Theorem: Plateau curvature-square matching

(Named theorem PlateauCurvatureSquareMatching.) Assume Gate V.

• Domain: double-plateau reading of the selected XVI-L complete-source two-edge graph (open set where ${\sigma }_{\epsilon }\equiv 1$ and ${\chi }_{\mathrm{I}}\equiv 1$ locally); coherent contact space ${\mathcal{C}}_{\mathrm{coh}}$ and regression-preserving subspace ${\mathcal{C}}_{\mathrm{reg}}$.

• Order: quadratic in the residual field $h$ on the flat background; total derivative degree $D=4$.

• Pairing: Gate V.

• Equivalence: literal local Euler-tensor representatives (ledger (1)), with the optional quadratic Gauss–Bonnet identification of $\int {\mathrm{Riem}}_{\mathrm{lin}}^2$ in ledger (2); EOM and field-redefinition equivalences are not used.

Then the Gate V Euler tensors of $\int {R_{\mathrm{lin}}}^2$ and of $\int {\mathrm{Ric}}_{\mathrm{lin}}^2$ each lie in $\mathrm{Collapse}({\mathcal{C}}_{\mathrm{coh}})$, with explicit exact preimages. In $\mathrm{Collapse}({\mathcal{C}}_{\mathrm{reg}})$ only the single combination $R2-\frac{188813}{88456}RC2$ is an image. The specified quadratic Einstein–Hilbert Euler tensor is not an image. These matchings are local to the double-plateau domain and do not complete inside the formal or $\chi$-plateau selected images (Proposition prop:completion). Audit status. The theorem still holds after the M.7 audit: the matching code uses neither the corrected divergence routine nor the equivariant vectors, the coefficient $188813/88456$ is unchanged on rerun, the targets agree with an independent metric-based construction, and the targets and their certified preimages satisfy the linear divergence identity ${\partial }_{\mu }{E}^{\mu \nu }=0$ (Proposition prop:M7-id).