Proposition: The two maps differ by a specified residual gauge

For every $Q\in \mathcal{Q}$, define a covector wave by

Then

Thus the maps are not generally identical, but their pairings with compact conserved symmetric test tensors agree. Their linearized curvature tensors also agree.

The field $\xi$ is smooth, but is not generally compactly supported. This comparison does not identify it with a gauge parameter in the smaller compact-gauge classes used above.

Proof. Both responses solve the same reduced wave equation, so their trace-reversed difference is homogeneous. For $\square \xi =0$,

At $t_0$, its only nonzero components are ${\overline{K\xi }}_{0i}=-V_i$. Its velocity components are

These are exactly the difference of the two table rows. Wave uniqueness proves (eq:mapdifference). A compact conserved test annihilates $K\xi$ by integration by parts, even when $\xi$ is noncompact. Linearized curvature annihilates every smooth pure gauge.