  We present an $\ell$-adic trace formula for saturated and admissible
dg-categories over a base monoidal dg-category. Moreover, we prove K\"unneth
formulas for dg-category of singularities, and for inertia-invariant vanishing
cycles. As an application, we prove a version of Bloch's Conductor Conjecture
(stated by Spencer Bloch in 1985), under the additional hypothesis that the
monodromy action of the inertia group is unipotent.
