Proposition: The witness fits the retained regular reserve

The spatial support in (eq:witnessdata) can be chosen so that $\mathrm{supp}w⋐\mathcal{L}\cap \{0<t<\epsilon \}$, inside the retained causal reserve. Moreover $u,v$ can be replaced by smooth free solutions with compact Cauchy data without changing (eq:positive).

Proof. Choose an event $(t_o,{\mathbf{x}}_o)\in \mathcal{O}$. By (eq:separation), $t_o>2\delta >2\epsilon$. A sufficiently small spatial ball $B$ around ${\mathbf{x}}_o$ has

Choose ${\chi }_0$ supported in $B$. The temporal support of $\psi$ is strictly inside $(0,\epsilon )$, so $\mathrm{supp}w$ is a compact subset of the regular chart.

For the second assertion, take a compact neighbourhood of $\mathrm{supp}w$ and its initial causal shadow. On $\Sigma$, choose smooth compact Cauchy data equal to those of $u$, respectively $v$, on a ball containing that shadow. Flat finite propagation makes the resulting free solutions equal to $u,v$ near $\mathrm{supp}w$. The local operator $Q_w$ is supported there, so the pairing is unchanged.

For either replaced metric solution $r$, set $b=\mathsf{C}r/2$ and take scalar and ghost fields zero. Equation (eq:operators) shows that this is a solution of the full free multiplet.