Proposition: Cauchy intertwiners

For the transformed jet (eq:hprime)–(eq:kprime),

The free input ${\varphi }_0$ is the same on both sides.

Proof. By Lemma lem:plateaucov, the right-hand side of (eq:scalarintertwiner) solves the transformed switched equation. Since $F_{\kappa }$ is the identity near ${\Sigma }_0$, its Cauchy data are exactly those of ${\varphi }_0$. Uniqueness in Lemma lem:scalarunique proves the first identity.

For the second identity, each component of ${\mathsf{X}}_{\mathbf{g}}$ is a scalar solution of the unswitched, massless density equation. Ordinary naturality of that equation implies that ${\mathsf{X}}_{\mathbf{g}}\circ F_{\kappa }$ solves the harmonic equation for $F_{\kappa }^{\ast }\mathbf{g}$. Its Cauchy data agree with (eq:harmonicdata), again because $F_{\kappa }$ is the identity near ${\Sigma }_0$. Lemma lem:coordinateunique gives the result.