Theorem: D1: causal structure remains additional

The inherited product ${\star }_c$, its wordwise coherence, and its calibrated constant coefficient do not by themselves supply a time-ordered or retarded system satisfying ${\mathcal{I}}_{\mathrm{TO}}$ or ${\mathcal{I}}_{\mathrm{ret}}$.

Proof. Assumption ass:free2 supplies actual free coefficient maps and their declared overlaps. It supplies no separate ${\mathsf{T}}_2$, no old time-ordered overlap equations, and no anti-time-ordering system.

Equation (eq:word) compares two existing free associations. It contains no causal-factorization identity relating a time-ordered map to ${\star }_c$. Section sec:D3 gives an explicit algebraic separation of these requirements.

Without ${\mathsf{T}}_2$, the difference in (eq:subtraction) is unavailable. Nor do the free statements supply the causal Green convention, higher intermediate domains, or argument-slot support proofs. These remain additional obligations. This is not a proof that a later causal extension cannot exist.