  A complete analysis of all heterotic Calabi-Yau compactifications based on
positive two-term monad bundles over favourable complete intersection
Calabi-Yau threefolds is performed. We show that the original data set of about
7000 models contains 91 standard-like models which we describe in detail. A
closer analysis of Wilson-line breaking for these models reveals that none of
them gives rise to precisely the matter field content of the standard model. We
conclude that the entire set of positive two-term monads on complete
intersection Calabi-Yau manifolds is ruled out on phenomenological grounds. We
also take a first step in analyzing the larger class of non-positive monads. In
particular, we construct a supersymmetric heterotic standard model within this
class. This model has the standard model gauge group and an additional
U(1)_{B-L} symmetry, precisely three families of quarks and leptons, one pair
of Higgs doublets and no anti-families or exotics of any kind.
