the scale at which gravity emerges, while in type II models, the emergent spacetime should probably be free of local observables from the start.

For type I models, the problem of diffeomorphism invariance parallels the decoupling problem for Lorentz invariance. One case is known in which a weak form of diffeomorphism invariance appears [@Girellib]. In this analog model, Nordstr[ø]{}m gravity emerges at lowest order, but conformal invariance of the matter fields makes the flat background metric unobservable, leaving only a background conformal structure. For other models, useful insights may come from the existing body of work on diffeomorphism invariance on a lattice. While some of this work directly addresses emergent models [@Wetterich3], much of it is in the context of lattice regularization of general relativity [@Dittrich0; @Dittrich; @Dittrich2; @Kempf; @Pullin; @Pullin2]. In particular, there are interesting ideas for obtaining an invariant lattice action—a “perfect action”—from a Wilsonian coarse-graining of the continuum [@Dittrich0; @Dittrich; @Dittrich2; @Kempf], which could point to a new type of emergent model.

A new problem arises in models in which the time evolution of the fundamental degrees of freedom depends on data in a finite region, as is the case in lattice models [@Wall]. Consider two disjoint 