Proposition: Linear divergence identities; not Ward identities

Let $E^{\mu \nu }$ denote Gate V Euler tensors; they carry upper indices because they are paired with the lowered $h_{\mu \nu }$, and $(\mathrm{div}E{)}^{\nu }:={\partial }_{\mu }{E}^{\mu \nu }$ involves no metric factor.

1. (A)

   Contracted Bianchi: ${\eta }^{\mu \rho }{\partial }_{\rho }{G^{\mathrm{lin}}}_{\mu \nu }=0$ identically in jets ($G$ with lower indices).

2. (B${}^{\prime }$)

   Curvature squares. For $I_w=\int w(\alpha {R_{\mathrm{lin}}}^2+\beta {R^{\mathrm{lin}}}_{\mu \nu }{R_{\mathrm{lin}}}^{\mu \nu })$ with constant $\alpha ,\beta$ and any fixed weight $w$ (in particular $w=1$ in the double plateau, $w={\sigma }_{\epsilon }$ in the $\chi$-plateau reading, $w={\chi }_{\mathrm{I}}{\sigma }_{\epsilon }$ formally),

   $${\partial }_{\mu }{E_{I_w}}^{\mu \nu }=0\text{identically.}$$

   Reason: $R_{\mathrm{lin}}$ and ${R^{\mathrm{lin}}}_{\mu \nu }$ are invariant pointwise under $\delta h_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$, so $0=\delta I_w=2\int E^{\mu \nu }{\partial }_{\mu }{\xi }_{\nu }=-2\int {\xi }_{\nu }{\partial }_{\mu }{E}^{\mu \nu }$; weight-derivative terms arise in the Euler tensor and cancel in its divergence. (For the weighted Einstein–Hilbert action the analogous weight-derivative terms are $\sigma G^{\mu \nu }+q^{\mu \nu }\square \sigma -{\nabla }^{\mu }{\nabla }^{\nu }\sigma$ of XII-C, lem:weighted-curvature, eq:weighted-full-variation; no landmark constructs the weighted $R^2/{\mathrm{Ric}}^2$ Euler tensors, which are internal to XVI-M.) This holds separately for $R^2$, ${\mathrm{Ric}}^2$ and for $R^2-\frac{188813}{88456}{\mathrm{Ric}}^2$ (exact assertions in all three readings).

3. (EH)

   For $I_w=\int w{h}^{\mu \nu }{G^{\mathrm{lin}}}_{\mu \nu }$: ${\partial }_{\mu }{E_{I_w}}^{\mu \nu }=({\partial }_{\mu }w){G_{\mathrm{lin}}}^{\mu \nu }[h]$ (indices raised with $\eta$), which vanishes identically for arbitrary $h$ iff $w$ is locally constant (double plateau). Reason: $\delta (h\cdot G)=2{\partial }_{\mu }{\xi }_{\nu }{G}^{\mu \nu }$ and ${\partial }_{\mu }{G}^{\mu \nu }=0$.

4. (B)

   Ultralocal. For the matched line $\int w(h_{\mu \nu }{h}^{\mu \nu }-\frac{89}{140}{h}^2)$: ${\partial }_{\mu }{E}^{\mu \nu }=2{\partial }_{\mu }[w(h^{\mu \nu }-\frac{89}{140}{\eta }^{\mu \nu }h)]\ne 0$ in every reading; this nonvanishing is genuine, because the ultralocal term is not invariant under $\delta h=\partial \xi +\partial \xi$.

The Euler tensors of $\int {R_{\mathrm{lin}}}^2$ and $\int {\mathrm{Ric}}_{\mathrm{lin}}^2$ are also built independently, from $g=\eta +ϵh$ by first-order expansion of the Christoffel symbols and Riemann tensor (no hand-written linearized formula), with sympy's euler_equations and the Gate V weights; they coincide exactly with the targets used in M.4/M.5 (and $R_{\mathrm{lin}}$, ${R^{\mathrm{lin}}}_{\mu \nu }$ agree with the formulas of Proposition prop:match with sign $+1$). The certified preimages of $E_{R2}$, $E_{RC2}$ (in ${\mathcal{C}}_{\mathrm{coh}}$) and of $E_{R2}-\frac{188813}{88456}{E}_{RC2}$ (in ${\mathcal{C}}_{\mathrm{reg}}$) collapse exactly to these independent targets and satisfy the identity. These are linear (free-field gauge) identities; none of (A), (B), (B${}^{\prime }$), (EH) is an interacting Ward identity (no ghost/gauge-fixed interacting sector is present; Target II unfinished).