Lemma: Boundary-data gauge identity

For every smooth vector field $\xi$,

If

then

In particular,

Proof. Expanding $P({\xi }^{\nu }{\partial }_{\nu }\varphi )$ gives

The last term is zero and the first two equal $-L_{K\xi }\varphi$, by (eq:DK). Subtracting the homogeneous solution with the same initial data gives the zero-data response. The final formula also follows directly from $K^{\ast }{J}_{\mathrm{raw}}=\mathbf{H}$.