Assumption: Future time-ordered lift and comparison

For this section only, assume:

1. A separate symmetric time-ordering map

   $${\mathsf{T}}_2:{\mathcal{D}}_2⟶{{\mathcal{V}}_{\Sigma }}^{[3]}$$

   exists on a specified actual tensor domain ${\mathcal{D}}_2\subset {{\mathcal{P}}_{\Sigma }}^{\otimes 2}$ stable under $\sigma$ and $\gamma$. The comparison domain

   $${\mathcal{E}}_2={\mathcal{D}}_2\cap D_{\mathrm{cmp}}$$

   contains every pair used in this section. Thus the two free orderings and their conjugates all belong to $D_{\mathrm{decl}}$.

2. Every time-ordered coefficient is admitted, respects actual-presentation relations, and agrees with the declared old time-ordered coefficients. Normalization, permitted contact terms, and the claimed continuity strength on named domains are specified. No actual-stage continuity is inferred from inherited F9a.

3. Both ${\mathsf{T}}_2(V,O)$ and $V{\star }_{c}O$ have classical term $VO$. On comparison pairs with

   $${\mathrm{supp}}_{\mathrm{fun}}O\cap J^{+}({\mathrm{supp}}_{\mathrm{fun}}V)=\varnothing ,$$

   causal factorization gives

   $${\mathsf{T}}_2(V,O)=V{\star }_{c}O.$$

4. Free involution reverses order:

   $$(V{\star }_{c}O{)}^{\ast }=O^{\ast }{\star }_c{V}^{\ast }.$$

   The binary anti-time-ordering identity on ${\mathcal{E}}_2$ is

   $${\overline{\mathsf{T}}}_2(F,G)=F{\star }_{c}G+G{\star }_{c}F-{\mathsf{T}}_2(F,G),$$

   and the time-ordering involution is

   $${\mathsf{T}}_2(F,G{)}^{\ast }={\overline{\mathsf{T}}}_2(F^{\ast },G^{\ast }).$$

   Every expression in these equations is defined on the specified domains. Equation (eq:futurefreeadjoint) is inherited; the additional time-ordering identities are not.

5. On a specified old retarded overlap

   $${\mathcal{E}}_{\mathrm{old}}\subset {\mathcal{E}}_2\cap D_{\mathrm{old}},$$

   the old unnormalized retarded convention is exactly $i({\mathsf{T}}_2-V{\star }_{c}O)$, using the unchanged old time-ordered and free coefficients.

The time-ordered existence, factorization, anti-time-ordering, and retarded-overlap hypotheses are not discharged by this manuscript.