Parent conversion, derivative orbit, and weighted classes

The class $\mathbb{R}r$ is the complete parent regulator conversion class in Assumption ass:VIID. It is not the span of all derivatives and is not a classification of arbitrary weighted or decorated extensions.

In the normal coordinates of (eq:density), $r$ has transverse scaling degree four. A nonzero pure normal derivative of order $k$ has degree $4+k$. This follows from the dilation of a delta distribution in four normal variables. It is not the test-order loss $64+d$, and it does not restrict arbitrary tangential coefficient functions to a finite-dimensional space.