Scope and the corrections required for a coherent sheet

The objective is to give a classical action and its identities before constructing any quantum metric. The relational prescription of X-A is retained, but no quantum conclusion is inferred from its classical covariance.

A production plan cannot require both a background de Donder term and an unbroken diffeomorphism identity for the resulting action. It also cannot identify a quadratic gravitational action with the nonlinear Einstein action at the next two coupling orders. Accordingly, the model below has one declared gauge-fixed action, together with its explicitly identified invariant part:

The identities for these two functionals are different and are printed separately.

Five corrections determine the release.

1. The Einstein–Hilbert action is specified before expansion. Its quadratic coefficient is Fierz–Pauli. Its cubic and quartic coefficients are retained in the classical sheet when required by coupling order. They are not evaluated as graphs.

2. The scalar action contains no metric derivatives. Its metric variation therefore contains $g$ but no $\partial g$, including at second order. Adding such terms would change the stated minimally coupled model.

3. The nonlinear Noether identity uses the formal adjoint of the metric pullback action. A bare partial divergence is only its leading flat contribution.

4. Covariance of a metric Euler derivative means covariance as a rank-two tensor density. Gauge fixing contributes a noncovariant background-dependent term.

5. An empty jet must have a specified input. The zero scalar and flat metric provide an immediate counterexample to universal emptiness.

The broader perturbative treatment of gravity distinguishes classical gauge symmetry, gauge fixing, and subsequent BV quantization BFR. Here those distinctions are enforced without opening a BV complex. Effective-field-theory discussions likewise do not justify deleting the classical Einstein self-interactions Donoghue.