Theorem: Unreduced Cauchy isomorphism

Every $(A,B,u,v)\in {\mathcal{X}}_{\mathrm{C}}$ determines a unique $(h,\phi )\in {\mathcal{S}}_{\mathrm{C}}$:

Evolution and Cauchy restriction are continuous linear inverses for the declared topologies. They are already continuous inverses on each corresponding support stage.

Proof. The metric formula is componentwise wave evolution, with the sine quotient equal to $\tau$ at zero frequency.

Let $d_{\nu }=(\mathsf{D}h{)}_{\nu }$. It obeys ${\square }_{\eta }{d}_{\nu }=0$, with

Both vanish by the constraints. Thus $\mathsf{D}h=0$.

The scalar formula has the specified data and solves its forced equation. Uniqueness holds first for $h$, then for $\phi$.

On a fixed stage, $\mathrm{supp}{L}_h\Phi \subset {\mathcal{C}}_n$. Propagation from the source cone and initial ball keeps the scalar in ${\mathcal{C}}_n$. Conversely, restriction of a stage solution gives data supported in ${\overline{\mathbb{B}}}_{r_n}$, satisfying the constraints.

On each finite interval, multiplication by the fixed derivatives of $\Phi$ maps the smooth seminorms of $h$ continuously to those of $L_h\Phi$. The forced Cauchy estimates prove stagewise continuity of evolution. Restriction and time differentiation prove continuity of its inverse. The universal property of the locally convex inductive limits then gives continuous inverse maps on the unions.