Definition: Time-only formal reading and the specialization S t

In the time-only formal reading, ${\sigma }_{\epsilon }=\sigma (t)$, $t=x^0$, and the symbols $\sigma ,{\sigma }^{\prime },{\sigma }^{\prime \prime },\dots$ are independent. The ${\chi }_{\mathrm{I}}$-jets ${\chi }_{\rho }$ are independent spacetime jets, and $h$ is a general spacetime field with independent jets. The time-only total derivatives are as follows: ${D_0}^{(t)}$ acts by Leibniz on $h$- and $\chi$-jets and maps ${\sigma }^{(k)}\mapsto {\sigma }^{(k+1)}$; ${D_i}^{(t)}$ ($i=1,2,3$) acts by Leibniz on $h$- and $\chi$-jets and annihilates every ${\sigma }^{(k)}$. The specialization $S_t$ from the formal reading sends ${\sigma }_{\alpha }\mapsto {\sigma }^{({\alpha }_0)}$ if $\alpha =({\alpha }_0,0,0,0)$, and ${\sigma }_{\alpha }\mapsto 0$ if $\alpha$ has a spatial index; it leaves the $h$- and $\chi$-jets unchanged. Put ${\mathrm{Collapse}}_t:=S_t\circ {\mathrm{Collapse}}_{\mathrm{formal}}$ and $H_t(C):=S_t({E_C}^{†}-E_C)$. The supplementary reading {time-only $\chi$-plateau} additionally sets ${\chi }_{\mathrm{I}}\equiv 1$, so that all $\chi$-derivatives vanish.