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We prove that TR is corepresentable by the reduced topological Hochschildhomology 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 ofspectra with Frobenius lifts, refining a result of Blumberg-Mandell. We definethe notion of an integral topological Cartier module using Barwick's formalismof spectral Mackey functors on orbital $\infty$-categories, extending the workof Antieau-Nikolaus in the $p$-typical setting. As an application, we show thatTR evaluated on a connective $\mathbf{E}_1$-ring admits a description in termsof the spectrum of curves on algebraic K-theory generalizing the work ofHesselholt and Betley-Schlichtkrull.

On curves in K-theory and TR