What fixed smooth regularization can prove

Applied to a sufficiently smoothed and spatially cut off $F_{1,f},F_{2,f}$, this lemma gives a regular comparison polynomial conditional on Q1–Q2 data. It does not prove a realization of the original noncompact Hessian, a regulator-removal limit, or a quantum gauge identity.

A fixed smoothing operator generally changes the classical prescription. Under a linear gauge transformation, the regularized linear functional has variation

using (eq:linear-ward). There is no reason for this residual to vanish unless the particular smoothing comparison is proved compatible.

Thus ordinary field-independent smoothing cannot be advertised simultaneously as harmless classical identification and as proved gauge-preserving regularization.