The actual type of a residual metric test

XIV-G's tested source uses

Its local operator representation is

The coefficients are smooth and supported in $\mathrm{supp}G$. These are the source's local differential-operator statements. They retain the actual source coefficients and ramp derivatives within that copied operator statement.

The test in (eq:Gtest) is a metric direction on $Z$. It is not a compact test section on all of $W$. Pulling it back by the projection to $Z$ gives a smooth object depending only on $z$, but does not generally give compact support in $a,b,y$. Moreover, the mixed residual output also retains a scalar slot at $y$. The metric direction alone therefore cannot be declared to be the full test space for the output coefficient.

This matters for duality. A vector-valued distribution with values in a bundle $E$ pairs with compact sections of the dual bundle, with an appropriate density. It does not, without a chosen fiber pairing, pair canonically with sections of the same bundle. A metric on $E$ could identify those two spaces, but no such metric on an output bundle is supplied by XIV-G.

The input metric bundle itself is not rank-mysterious. In four spacetime dimensions a symmetric covariant two-tensor has ten components. What is not identified in XIV-G is a separate bundle called $E_{\mathrm{res}}$ which packages the output residual species with the required value/test duality. The two questions must not be confused.