Why compact translations do not supply it

For comparison, define

These constant directional derivatives commute: $[T_{\xi },T_{\zeta }]=0$. They are not the representation in (eq:representation).

They also fail equation-ideal preservation. For a compact test $q$,

but

can be a nonzero constant.

Indeed, if a nonzero compact $\xi$ had $\square K\xi =0$, Cauchy uniqueness would give $K\xi =0$. The Killing equation in Minkowski space then gives ${\partial }_{\mu }{\partial }_{\nu }{\xi }_{\rho }=0$, so $\xi$ is affine and cannot be nonzero with compact support. An appropriate compact component test $q$ therefore detects $\square K\xi$ BGP (Chapter 3).

The natural tensor pullback rule is not a remedy on the fixed free sheet. With the convention

it has the ordinary vector-field representation sign, but fails shell descent. At the free configuration $\gamma =\eta$, $b=0$, its metric variation is $-K\xi$, and the same equation test is nonzero. The fixed flat wave operator is not preserved by an arbitrary compact diffeomorphism.

These comparisons do not enter the cocycle calculation.