Assumption: Same host, chart, and completion

Retain the X-A-ST instance without modification:

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

2. the supported developing embedding

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

   and the regular compact reserve used by the predecessors;

3. the separate analytical completion

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

4. the signature and time ledger

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

   The host signature remains $(-,+,+,+)$; the analytical equations retain $(+,-,-,-)$;

5. the 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 host's regular part is not used as a globally hyperbolic Cauchy spacetime. All global free fields and Green operators in this manuscript belong to $\hat{M}$.