Proof: Proof of Theorem thm:main , items (i)–(v)

Proof. Item (i) is Proposition prop:coordeq; (ii) is Proposition prop:preflight; (iii) is Theorem thm:coordinate.

For (iv), formal inversion of (eq:intertwiner) gives ${{\mathsf{X}}_{{\mathsf{F}}^{\ast }\mathsf{q}}}^{-1}=F^{-1}\circ {{\mathsf{X}}_{\mathsf{q}}}^{-1}(\mathrm{mod}{\kappa }^3)$. With $F^{\ast }\varphi =\varphi \circ F$,

pointwise in the label $z$. Smearing with the fixed $f$ gives (iv). A direct coefficient verification of this cancellation, with all quadratic gauge terms retained, is in Appendix app:direct.

For (v), Proposition prop:scalar and (iv) give