Proposition: Same-patch diffeomorphism and gauge invariance

For every $f,h\in {\mathcal{H}}_{\Gamma }$,

Proof. The transformed extension is

It has trace $f$: the transformation is the identity on $\Gamma$, and a boundary diffeomorphism fixing $\Gamma$ maps its complement to itself. The connection derivative transforms as

Unitarity and change of variables prove equality of energies and pairings. The weak harmonic equations and uniqueness in Lemma lem:DNwell identify the transformed solutions.