  We prove that on a compact Riemannian manifold of dimension $3$ or higher,
with positive Ricci curvature, the Allen--Cahn min-max scheme (implemented by
the first author and N. Wickramasekera in 2020), with prescribing function
taken to be a non-zero constant $\lambda$, produces an embedded hypersurface of
constant mean curvature $\lambda$ ($\lambda$-CMC). More precisely, we prove
that the interface arising from said min-max contains no even-multiplicity
minimal hypersurface and no quasi-embedded points (both of these occurrences
are in principle possible in the conclusions of the aforementioned work by the
first author and N. Wickramasekera). The immediate geometric corollary is the
existence (in ambient manifolds as above) of embedded, closed $\lambda$-CMC
hypersurfaces (with Morse index $1$) for any prescribed non-zero constant
$\lambda$, with the expected singular set when the ambient dimension is $8$ or
higher.
