Theorem: Supported unreduced Cauchy isomorphism

Evolution and time-zero restriction are continuous inverse linear maps

In particular, every admissible datum determines one supported conical-laboratory solution.

Proof. First evolve $A,B$ componentwise by the wave equation. Then solve the scalar forced equation with data $u,v$. Both are furnished by Lemma lem:evolution.

The field $d_{\nu }=(Dh{)}_{\nu }$ obeys $\square d_{\nu }=0$, with

Both vanish by (eq:constraints), hence $Dh=0$.

Uniqueness holds first for $h$, then for $\phi$. The source $L_h\Phi$ is supported in ${\mathcal{C}}_{r_{\ast }}$. Multiplication by the fixed smooth derivatives of $\Phi$ is continuous in the relevant seminorms. The energy estimates prove continuity of evolution. Restriction and differentiation prove continuity of the inverse.

The zero collar from (eq:margin) gives the claimed conical-laboratory interpretation.