  We discuss theoretical issues pertaining to baryonic matter in the combined
heavy-quark and large $N_c$ limits of QCD. Witten's classic argument that
baryons and interacting systems of baryons can be described in a mean-field
approximation with each of the quarks moving in an average potential due to the
remaining quarks is heuristic. It is important to justify this heuristic
description for the case of baryonic matter since systems of interacting
baryons are intrinsically more complicated than single baryons due to the
possibility of hidden color states---states in which the subsystems making up
the entire baryon crystal are not color-singlet nucleons but rather colorful
states coupled together to make a color-singlet state. In this work, we provide
a formal justification of this heuristic prescription. In order to do this, we
start by taking the heavy quark limit, thus effectively reducing the problem to
a many-body quantum mechanical system. This problem can be formulated in terms
of integrals over coherent states, which for this problem are simple Slater
determinants. We show that for the many-body problem, the support region for
these integrals becomes narrow at large $N_c$, yielding an energy which is
well-approximated by a single coherent state---that is a mean-field
description. Corrections to the energy are of relative order $1/N_c$. While
hidden color states are present in the exact state of the heavy quark system,
they only influence the interaction energy at sub-leading order in $1/N_{c}$.
