Why this does not contradict geometric descent

For the geometric action, the transformation of the first scalar coefficient is

In the witness, the switched retarded assignment instead gives

It therefore fails to implement the scalar part of that action.

Theorem thm:descent says that the complete solution jet transforms correctly. Theorem thm:plateau-obstruction says that a particular switched assignment does not furnish such a transformation from the unchanged free input. These are compatible statements.

The obstruction is not a theorem excluding relational observables in perturbative gravity. It excludes the stated combination of:

1. the fixed unswitched coordinate dressing $w[h]$;

2. the compactly switched response $-G_{\mathrm{R}}{\mathsf{Q}}_{gh}{\varphi }_0$;

3. all gauge parameters in ${\mathfrak{g}}_T$;

4. the representative identification (eq:free-representative-identification).

Changing the admissible gauge support, treating the switch as an additional transforming background, or modifying the interacting relational prescription changes the problem and requires a new proof.