Theorem: Failure of the naive compact gauge translation

The ordinary affine transformation

preserves the free equation relations only if $\mathcal{B}K\xi =0$. For compact $\xi$ with $K\xi \ne 0$, this condition fails. Thus the full compact geometric gauge action is not supplied as an ordinary translation action on the gauge-fixed free equation quotient.

Proof. Applying the proposed map to $\hat{\Phi }(\mathcal{B}f)=0$ leaves the scalar

It vanishes for every compact $f$ exactly when $\mathcal{B}K\xi =0$.

If that equation held, the compactly supported section $K\xi$ would be a homogeneous solution of $\mathcal{L}$. Its zero Cauchy data before its support force $K\xi =0$, contradicting the hypothesis. Such nontrivial compact descriptors exist, for example time-dependent compact spatial vector fields.