The fiber of the cyclotomic trace for the sphere spectrum

Let $pin mathbb Z$ be an odd prime. We identify the homotopy type of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic $K$-theory of $mathbb Z$. To do this, we prove a spectral version of Tate-Poitou duality for the algebraic $K$-theory spectra of number rings with $p$ inverted. This identifies the homotopy type of the fiber of the cyclotomic trace $K(mathcal O_{F})^{scriptscriptstylewedge}_{p} to TC(mathcal O_{F})^{scriptscriptstylewedge}_{p}$ after taking a suitably connective cover. As a further application, we prove a conjecture of Calegari about the completed cohomology of congruence subgroups.