Theorem: D3: constant projection of an existing word, but no causal lift

The constructed free word satisfies

This scalar assignment does not by itself define ${\mathsf{T}}_2$, ${\mathsf{R}}_1$, or a higher retarded map on the intermediate actual-functional stages.

Proof. The first assertion is the constant coefficient of the inherited equality (eq:word). Thus the scalar is now a projection of an existing word, not a projection of an unconstructed association.

Nevertheless, $\Pi$ has a large kernel. For example, $\Pi (\hslash F)=0$ for every $F\in {\mathcal{P}}_{\Sigma }$. Changing a lower-order nonconstant coefficient leaves the displayed constant projection fixed. This is an information-loss statement; it is not permission to change the locked coefficients of ${\star }_c$.

IX-Free-2 supplies the nonconstant free coefficients lost by this projection. It supplies no separate time-ordering coefficients, causal factorization, old time-ordered or retarded overlaps, or anti-time-ordering identities. In particular, the scalar does not select

The existence of the free word therefore does not convert its constant projection into a causal map.