Products needing two noncompact legs

Even for smooth $z$, the first derivative ${F_f}^{(1)}[z]$ generally has a noncompact dipole component. In adjoint notation its test profiles include

whose spatial decay is $O(r^{-2})$.

Therefore

requires an extension of the covariance to pairs of such noncompact profiles. Assumption ass:external concerns one compact-tested endpoint and does not supply that extension.

A sufficient additional domain statement is a continuous bilinear covariance pairing on

with cutoff-independent pairings and the native antisymmetric part. This property must be proved for the chosen representative; it is not imposed here as an authenticated fact.

Thus products of two native lapse insertions are left undefined outside a separately verified two-endpoint domain.