Proposition: Properties of the companion

1. ${\delta }_{\mathrm{gf}}[s]=-2\mathcal{R}{{\mathsf{E}}_0}^{\mathrm{R}}Ds=K{\zeta }_{\mathrm{gf}}[s]$: the companion is pure gauge.

2. ${\zeta }_{\mathrm{gf}}[s]$ has zero Cauchy data and $\mathrm{supp}{\zeta }_{\mathrm{gf}}[s]\subset J^{+}(\mathrm{supp}s)\cap M_T\subset \{t>\epsilon \}$; it is not in ${\mathfrak{g}}_U$.

3. $D(s+K{\zeta }_{\mathrm{gf}}[s])=0$: ${\zeta }_{\mathrm{gf}}$ is the second-order parameter restoring the de Donder divergence of $s$.

4. The divergence of the source is (eq:divgf); the response satisfies $\square {\zeta }_{\mathrm{gf}}=-Ds$.

Proof. (a) $-{G_{\mathrm{gr}}}^{\mathrm{R}}(2D^{\ast }Ds)=2\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}{D}^{\ast }Ds=2\mathcal{P}{D}^{\ast }{{\mathsf{E}}_0}^{\mathrm{R}}Ds=-2\mathcal{R}{{\mathsf{E}}_0}^{\mathrm{R}}Ds=K(-{{\mathsf{E}}_0}^{\mathrm{R}}Ds)$, using (eq:algebra) and $\mathcal{R}=\frac{1}{2}K$. (b) Zero-data retarded solution of a source supported in $U\subset \{t>\epsilon \}$; its support is not compact in $U$. (c) $D(s+K{\zeta }_{\mathrm{gf}})=Ds+\square {\zeta }_{\mathrm{gf}}=Ds-Ds=0$. (d) is immediate.