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

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 $TC(\mathbb{Z}{)}_p^{\wedge }\simeq j\vee \mathrm{\Sigma }{j}^{\mathrm{\prime }}\vee Z$ in terms of units in the $p$-cyclotomic tower of ${\mathbb{Q}}_p$.