  We study the stability of heterotic compactifications described by (0,2)
gauged linear sigma models with respect to worldsheet instanton corrections to
the space-time superpotential following the work of Beasley and Witten. We show
that generic models elude the vanishing theorem proved there, and may not
determine supersymmetric heterotic vacua. We then construct a subclass of
linear models for which a vanishing theorem holds, generating an extensive list
of consistent heterotic backgrounds.
