Corollary: The unchanged tree identity is not refuted

In the frozen admitted tree carrier, at ${\hslash }^0$,

This statement and Theorem thm:obstruction are compatible.

Proof. For the first metric coefficient $h+r$, a complete second-slot transformation would contain $k^{\prime }+{{\mathcal{L}}^{\mathrm{cov}}}_{\xi }r$. The affine identity in (eq:tests) and (eq:scalaridentity) therefore leave the scalar-completed expression with defect $-{\kappa }^2{\hat{A}}_1[{{\mathcal{L}}^{\mathrm{cov}}}_{\xi }\mathbf{r}]$.

Equation (eq:metricjet) makes the mixed tree vary by $+{\kappa }^2{\hat{A}}_1[{{\mathcal{L}}^{\mathrm{cov}}}_{\xi }\mathbf{r}]$, modulo the explicit graviton-slot equation image. The matter and self trees are background independent and inert. These are exactly the inherited cancelling terms, not new copies of them.

The obstruction instead chooses the background $K{\xi }_{\mathrm{on}}$ inside the onset layer and requires agreement of the entire metric field map on every free input. That background is not a plateau transformation. The two claims have different quantifiers.