Assumption: Exact host retention

Retain the following X-D-ST/X-A-ST instance without modification:

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

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

   and its retained compact reserve, disjoint from $\mathcal{W}$;

3. the independently 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 signature and time ledger

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

   the host retains signature $(-,+,+,+)$, and the analysis retains $(+,-,-,-)$;

5. the observation region

   $$\mathcal{O}⋐(0,T_{\ast })\times {\mathbb{R}}^3,D_{\ast }:=J_{\eta }^{-}(\overline{\mathcal{O}})\cap M_{T_{\ast }}⋐\mathcal{L}.$$

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

The regular conical product is not treated as a globally hyperbolic Cauchy spacetime. All global free fields and Green operators in this paper are on $\hat{M}$.