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Let $p\in \mathbb Z$ be an odd prime. We show that the fiber sequence for thecyclotomic trace of the sphere spectrum $\mathbb S$ admits an "eigensplitting"that generalizes known splittings on $K$-theory and $TC$. We identify thesummands in the fiber as the covers of $\mathbb Z_{p}$-Anderson duals ofsummands in the $K(1)$-localized algebraic $K$-theory of $\mathbb Z$. Analogousresults hold for the ring $\mathbb Z$ where we prove that the $K(1)$-localizedfiber sequence is self-dual for $\mathbb Z_{p}$-Anderson duality, with theduality permuting the summands by $i\mapsto p-i$ (indexed mod $p-1$). Weexplain an intrinsic characterization of the summand we call $Z$ in thesplitting $TC(\mathbb Z)^{\wedge}_{p}\simeq j \vee \Sigma j'\vee Z$ in terms ofunits in the $p$-cyclotomic tower of $\mathbb Q_{p}$.

The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum