Conditional audit of the inherited switch representative

This section concerns the representative formula reproduced from XI-G in XVI-H-v3. It is not needed for the kernel construction, continuity, or witness matching.

The following are inherited operator inputs:

1. The ambient causal propagator is

   $${\mathsf{E}}_{\mu }={{\mathsf{E}}_{\mu }}^{\mathrm{R}}-{{\mathsf{E}}_{\mu }}^{\mathrm{A}}.$$

   Both Green inverses invert $P_{\mu }=\square +{\mu }^2$ on the appropriate compact distributional inputs. Hence ${\mathsf{E}}_{\mu }{P}_{\mu }r=0$ for $r\in {\mathcal{E}}^{\prime }(M)$.

2. The distributional slot convention is

   $$⟨{\mathsf{E}}_{\mu ,i}w,\phi ⟩=-⟨w,{\mathsf{E}}_{\mu ,i}\phi ⟩.$$

   This is the transpose convention for the causal propagator, not an additional sign in ${\mathsf{E}}_{\mu }{P}_{\mu }r=0$.

3. The temporal functions ${\zeta }_i$ are past-to-future switches:

   $${\zeta }_i(t)=0(t\le -T_0),{\zeta }_i(t)=1(t\ge T_0).$$

   They are not compactly supported.

4. The inherited slot calculus supplies the Green identities, causal support, and commuting distinct-slot operations used below for compact kernels.

These are the reproduced interfaces attributed to XI-G, lem:spcompact and lem:kernel. The complete XI-G source is not independently checked here XVIHvThree,XIG.