  We prove the Borcherds Products Everywhere Theorem, Theorem 6.6, that
constructs holomorphic Borcherds Products from certain Jacobi forms that are
theta blocks without theta denominator. The proof uses generalized valuations
from formal series to partially ordered abelian semigroups of closed convex
sets. We present nine infinite families of paramodular Borcherds Products that
are simultaneously Gritsenko lifts. This is the first appearance of infinite
families with this property in the literature.
