Theorem: Partial-product reversal

The declared domain is dagger stable and, for $0\le j\le 3$,

Consequently every admissible pair has an admissible reversed conjugate pair and

The formula extends coefficientwise to formal admissible pairs, with ${\hslash }^{\ast }=\hslash$.

Proof. Reality of the smooth operators gives

The inherited first-stage exchange gives

Thus dagger exchanges the two first-stage domains and the corresponding left and right second-stage domains. The old domain is dagger stable by its inherited convention.

On the descended degree-four blocks, the inherited graph exchanges give

For example, ${\mathsf{G}}_{13}+{\mathsf{G}}_{23}$ exchanges with ${\mathsf{G}}_{13}+{\mathsf{G}}_{12}$, and ${\mathsf{G}}_{13,23}$ exchanges with ${\mathsf{G}}_{12,13}$. The inherited degree-two reversal gives the same identity on those blocks. The scalar exchange uses real $c$ and the parent dagger symmetry. Classical multiplication and the zero coefficients obey the identity as well.

It holds on the old domain by Assumption ass:old. Unique gluing on $D_{\mathrm{decl}}$ extends these component identities to every declared tensor.