Lemma: Compact equation representatives

For $\xi ,\zeta \in {\mathfrak{g}}_p$, let $Z_{\xi }=X_{\zeta }+\frac{1}{2}{X_{\xi }}^2$. The completed tests satisfy

All preimages displayed inside $P_m$ are compactly supported. The associated classical relational scalar is invariant modulo ${\kappa }^3$, including on ${{\mathfrak{C}}^{[2]}}_{{\varphi }_0}$.

Proof. The ramp coordinate density transforms naturally on the support of $F-\mathrm{id}$, since $\sigma =g=1$ there, and is unchanged elsewhere. The transformation is the identity near $\Sigma$. Cauchy uniqueness gives $X[q^{\prime }]=X[q]\circ F$ coefficientwise. The inverse-coordinate smearing density therefore obeys $c[q^{\prime }]=F^{♯}c[q]$.

Let $E[q]=P_m+L[q]$ be the positive scalar Euler operator, and let $s={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}e[q]$. Then $E[q]s=c[q]$. Plateau covariance gives

The advanced solution for the transformed source is therefore $s^{\prime }=F^{\ast }s$, by coefficientwise advanced uniqueness. Applying $P_m$ gives

Expand through second order. The generators have compact support, so the indicated preimages are compact smooth functions.

For a smooth free input $u$, pairing with $u$ annihilates these equation tests. The formal inverse coordinates give the same completed scalar functional; alternatively this is the retarded scalar recursion encoded in (eq:completed).