The separate switched-gauge gate

No gauge transformation is needed to construct ${\mathcal{P}}_{\Sigma }$ or ${\tau }_{\Sigma }$. Nevertheless the distinction from an interacting relational realization can be stated directly.

For a compact gauge vector $\xi$ supported strictly after $\Sigma$,

Hence the sharp bare dressing changes by

Suppose a compactly switched forced scalar response depends only on $gh$. Choose $\xi$ constant on a neighborhood of $\mathrm{supp}g$, with compact cutoff to zero outside that neighborhood. Then

while (eq:gaugeB) can be nonzero. The forced response is unchanged, but the unswitched dressing is not.