Proposition: Domain stability

The domain (eq:Dtwo) is contained in $D_{\mathrm{cmp}}$ and is stable under $\sigma ,\gamma ,†$. An old time-ordered map on a nonzero comparison space is not supplied by declaring $O_2=0$.

Proof. Lemma lem:comparison puts every generator in $D_{\mathrm{cmp}}$. The involutions permute the generators and preserve their complex span. The zero seed defines only the zero map on the zero space.