Proposition: Time-only matching probes

Assume Gate V. Take the weight $w={\chi }_{\mathrm{I}}(x){\sigma }_{\epsilon }(t)$, or $w={\sigma }_{\epsilon }(t)$ in the $\chi$-plateau variant, with all preparation and switch derivatives retained in the Euler tensors.

1. $D=0$: $\mathrm{span}\{E_{w{P}_1},E_{w{P}_2}\}\cap \mathrm{im}{\mathrm{Collapse}}_t$ is one-dimensional, spanned by $P_1-\frac{89}{140}{P}_2$, with preimage $C[N=4,\{33,33\},()]=3/70\in {\mathcal{C}}_{\mathrm{reg}}$ (regression-preserving), in both ${\mathcal{C}}_{\mathrm{coh}}$ and ${\mathcal{C}}_{\mathrm{reg}}$. Neither $P_1$ nor $P_2$ is an image by itself. The Helmholtz check $\mathrm{EL}(\frac{1}{2}⟨h,E_{C}h⟩)=E_C$ is verified by two independent routes (time-only jets; euler_equations with $\sigma (x^0)$, $\chi (x)$, $h(x)$).

2. $D=2$: the weighted Einstein–Hilbert Euler tensor is not in $\mathrm{im}{\mathrm{Collapse}}_t$, in either ${\mathcal{C}}_{\mathrm{coh}}$ or ${\mathcal{C}}_{\mathrm{reg}}$.

3. $D=4$: $\mathrm{span}\{E_{w{R}^2},E_{w{\mathrm{Ric}}^2}\}\cap \mathrm{im}{\mathrm{Collapse}}_t=0$, in both ${\mathcal{C}}_{\mathrm{coh}}$ and ${\mathcal{C}}_{\mathrm{reg}}$. The double-plateau matchings of Proposition prop:match(3) and Theorem thm:M5-plateau-R2 therefore do not survive the time-only reading: once ${\sigma }_{\epsilon }^{\prime }(t)$, ${\sigma }_{\epsilon }^{\prime \prime }(t),\dots$ are retained, the weighted $R^2$ and ${\mathrm{Ric}}^2$ Euler tensors are not images, even with ${\chi }_{\mathrm{I}}\equiv 1$.

4. Divergence identities (upper-index $E$, no metric factor; time-only jets): ${\partial }_{\mu }{E}^{\mu \nu }=0$ for $\int w{R}_{\mathrm{lin}}^2$ and $\int w{\mathrm{Ric}}_{\mathrm{lin}}^2$; ${\partial }_{\mu }{E_{\mathrm{EH}}}^{\mu \nu }=({\partial }_{\mu }w){G_{\mathrm{lin}}}^{\mu \nu }$; nonzero for the ultralocal $P_1$.

Each weighted target is checked to be self-adjoint under ${†}_t$. Membership is decided by the exact verified kernel of the $(D,\text{parity }0)$ block augmented by the targets.