K3 double structures on Enriques surfaces and their smoothings

Let Y be a smooth Enriques surface. A K3 carpet on Y is a double structure on Y with the same invariants as a smooth K3 surface (i.e., regular and with trivial canonical sheaf). The surface Y possesses an étale K3 double cover image. We prove that π can be deformed to a family image of projective embeddings of K3 surfaces and that any projective K3 carpet on Y arises from such a family as the flat limit of smooth, embedded K3 surfaces.