Proposition: Matching probes

Assume Gate V. Membership in a linear space does not depend on normalization, so the target normalizations below only fix the preimage coefficients, which are not Wilson coefficients. Targets, at quadratic order on the flat background, with $R_{\mathrm{lin}}={\partial }^{\mu }{\partial }^{\nu }{h}_{\mu \nu }-\square h$ and ${R^{\mathrm{lin}}}_{\mu \nu }=\frac{1}{2}({\partial }^{\rho }{\partial }_{\mu }{h}_{\rho \nu }+{\partial }^{\rho }{\partial }_{\nu }{h}_{\rho \mu }-\square h_{\mu \nu }-{\partial }_{\mu }{\partial }_{\nu }h)$:

Here the scalar weight is $w={\chi }_{\mathrm{I}}{\sigma }_{\epsilon }$ in the formal reading, $w={\sigma }_{\epsilon }$ in the $\chi$-plateau reading, and $w=1$ in the double-plateau reading. $\int {\mathrm{Riem}}_{\mathrm{lin}}^2$ is not separate: at quadratic order $\int ({\mathrm{Riem}}^2-4{\mathrm{Ric}}^2+R^2{)}_{\mathrm{lin}}$ is a total derivative.

1. In all three readings at $D=0$: $\mathrm{span}\{E_{P_1},E_{P_2}\}\cap \mathrm{im}\mathrm{Collapse}$ is one-dimensional, spanned by the Euler tensor of $P_1-\frac{89}{140}{P}_2$. The preimage is the single coordinate $C[N=4,\{33,33\},()]=3/70\in {\mathcal{C}}_{\mathrm{reg}}$. Neither $P_1$ nor $P_2$ alone, nor the Fierz–Pauli combination $P_1-P_2$, is an image.

2. Non-membership (double plateau). Under Gate V, the specified quadratic Einstein–Hilbert Euler tensor $E_{\mathrm{EH}}\propto G^{\mathrm{lin}}$ is not in the selected double-plateau collapse image ($D=2$), either in ${\mathcal{C}}_{\mathrm{coh}}$ or in ${\mathcal{C}}_{\mathrm{reg}}$.

3. Constructed (double plateau). $E_{R2}$ and $E_{RC2}$ are each collapse images ($D=4$), with explicit exact preimages in ${\mathcal{C}}_{\mathrm{coh}}$ ($286$ nonzero coefficients each). In ${\mathcal{C}}_{\mathrm{reg}}$ only the single combination $R2-\frac{188813}{88456}RC2$ is an image. Matching $R^2$ or ${\mathrm{Ric}}^2$ individually therefore requires coherent changes on the ${\mathcal{V}}_{\mathrm{K}}$ block, which do not preserve regression.

4. In the formal reading no target with $D\ge 1$ is an image of a variational collapse (Theorem thm:M4(i)). The double-plateau matchings hold locally on the open set where ${\sigma }_{\epsilon }\equiv 1$ and ${\chi }_{\mathrm{I}}\equiv 1$; they fail once switch and preparation derivatives are retained. This is a separation of plateau matching from full switched matching. In the time-only reading the weighted $R^2$ and ${\mathrm{Ric}}^2$ Euler tensors are not images either, even with ${\chi }_{\mathrm{I}}\equiv 1$ (Proposition prop:to-match(3)): the double-plateau matching does not survive the time-only reading.

Proof. For each target $T$: $T$ is self-adjoint (checked). Membership is decided by the exact verified kernel of the matrix whose columns are the $\mathrm{Collapse}$ columns of the $(D,\text{parity }0)$ block, augmented by $T$. Each preimage is reassembled and compared with $T$ exactly; regression_M.py re-verifies the $R2$ preimage with an independently coded $R_{\mathrm{lin}}$. Targets are Lorentz scalars, hence of reflection parity $0$, so only parity-$0$ blocks contribute. The target construction is checked by the linearized Bianchi identity in jets. Audit (Proposition prop:M7-id): $E_{R2}$ and $E_{RC2}$ are rebuilt independently from the metric linearization and coincide exactly with the targets; both targets and the certified preimages (after $\mathrm{Collapse}$) satisfy ${\partial }_{\mu }{E}^{\mu \nu }=0$.