Corollary: The fixed-algebra identity passes to the limit

With the free field held fixed,

Thus the limiting completed observable is background-gauge invariant through ${\kappa }^2$ on the unchanged algebra.

Proof. Convergence holds for both background pairs. Use X-C's existing three-term variation on the plateau, where its coordinate and switch defects vanish:

The coordinate and interaction plateau conditions hold for every $\epsilon <\delta$.

Green-operator continuity gives smooth convergence of $w_{\epsilon }$ on the compact generator support. Every positive-order coefficient of $(F^{\ast }-1)w_{\epsilon }$ has that fixed compact support and converges as a compact test. Hence

Explicitly, with $X=X_{\xi }$ and $T_2=X_{\zeta }+\frac{1}{2}{X_{\xi }}^2$,

and

The functions inside $P_m$ are compactly supported. These are explicit representatives in the unchanged equation ideal, so their free-field classes vanish.