  We present a counterexample to the Nelson-Seiberg theorem and its extensions.
The model has 4 chiral fields, including one R-charge 2 field and no R-charge 0
filed. Giving generic values of coefficients in the renormalizable
superpotential, there is a supersymmetric vacuum with one complex dimensional
degeneracy. The superpotential equals zero and the R-symmetry is broken
everywhere on the degenerated vacuum. The existence of such a vacuum disagrees
with both the original Nelson-Seiberg theorem and its extensions, and can be
viewed as the consequence of a non-generic R-charge assignment. Such
counterexamples may introduce error to the field counting method for surveying
the string landscape, and are worth further investigations.
