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