Lemma: Plateau covariance of the regulated coordinate operator

For $F\in {\mathcal{G}}_U$ and every formal scalar $\psi$,

Only the condition $\overline{U}\subset \{\epsilon <t<T\}$ is used.

Proof. Write ${\mathfrak{A}}_{\epsilon }[q]=(1-{\sigma }_{\epsilon }){\eta }^{-1}+{\sigma }_{\epsilon }{\mathfrak{A}}_q$. By Lemma lem:group(b) and (eq:nat1),

so $F^{\ast }{\mathfrak{A}}_{\epsilon }[q]-{\mathfrak{A}}_{\epsilon }[F^{\ast }q]=(1-{\sigma }_{\epsilon })(F^{\ast }{\eta }^{-1}-{\eta }^{-1})$. On $U$ the factor $1-{\sigma }_{\epsilon }$ vanishes by (eq:U); outside $U$, $F=\mathrm{id}$ and $DF=\mathrm{id}$, so $F^{\ast }{\eta }^{-1}={\eta }^{-1}$. Hence $F^{\ast }{\mathfrak{A}}_{\epsilon }[q]={\mathfrak{A}}_{\epsilon }[F^{\ast }q]$, and (eq:nat2) gives (eq:plateaucoord).