  We prove that any limit-interface corresponding to a locally uniformly
bounded, locally energy-bounded sequence of stable critical points of the van
der Waals--Cahn--Hilliard energy functionals with perturbation parameter
tending to 0 is supported by an embedded smooth stable minimal hypersurface in
low dimensions and an embedded smooth stable minimal hypersurface away from a
closed singular set of co-dimension at least 7 in general dimensions. This
result was previously known in case the critical points are local minimizers of
energy, in which case the limit-hypersurface is locally area minimizing and its
(normalized) multiplicity is 1 a.e. Our theorem uses earlier work of the first
author establishing stability of the limit-interface as an integral varifold,
and relies on a recent general theorem of the second author for its regularity
conclusions in the presence of higher multiplicity.
