Relation to density switching

The quadratic functional

has Hessian

not ${\sigma }_{\epsilon }{Q}_w$. Actual switching of a selected density before variation similarly supplies ramp-derivative terms, with details depending on the density representative.

These are classical corrections. They cannot be hidden inside a finite renormalization that starts at $\hslash$.

For $w=K\xi$ with $\xi \in {\mathfrak{g}}_U$, the relevant support lies where ${\sigma }_{\epsilon }=1$. The witness of this section does not occur in X-F's particular plateau variation. Therefore the direct three-tree identity of X-F is not refuted.

What fails is the stronger identification of its entire source-switched field map with the classical part of one variational Bogoliubov map for all permitted backgrounds.