invariant action with no additional background structures, then the action can depend only on the combination $\eta_{ab}+h_{ab}$. This largely determines the form of the nonlinearities to be those of general relativity, thus fixing the self-coupling.

Diffeomorphism invariance I
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Diffeomorphism invariance is a notoriously slippery concept in general relativity [@Anderson; @Giulini]. The rather heuristic form I will use is the absence of any nondynamical background structure that could define a preferred reference frame. Most type I models have a nondynamical background, so the issue is again one of decoupling. Most type II models do not, but one must show that the emergent spacetime is enough like a smooth manifold for diffeomorphism invariance to make sense at all. Note that while diffeomorphism invariance and Lorentz invariance are conceptually distinct, they are not completely unrelated: in an “already Lorentz invariant” model, the only invariant background structures are a flat metric and a volume element, so the possible forms of diffeomorphism noninvariance are restricted.

As Witten has stressed [@Witten], diffeomorphism invariance also requires the absence of local observables [@Torre; @Hartle]. This presents yet another decoupling problem [@Elvang]: in type I models, all local observables, including any fundamental stress-energy tensor, must be invisible at 