Theorem: Prepared witness-response package

Assume the explicit source input (eq:Slambda)–(eq:d1), the reported existence of the real diagonal responses (eq:response), and the compact-interior scalar Green theorem (eq:compactGreen).

For every $\lambda >0$ and $0<\epsilon <T$:

1. $S_{\lambda }$ is past compact relative to compact observation sets but is not a compact-interior source.

2. Real polarization has a unique complex bilinear extension, and its witness value is $U_D[{{\gamma }_1}^{\lambda },{{\gamma }_2}^{\lambda }]=-\mathcal{P}{V}_{\lambda }$.

3. The exact spatial traceless response contraction is (eq:contraction).

4. The scalar completion field is verified on the witness.

5. The coordinate Cauchy problem has the unique smooth solution (eq:Xcand), and the restricted auxiliary extension (eq:restricted-extension) exists.

6. If the inherited retarded coordinate prescription satisfies the hypothesis of Proposition prop:coordinate-compatibility, its direct witness completion equals $I_{\mathrm{cand}}$.

The conclusions require no spatial decay of the witness or of the compared smooth solutions.

Proof. Assertion (1) is Proposition prop:source-domain. Assertion (2) follows from Lemmas lem:realB, lem:BC, and Proposition lem:identify, using the independent uniqueness proof Lemma lem:uniqueness. Assertion (3) is Proposition lem:contraction. Assertions (4) and (5) are Propositions lem:scalar, lem:Xcand, and prop:restricted-extension. Assertion (6) is Proposition prop:coordinate-compatibility.