Lemma: Fixed-support closed-range test

For $x=(A,B,u,v)\in {\mathcal{X}}_{r_{\ast }}$,

When the equality holds, every reconstructed descriptor is supported in ${\overline{B}}_{r_{\ast }}$. The range $\mathfrak{g}({\mathcal{G}}_{r_{\ast }})$ is closed.

Proof. For true compact gauge data, (eq:gauge-data) gives

Newton inversion recovers $a$, and the remaining equations recover $b$.

Conversely suppose the stated equality holds. All Newton potentials in (eq:reconstruction) decay at infinity. Outside ${\overline{B}}_{r_{\ast }}$, $A_{ij}=0$, so

Differentiation and index permutation give

The exterior of a ball is connected. Thus the spatial $a_i$ are affine there, and decay makes them zero.

Also $b_0=0$ outside the ball, and $A_{0i}=0$ gives $b_i=-{\partial }_i{a}_0$. The equation $B_{ij}=0$ then gives ${\partial }_i{\partial }_j{a}_0=0$ there. Again decay forces $a_0=0$, and hence $b_i=0$. The scalar equations in the assumed equality provide exactly (eq:scalar-data).

Finally, Newton convolution from a fixed compact smooth support stage into $C^{\infty }({\mathbb{R}}^3)$, with compact-open topology, is continuous. Derivatives can be transferred to the source; the Newton kernel is locally integrable. Therefore

is a continuous map into a finite product of Hausdorff smooth spaces. Its kernel is precisely the gauge range, so that range is closed.