Why the regulator is nontrivial

The regulator smooths every original metric input slot before the harmonic problem is solved. Its joint kernels are the smooth parameter families of Section sec:kernels. It does not merely move an initial surface and does not average identical solutions.

The strict identity (eq:positiveenergy) proves that this regularization is nontrivial on an admitted gauge direction. That nontrivial action is exactly what breaks the required right-inverse identity.