that this functor restricts to an equivalence between full subcategories of appropriately bounded-below objects. As an application, we compute the real topological cyclic homology of perfect $\mathbb{F}_p$-algebras for all primes $p$.