Proposition: Scalar and coordinate Cauchy intertwiners

For $\xi ,\zeta \in {\mathfrak{g}}_U$,

In coefficient form,

The free input ${\varphi }_0$ is unchanged.

Proof. Lemma lem:plateaucov shows that the pulled-back scalar solves the transformed equation. The pullback is the identity near ${\Sigma }_0$, so all its prescribed Cauchy data agree with those of the transformed scalar assignment. Proposition prop:scalar proves equality.

The unswitched massless density equation is natural by the same change-of-variables argument, without the switch discrepancy. Each pulled-back harmonic label has the prescribed initial data. Proposition prop:coordinates therefore identifies it with the transformed coordinate solution. Expansion proves (eq:intertwiner-coeff).