An unswitched sourced model as a proposed next problem

For a concrete source convention, take the minimally coupled scalar action together with Einstein–Hilbert coefficient $2/{\kappa }^2$. Write

where

Then the Einstein equation is

This normalization is compatible with the ADM subsection. It has no role in the preceding quotient proofs.

Set

The invariant order-${\kappa }^2$ metric equation is

where $G^{(2)}[h,h]$ is defined as the coefficient of ${\kappa }^2$ in $G(\eta +\kappa h)$. The complete successor also includes the scalar equation through that order, its gauge action, and its constraints.

By the linearized Bianchi identity, a necessary compatibility condition for (eq:sourced-successor) is

This must be checked for the actual model. For the unswitched minimal scalar,

This elementary conservation fact is not a sourced Cauchy-existence, constraint-solving, or quotient theorem. Switching the action or source changes the compatibility audit.