Two distinct uses of Newton reconstruction

In Section sec:product, $\mathcal{N}$ defines the particular Poisson-dressed symbol being tested. In Lemma lem:closed, it tests whether compact data are in a specified gauge range.

The first construction may have tails. The second permits tails only during reconstruction and proves their disappearance when the membership equation holds. Neither admits noncompact gauge data merely because a Newton integral was written.