Theorem: Time-only classification

Assume Gate V. In the time-only formal reading the following dimensions are exact (Lemma lem:to-comp):

Here $k_C=\dim \ker {\mathrm{Collapse}}_t$, $i_C=\dim \mathrm{im}{\mathrm{Collapse}}_t$, $k_H=\dim \ker H_t$, $v=\dim {\mathrm{Collapse}}_t(\ker H_t)$ (variational image), ${k^{\mathrm{reg}}}_{\bullet }$, ${k^{\mathrm{K}}}_{\bullet }$ are the dimensions of the intersections with ${\mathcal{C}}_{\mathrm{reg}}$ and ${\mathcal{C}}_{\mathrm{K}}$, and $v^{\mathrm{reg}}=\dim {\mathrm{Collapse}}_t(\ker H_t\cap {\mathcal{C}}_{\mathrm{reg}})$ (regression-preserving variational image). Thus ${\mathrm{Collapse}}_t$ is injective for every $D$, and the time-only variational image is exactly the $26$-dimensional ultralocal family $E^{\mu \nu }={\chi }_{\mathrm{I}}{\sigma }_{\epsilon }{\sum }_{a\le b}{K^{\mu \nu }}_{ab}{h}_{ab}$ with $w_{\mu \nu }{K^{\mu \nu }}_{ab}=w_{ab}{K^{ab}}_{\mu \nu }$. The family is supported on $N=4$, $\gamma =0$, lies entirely in ${\mathcal{C}}_{\mathrm{reg}}$ (so it preserves regression), and coincides with the formal SA family: the $26$ formal SA vectors lie in $\ker H_t$, and $\dim \ker H_t=26$. There is no time-only variational direction of total derivative degree $D\ge 1$ for a variable (formal) switch.