  The compactification from the eleven-dimensional Ho\v{r}ava-Witten orbifold
to five-dimensional heterotic M-theory on a Schoen Calabi-Yau threefold is
reviewed, as is the specific $SU(4)$ vector bundle leading to the "heterotic
standard model" in the observable sector. Within the context of strongly
coupled heterotic M-theory, a formalism for consistent hidden-sector bundles
associated with a single line bundle is presented, and a specific line bundle
is introduced as a concrete example. Anomaly cancellation and the associated
bulk space five-branes are discussed in this context, as is the constraint that
the hidden sector bundle be compatible with the slope-stability requirements of
the observable sector $SU(4)$ gauge bundle. The further compactification to a
four-dimensional effective theory on a linearized BPS double domain wall is
then presented to order $\kappa_{11}^{4/3}$. Specifically, the generic
constraints required for anomaly cancellation and the restrictions imposed by
positive squared gauge couplings to order $\kappa_{11}^{4/3}$ are presented in
detail. Three additional constraints are imposed, one guaranteeing that the
$S^{1}/{\mathbb{Z}}_{2}$ orbifold length is sufficiently larger than the
average Calabi-Yau radius, and two enforcing that the hidden sector be
compatible with both the unification mass scale and unified gauge coupling of
the $SO(10)$ group in the observable sector. Finally, the expression for the
Fayet-Iliopoulos term associated with an anomalous $U(1)$ symmetry is presented
and its role in $N=1$ supersymmetry in the low-energy effective theory is
discussed. It is shown that $N=1$ supersymmetry can be preserved by cancelling
the tree-level and genus-one contributions against each another.
