Proposition: Time-only χ -plateau: new spatial variational directions

Assume Gate V. With ${\chi }_{\mathrm{I}}\equiv 1$ and $\sigma =\sigma (t)$ formal, ${\mathrm{Collapse}}_t$ is injective, $\ker H_t$ has dimensions $(26,0,156,0,390)$ by $D$ ($572$ in total), all of it lies in ${\mathcal{C}}_{\mathrm{reg}}$, $\ker H_t\cap {\mathcal{C}}_{\mathrm{K}}=0$, and ${\mathrm{Collapse}}_t(\ker H_t)$ has the same dimensions. Exactly,

so that for $D=0,2,4$ one has $E_C=\sigma (t)K{\partial }^{\gamma }h$ with $\gamma$ purely spatial; here $26\times 6=156$ and $26\times 15=390$, and the dimensions $(26,0,156,0,390)$ are unchanged. Reason: for $E[h]=\sigma (t)K{\partial }_{1}h$ one has $E^{†}=-E$, since ${\partial }_1\sigma =0$ (odd spatial order is anti-self-adjoint); likewise for every odd-order purely spatial multi-index. Even-order purely spatial derivatives are formally self-adjoint and commute with $\sigma (t)$, so the even-$D$ directions are self-adjoint; they are variational only by virtue of the time-only structure. The $D=0$ ultralocal family is the formal self-adjoint family already recorded above; the sentence that none is self-adjoint in the spacetime-formal $\chi$-plateau reading refers to the extra $D=2,4$ directions (checked), so those are time-only-specific. All even-$D$ directions remain self-adjoint in the double plateau. For all $546$ directions with $D=2$ or $D=4$, $\mathrm{EL}(\frac{1}{2}⟨h,E_{C}h⟩)=E_C$ is verified by an independent sympy time-only jet route, and for samples also by euler_equations on functions. They are spatial-only and not Lorentz covariant, and they are not Euler tensors of $\int \sigma R$, $\int \sigma R^2$ or $\int \sigma {\mathrm{Ric}}^2$ (Proposition prop:to-match).