  Soft threshold factorization has been used extensively to study hadronic
collisions. It is derived in the limit where the momentum fractions $x_{a,b}$
of both incoming partons approach $x_{a,b}\to 1$. We present a generalized
threshold factorization theorem for color-singlet processes, which holds in the
weaker limit of only $x_a \to 1$ for generic $x_b$ (or vice versa),
corresponding to the limit of large rapidity but generic invariant mass of the
produced color singlet. It encodes the complete soft and/or collinear singular
structure in the partonic momentum fractions to all orders in perturbation
theory, including in particular flavor-nondiagonal partonic channels at leading
power. It provides a more powerful approximation than the classic soft
threshold limit, capturing a much larger set of contributions. We demonstrate
this explicitly for the Z and Higgs rapidity spectrum to NNLO, and we use it to
predict a nontrivial set of its N3LO contributions. Our factorization theorem
provides the relevant resummation of large-$x$ logarithms in the rapidity
spectrum required for resummation-improved PDF fits. One of our factorization
ingredients is a new beam function closely related to the N-jettiness beam
function. As a byproduct, we identify the correct soft threshold factorization
for rapidity spectra among the differing results in the literature.
