Theorem: Variational subspace of the contact collapse

Assume Gate V. These are reported exact computations (Lemma lem:M4comp):

${}^{\ast }$var.\ image $=\dim ({\mathcal{T}}_{\mathrm{variational}}\cap \mathrm{im}\mathrm{Collapse})=\dim \mathrm{SA}-\dim \ker \mathrm{Collapse}$; all entries are dimensions. ${}^{†}$universal reading with formal ${\chi }_{\mathrm{I}}$-jets (exact for that reading; not a statement about the actual fixed ${\chi }_{\mathrm{I}}$). ${}^{‡}$universal time-only readings, ${\sigma }_{\epsilon }=\sigma (t)$ with formal ${\sigma }^{(k)}$ (SA $=\ker H_t$; Section sec:to-timeonly).

In particular:

1. (i)

   Formal-reading vanishing (not a global obstruction). Under Gate V, the formal universal-jet collapse image of this selected graph contains no nonzero variational tensor of total derivative degree $D\ge 1$. Every variational collapsed tensor in that image is an ultralocal multiplication operator $E^{\mu \nu }={\chi }_{\mathrm{I}}{\sigma }_{\epsilon }{\sum }_{a\le b}{K^{\mu \nu }}_{ab}{h}_{ab}$ with the Gate V weighted symmetry $w_{\mu \nu }{K^{\mu \nu }}_{ab}=w_{ab}{K^{ab}}_{\mu \nu }$ (a $26$-dimensional family, all in ${\mathcal{C}}_{\mathrm{reg}}$, all from $N=4$, $\gamma =0$). (Certificate A checks $w_{00}{K^{00}}_{02}=-56/3=w_{02}{K^{02}}_{00}$; the index-reversed form fails.) The matched functionals are $I_C=\frac{1}{2}\int {\chi }_{\mathrm{I}}{\sigma }_{\epsilon }⟨h,Kh⟩$. This is a statement about the selected two-edge graph in the formal reading; it does not claim absence of derivative counterterms in a completed gravitational sector, failure for every fixed preparation, or impossibility after adding other diagrams.

2. (ii)

   The $\chi$-plateau reading has the same $26$-dimensional SA basis.

3. (iii)

   In the double-plateau reading, derivative counterterms do occur: $271$ second-order, $8$ third-order, and $1020$ fourth-order variational directions in the image (variational per-$D$ dimensions $(26,0,271,8,1020)$).