Requirement: Time-ordered interface I TO

A proposed bosonic time-ordering system must specify:

1. Actual-functional domains

   $${\mathcal{D}}_2\subset {{\mathcal{P}}_{\Sigma }}^{\otimes 2},{\mathcal{D}}_3\subset {{\mathcal{P}}_{\Sigma }}^{\otimes 3},$$

   the coefficient range and truncation, symmetry, normalization ${\mathsf{T}}_1(F)=F$, and classical order-zero multiplication. Every required reversed, conjugate, or intermediate domain must be named.

2. Admission and actual-presentation descent for every coefficient and required directional descendant.

3. Causal factorization:

   $${\mathsf{T}}_2(F,G)=F{\star }_{c}G\text{if }F\gtrsim G,$$

   where $F\otimes G$ belongs both to the time-ordering domain and to $D_{\mathrm{decl}}$. The comparator is specifically the IX-Free-2 product. Outside $D_{\mathrm{decl}}$, this identity requires a separately constructed free extension. Higher factorization must name every intermediate free product used.

4. Exact declared old time-ordered overlaps and a specified class of permitted contact terms, with admission and descent tests for those terms.

5. The appropriate involution or unitarity identities, including anti-time ordering rather than an unjustified self-adjointness requirement.

6. A continuity statement on a named domain at a specified strength. F9a must not be promoted to F9b or quantum hypocontinuity.