Assumption: Inherited supported presentation

Retain exactly the X-A-ST/X-C-ST host instance:

1. the regular time-oriented conical product $({\mathcal{M}}^{\circ },G)$, its stipulated worldsheet $\mathcal{W}$, and its supplied future relation;

2. the supported injective developing chart

   $$\iota :\mathcal{L}⟶{\mathcal{M}}^{\circ },$$

   whose retained compact reserve stays off $\mathcal{W}$;

3. the separately declared analytical completion

   $$\hat{M}={\mathbb{R}}^{1+3},\eta =\mathrm{diag}(1,-1,-1,-1),M_{T_{\ast }}=[0,T_{\ast }]\times {\mathbb{R}}^3,\Sigma =\{t=0\};$$

4. the inherited signature and time ledger

   $${\iota }^{\ast }G=-\eta ,t=c_{\mathrm{host}}(T-T_{\Sigma }).$$

   The host keeps signature $(-,+,+,+)$, while the analytical equations use $(+,-,-,-)$;

5. a retained observation region $\mathcal{O}⋐(0,T_{\ast })\times {\mathbb{R}}^3$, with

   $$D_{\ast }:=J_{\eta }^{-}(\overline{\mathcal{O}})\cap M_{T_{\ast }}⋐\mathcal{L}.$$

   Compact laboratory tests are transported to $\hat{M}$ and extended by zero.

The regular conical product is not globally hyperbolic. All global Green operators and the free scalar field used below belong to $\hat{M}$, not to that regular product.