Theorem: The limiting background identity

For $F\in {\mathcal{G}}_p$, generated by $\xi ,\zeta \in {\mathfrak{g}}_p$, the limiting completed field

is background invariant through order ${\kappa }^2$, with the free field held fixed.

More explicitly,

The functions inside $P_m$ are compactly supported.

Proof. For every $\epsilon <\delta$, the generators lie where both $g$ and ${\sigma }_{\epsilon }$ equal one. The coordinate density transforms naturally there, and the transformation is the identity elsewhere. The fixed coordinate Cauchy data are unchanged. Coefficientwise Cauchy uniqueness therefore gives the inherited coordinate intertwiner and ${\Theta }_{F,\epsilon }=0$. The switch defect also vanishes.

Lemma lem:variation becomes

Theorem thm:convergence applies separately to both fixed background pairs. Green continuity gives smooth convergence of ${\mathsf{E}}_{\mathrm{A}}{e}_{\epsilon }$ on the compact generator support. Multiplication and differentiation by $F^{\ast }-1$ then give convergence in one compact-test space. Passing to the limit proves

Expanding (eq:pullbacks) yields (eq:limit-ideal). Their classes vanish in $Q_m$.