Cutoff, mollifier, and chart issues are different

Proposition prop:cutoff gives eventual exact equality for different cutoffs that equal one near the relevant relative image. It also gives exact compatibility on compact overlaps for the same flat relative coordinates and the same mollifier.

Different presentations of the same internal distribution give identical regularized values for every $\delta$. Thus changing internal adapted charts through valid reconstruction identities does not change the family.

A change of the coordinates in which the crossed distribution is mollified is a different auxiliary construction. General coordinate invariance of that smoothing procedure is not claimed.