Proposition: Equivariant SA, exact; Gate V

$V$ is the exact rational kernel of the adjoint-defect map $x\mapsto {E_{C(x)}}^{†}-E_{C(x)}$ on the $14$ equivariant coordinates:

1. Formal and $\chi$-plateau: $V$ is the line of $C_{2,0}$ at $N=4$ (${\square }^2$, $\gamma =0$), contained in ${\mathcal{C}}_{\mathrm{reg}}$; $\mathrm{Collapse}(V)$ is spanned by the Euler tensor of $P_1-\frac{89}{140}{P}_2$ (ratio $17920$ in the formal reading), i.e.\ the matched ultralocal line of Proposition prop:match.

2. Double plateau: $\dim V=12$, of which $3$ dimensions lie in $\ker \mathrm{Collapse}$. ${\pi }_K(V)$ is all of ${\mathrm{Equiv}}_K$ ($5$), but only $V\cap {\mathcal{C}}_{\mathrm{K}}$ ($3$) consists of SA assignments; the remaining projected directions have nonzero adjoint defect and are not SA assignments. Likewise ${\pi }_{\mathrm{reg}}(V)={\mathrm{Equiv}}_{\mathrm{reg}}$ ($9$) while $V\cap {\mathcal{C}}_{\mathrm{reg}}$ has dimension $7$. Check: $3+7=10\le 12$.

Every basis vector of $V$, $V\cap {\mathcal{C}}_{\mathrm{K}}$ and $V\cap {\mathcal{C}}_{\mathrm{reg}}$ is re-verified from scratch against all constraints: equivariance (six generators, all grades) and zero adjoint defect; regression_M.py re-verifies them independently, including the Helmholtz route $\mathrm{EL}(\frac{1}{2}⟨h,E_{C}h⟩)=E_C$. Mod-$p$ ranks at three primes agree and are supplementary only. Prescription-covariant variational contacts exist but do not by themselves produce a Ward identity.