Lemma: Formal inverse

The map ${\mathsf{X}}_q$ has the unique formal inverse

Proof. Substitute into ${\mathsf{X}}_q({{\mathsf{X}}_{\mathsf{q}}}^{-1}(z))$: the first-order terms cancel, and at second order the Taylor term $-{{\mathsf{X}}_1}^{\nu }{\partial }_{\nu }{\mathsf{X}}_1$ is cancelled by the displayed coefficient. Uniqueness is coefficientwise. No numerical nonperturbative inverse is asserted.