  Counterexample models to the Nelson-Seiberg theorem have been discovered, and
their features have been studied in previous literature. All currently known
counterexamples have generic superpotentials respecting the R-symmetry, and
more R-charge 2 fields than R-charge 0 fields. But they give supersymmetric
vacua with spontaneous R-symmetry breaking, thus violate both the
Nelson-Seiberg theorem and its revisions. This work proves that the other type
of counterexamples do not exist. When there is no R-symmetry, or there are no
more R-charge 2 fields than R-charge 0 fields in models with R-symmetries,
generic superpotentials always give supersymmetric vacua. There exists no
specific arrangement of R-charges or non-R symmetry representations which makes
a counterexample with a supersymmetry breaking vacuum. This nonexistence
theorem contributes to a refined classification of R-symmetric Wess-Zumino
models.
