Tate-Poitou duality and the fiber of the cyclotomic trace for the sphere spectrum

Let $p\in \mathbb Z$ be an odd prime. We prove a spectral version of Tate-Poitou duality for the algebraic $K$-theory spectra of number rings localized at a finite set $S$ of primes including the divisors of $p$. When $S$ consists of the divisors of $p$, this identifies the homotopy type of the fiber of the cyclotomic trace $K(\mathcal O_{F})^\wedge_p \to TC(\mathcal O_{F})^\wedge_p$ after taking a suitably connective cover. As an application, we identify the homotopy type of the homotopy fiber of the cyclotomic trace for the sphere spectrum at primes that satisfy the Kummer-Vandiver condition.