  The combined work of Guaraco, Hutchinson, Tonegawa and Wickramasekera has
recently produced a new proof of the classical theorem that any closed
Riemannian manifold of dimension $n + 1 \geq 3$ contains a minimal hypersurface
with a singular set of Hausdorff dimension at most $n-7$. This proof avoids the
Almgren--Pitts geometric min-max procedure for the area functional that was
instrumental in the original proof, and is instead based on a considerably
simpler PDE min-max construction of critical points of the Allen--Cahn
functional. Here we prove a spectral lower bound for the hypersurfaces arising
from this construction. This directly implies an upper bound for the Morse
index of the hypersurface in terms of the indices of the critical points,
provided it is two-sided. In particular, two-sided hypersurfaces arising from
Guaraco's construction have Morse index at most $1$. Finally, we point out by
an elementary inductive argument how the regularity of the hypersurface follows
from the corresponding result in the stable case.
