The quadratic normalization

two rejected replacements

Replacing the full Einstein action by the Fierz–Pauli coefficient does not preserve the nonlinear Noether identity. The fixed gauge-fixing term is not invariant under ${\mathcal{G}}_P$.

Proof. For the first assertion, XI-A's order-one vacuum identity contains

For a tensor equal to $\psi (t)\eta$ near a point,

Deleting $H_2$ therefore leaves a generally nonzero term.

For the second assertion, take the same compact spatial parameter used in XI-A, $\xi =f{\partial }_1$, $0\ne f\in {C_c}^{\infty }(U)$. Then

A nonzero spacetime-compact homogeneous wave is excluded by Cauchy uniqueness, so $\square f\not\equiv 0$. These are the existing XI-A checks, now inside the retained regular chart CheungXIA.

No gauge algebra is recovered by imposing both compact spacetime support and $\square \xi =0$: that restriction would leave only the zero parameter.

tensor-density covariance

For an orientation-preserving diffeomorphism $F$, define

For $F\in {\mathcal{G}}_P$, put $q^{\prime }=F^{\ast }q$, ${\varphi }^{\prime }=F^{\ast }\varphi$, and ${\gamma }^{\prime }=(q^{\prime }-\eta )/\kappa$. Then

The formal identity retains its coefficientwise meaning through order ${\kappa }^2$ when projection is performed after the required coefficients have been retained. It is not valid in general with $e+e_{\mathrm{gf}}$ in place of $e$.

For a compact near-identity flow not required to preserve $g$, its metric covariance defect is

Proof. This is XI-A :covariance and :covariance-defect CheungXIA. The two tensor densities in (eq:eexact) are natural. Their switched matter terms differ only by $g-F^{\ast }g$, giving (eq:covariance-defect). The plateau support gives $F^{\ast }g=g$. The explicit gauge-fixing operator (eq:egf) retains the fixed background and fails the corresponding assertion, as already diagnosed in Restatement rest:rejections. Lemma lem:transfer supplies the local host reading.

The infinitesimal covariance defect involves ${\xi }^{\mu }{\partial }_{\mu }g$. The source-divergence defect involves $({\partial }_{\mu }g){T_0}^{\mu \nu }$. They are related background-switch effects, not identical formulas.