Definition: Declared flat-completion prescription

Fix a regular developing patch $U\subset M$, and write $D=\mathrm{dev}{|}_U$. The analytic field models are defined on a separately declared auxiliary Minkowski spacetime

All global wave evolutions, Fourier modes, TT projectors, Newton inverses, states, and cutoffs are evaluated on $\tilde{M}$.

A compact scalar test $f_U$ with support in $U$ corresponds to $f=D_{\ast }{f}_U$, extended by zero outside $D(U)$. Tensor tests are transported with their tensor type and the volume pairing. A represented field or response $q$ has the patch interpretation $D^{\ast }q$.