Theorem: Algebraic extension criterion

A complex-linear map on $D_2$ satisfying the prescribed values, symmetry, classical multiplication, and (eq:antitime) exists algebraically if and only if, for every finite relation,

This conclusion includes no sharp distributional construction or continuity theorem.

Proof. Write ${\mathsf{T}}_2=A+X$. Symmetry requires $X$ to descend to $Q$; (eq:antitime) requires $X(\gamma q)=-X(q{)}^{\ast }$. Thus its forced values are $X([d_i])=x_i$ and $X([\gamma d_i])=-x_i^{\ast }$, giving necessity.

Conversely, (eq:relationtest) defines such a map on the conjugation-stable span of these classes. Multiplying its range by $i$ makes it conjugation-equivariant. Extend the resulting real-linear map from the real fixed-point subspace to a real complement in $Q$, then complexify and multiply by $-i$. Pull back to $D_2$ and add $A$. The extension takes values in $\hslash {\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$, so does not change the classical coefficient. All values are finite linear combinations of admitted elements. No continuity of the chosen complement is implied.