  We study the boundary behavior of any limit-interface arising from a sequence
of general critical points of the Allen-Cahn energy functionals on a smooth
bounded domain. Given any such sequence with uniform energy bounds, we prove
that the limit-interface is a free boundary varifold which is integer
rectifiable up to the boundary. This extends earlier work of Hutchinson and
Tonegawa on the interior regularity of such limit-interface. A key novelty in
our result is that no convexity assumption of the boundary is required and it
is valid even when the limit-interface clusters near the boundary. Moreover,
our arguments are local and thus works in the Riemannian setting. This work
provides the first step towards the regularity theory for the Allen-Cahn
min-max theory for free boundary minimal hypersurfaces, which was developed in
the Almgren-Pitts setting by the first-named author and Zhou.
