Lemma: Intertwining

Independent of Input (S-bv). For every polynomial $F$ in the formal jets, $S_t(D_{\mu }F)={D_{\mu }}^{(t)}{S}_t(F)$ for $\mu =0,\dots ,3$. Hence $S_t\circ {†}_{\mathrm{formal}}={†}_t\circ S_t$, where ${†}_t$ is the Gate V formal adjoint built from $D^{(t)}$, and $H_t(C)={E_{C,t}}^{{†}_t}-E_{C,t}$ with $E_{C,t}={\mathrm{Collapse}}_t(C)$.

Proof. Both sides are derivations that agree on generators. On $h$- and $\chi$-jets $S_t$ is the identity. For ${\sigma }_{\alpha }$: $D_{\mu }{\sigma }_{\alpha }={\sigma }_{\alpha +e_{\mu }}$. If $\mu =0$, then $\alpha +e_0$ is time-only exactly when $\alpha$ is, and both sides equal ${\sigma }^{({\alpha }_0+1)}$ or $0$. If $\mu =i\ge 1$, then $\alpha +e_i$ is spatial, so the left side is $0$, and the right side is ${D_i}^{(t)}{S}_t({\sigma }_{\alpha })=0$. The adjoint is the composite of multiplications and iterated total derivatives, so it commutes with the ring homomorphism $S_t$.