The evaluated sign check

Take

in a sufficiently large causal cylinder, with $a\ge 0$, $a\not\equiv 0$, compactly supported above the ramp. A compact spatial cutoff and plateau make $\xi$ admissible without changing the calculation.

For $r={\mathcal{L}}_{\xi }\gamma$,

Hence

and

For a nonnegative observation after the support of $a$,

The held-fixed change contributes $-I$; the restored first coefficient contributes $+I$. Their sum is zero.

The same tensor $r$ is not generally free. In the cylinder its spatial wave operator contains $-a^{\prime \prime }{\delta }_{ij}$, which is not identically zero for a nonzero compact bump. Thus the co-transformation is a legitimate off-shell coefficient operation, not an automorphism of XI-B's free solution algebra.