  We begin with an $E_{8}\times E_{8}$ Heterotic model broken to an
$SU(5)_{gauge}$ and a mirror $SU(5)_{gauge}$, where one $SU(5)$ and its
spectrum is identified as the visible sector while the other can be identified
as a hidden mirror world. In both cases we obtain the minimal supersymmetric
standard model spectrum after Wilson-line symmetry-breaking enhanced by a low
energy R-parity enforced by a local (or global) $U(1)_{X}$-symmetry. Using
Heterotic/$F$-theory duality, we show how to eliminate the vector-like exotics
which were obtained in previous constructions. In these constructions, the
Calabi-Yau {[}CY{]} four-fold was defined by an elliptic fibration with section
over a base $B_{3}$ and a GUT surface given by $K3/\mathbb{Z}_{2}=$ Enriques
surface. In the present paper we construct a quotient CY four-fold fibered by
tori with two elliptic structures given by a a pair of sections fibered over
the Enriques surface. Using Heterotic/$F$-theory duality we are able to define
the cohomologies used to derive the massless spectrum.
  Our model for the 'correct' $F$-theory dual of a Heterotic model with
Wilson-line symmetry-breaking builds on prior literature but employs the
stack-theoretic version of the dictionary between the Heterotic semi-stable
$E_{8}$-bundles with Yang-Mills connection and the $dP_{9}$-fibrations used to
construct the $F$-theory dual.
