Theorem: The free release does not supply either causal interface

Assumption ass:free2 supplies neither a system satisfying ${\mathcal{I}}_{\mathrm{TO}}$ nor one satisfying ${\mathcal{I}}_{\mathrm{ret}}$. In particular, (eq:word) is not causal factorization.

Proof. The free assignment has a specified free tensor domain and coefficient values. The time-ordered maps, ordered restrictions, anti-time-ordering equations, old time-ordered overlaps, and retarded slot distributions in the requirements are additional objects. They are not among its outputs. The explicit nonimplication witness in Lemma lem:separation below shows that even adjoining a symmetric admitted auxiliary multiplication does not supply the missing causal axiom.