  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 $\mathbb Z^{\wedge}_{p}$-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 $\mathbb
Z^{\wedge}_{p}$-Anderson duality, with the duality permuting the summands by
$i\mapsto 1-i$ (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 $\mathbb
Q^{\wedge}_{p}$.
