Dependence on the relational prescription

Changing coordinate Cauchy data is not a gauge transformation in the group used above. For example, if $w$ is replaced by $w+a$, where ${\square }_{\eta }a=0$, then at first order

This need not vanish.

The following operations must therefore remain separate:

1. applying a diffeomorphism in the named gauge group;

2. changing a gauge-fixing term used for a perturbative construction;

3. changing coordinate initial data or the relational prescription;

4. changing solution Cauchy data.

Theorem thm:descent concerns only the first operation with the other choices fixed.

Figure fig:VI-prescription-choices separates gauge descent from changes of the relational or dynamical data.