Lemma: Classical descent is not regulator removal

Classical covariance or descent of a functional does not imply existence of a regulator-removal limit, even when every positive-regulator member is smooth and covariant.

Proof. Let a gauge group act on any nonempty configuration space and let $C$ be the constant scalar field $1$ on a spacetime region. For $\delta >0$, set

Every $Z_{\delta }$ is smooth and gauge invariant. For a compact test $f$ with $\int f{\mathrm{d}}^{4}x\ne 0$,

has no finite limit. Hence covariance does not imply distributional regulator removal.