Configurations, labels, and orders

Let $s$ be a dimensionless formal perturbation parameter. It is not automatically an independently normalized gravitational coupling. Freeze the factorial convention

All tensor components are retained.

The classical coefficients are smooth and, on each compact time interval used in the calculation, have a common compact spatial support. They are not required to solve the perturbed Euler equations. This is an off-shell classical observable prescription on an on-shell background.

Choose a compact label-time interval $J⋐I$, and perform the construction on a slightly larger interval $J_{+}⋐I$. This leaves room for the formal time displacement.

The external label coordinates are $(\tau ,\mathbf{x})$. The clock calibration is fixed by

The raw lapse is calibrated to the matter scalar ${\varphi }_s$, not to a Euclidean norm and not to a separately rescaled clock:

The positive timelike branch is used.

Formal square roots and inverses are well defined around the positive background coefficient. No assertion of a nonlinear global coordinate chart for arbitrary large perturbations is needed.

The proposed quantum target is the coefficient range

A formal truncation is not a numerical error estimate. The classical local expressions below use at most second derivatives of the elementary fields before the displayed spatial inverses. A proposed local counterterm search may be restricted to four derivatives, but such a restriction is not declared exhaustive for this nonlocal observable.