Proposition: Actual preparation versus universal switch; explicit hypotheses

Status: a proposition with explicit hypotheses; it is not counted as a Constructed fixed-data result. Assume Gate V and:

1. (H1)

   ${\sigma }_{\epsilon }$ is the actual XII-C ramp of Lemma lem:to-indep, and $U$ is a nonempty open set with $U\subset (0,\epsilon )\times {\mathbb{R}}^3$;

2. (H2)

   either the switch is treated universally on $U$ (the identities $E_C=0$, resp.\ $E_C={E_C}^{†}$, are required for all ${\chi }_{\mathrm{I}}$), or $U\subset \{{\chi }_{\mathrm{I}}=1{\}}^{\circ }$ for the actual ${\chi }_{\mathrm{I}}$;

3. (H3)

   on $U$ every coefficient of ${\mathrm{Collapse}}_t(C)$ and of $H_t(C)$ is a finite sum $\sum c_{k,\rho ,\beta }{\sigma }^{(k)}(t){\chi }_{\rho }(x){\partial }^{\beta }h$ with $k\le 4$, linear in the $\sigma$-jet (each graph coefficient carries exactly one $\sigma$-jet factor, Proposition prop:M1(3), and collapse and adjoint preserve this; Lemma lem:joint).

Then, with a universal switch, $\mathrm{Collapse}$ and $\mathrm{SA}$ on $U$ are exactly those of Theorem thm:to-timeonly; and if $U\subset \{{\chi }_{\mathrm{I}}=1{\}}^{\circ }\cap (0,\epsilon )\times {\mathbb{R}}^3$, with actual ${\sigma }_{\epsilon }$ and actual ${\chi }_{\mathrm{I}}$, they are those of Proposition prop:to-chiplateau (vacuous if that set is empty).

Proof. By (H3), vanishing of a coefficient for all $h$, and (in the universal-switch case) for all ${\chi }_{\mathrm{I}}$, forces ${\sum }_k{c}_{k,\rho ,\beta }{\sigma }^{(k)}=0$ on the time projection of $U$, hence $c=0$ by Lemma lem:to-indep. In the plateau case the ${\chi }_{\rho }$ with $\rho \ne 0$ vanish identically on $U$, and the same argument applies.