  The $N$-representability problem is the problem of determining whether or not
there exists $N$-particle states with some prescribed property. Here we report
an affirmative solution to the fermion $N$-representability problem when both
the density and paramagnetic current density are prescribed. This problem
arises in current-density functional theory and is a generalization of the
well-studied corresponding problem (only the density prescribed) in density
functional theory. Given any density and paramagnetic current density
satisfying a minimal regularity condition (essentially that a von
Weiz\"acker-like the canonical kinetic energy density is locally integrable),
we prove that there exist a corresponding $N$-particle state. We prove this by
constructing an explicit one-particle reduced density matrix in the form of a
position-space kernel, i.e.\ a function of two continuous position variables.
In order to make minimal assumptions, we also address mathematical subtleties
regarding the diagonal of, and how to rigorously extract paramagnetic current
densities from, one-particle reduced density matrices in kernel form.
