Proposition: Regular ordering correction

For an admissible smooth change $d=H^{\prime }-H$,

Every component of ${\ell }_d$ is compact and smooth. Only admissible metric–metric and metric–scalar contractions can contribute.

Proof. The second functional derivative of a cubic contributes $3\cdot 2$; the ordering generator contains $\hslash /2$. No higher even derivative remains.

A nonzero wavefront point of the projected linear kernel would require a tuple $(0,0,p)$ in the cubic wavefront set. The sum-zero bound forces $p=0$, a contradiction. Compactness follows from proper projection of the compact cubic support. There is only one scalar slot, and no odd species, so the stated pair types exhaust the possibilities.