On curves in K-theory and TR

We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line $\mathbf{S}[t]$ as a functor defined on the $\infty$-category of cyclotomic spectra with values in the $\infty$-category of spectra with Frobenius lifts, refining a result of Blumberg-Mandell. We define the notion of an integral topological Cartier module using Barwick's formalism of spectral Mackey functors on orbital $\infty$-categories, extending the work of Antieau-Nikolaus in the $p$-typical setting. As an application, we show that TR evaluated on a connective $\mathbf{E}_1$-ring admits a description in terms of the spectrum of curves on algebraic K-theory generalizing the work of Hesselholt and Betley-Schlichtkrull.