Gates

Input (S-bv) (Input inp:Sbv) was introduced as an explicit input of XVI-M. Certification had to establish the representation for the actual predecessor kernel $S=h_0+i{E}_0^{\mathrm{A}}$ on the punctured space $\Omega$, including across the nonzero null cone—not only for a model kernel with the same wave equation, and not only off the cone. (The alternative route, a direct certification of the relations of M.2 and of their completeness, is no longer needed.) Earlier drafts proceeded conditionally on (S-bv); no downstream matching result was used to remove that condition. The unconditional general-$h$ assembly (M.1) and the wave-operator bound (Proposition prop:wave) are stated separately and never depended on this gate. Gate S-bv certified (Theorem thm:wpA-Sbv): $S=\frac{1}{4{\pi }^2}(Q-i0{)}^{-1}$ in ${\mathcal{D}}^{\prime }(\Omega )$, i.e.\ $c=1/(4{\pi }^2)$ with boundary side $Q-i0$ (Section sec:wpA-Sbv, Remark rem:Sbv-status).

Chosen Phase II convention; it is not certified by the normalization ledger.

1. Field variable: the symmetric lowered residual field $h_{\mu \nu }$, with no rescaling and no trace reversal.

2. Variation: $h\mapsto h+t$ with $t$ symmetric and compactly supported.

3. Pairing: $\delta I[h](t)=\int d^{4}y{t}_{\mu \nu }{E_I}^{\mu \nu }[h]$ with the full symmetric-index contraction over all $16$ pairs $(\mu ,\nu )$. In the ten-component coordinates $h_{\mu \nu }$, $\mu \le \nu$:

   $$t_{\mu \nu }{E}^{\mu \nu }=\sum _{\mu \le \nu }{w}_{\mu \nu }{t}_{\mu \nu }{E}^{\mu \nu },w_{\mu \mu }=1,w_{\mu \nu }=2(\mu <\nu ),$$

   so the Euler tensor of a density $\mathcal{L}$ in the independent components is

   $${E_I}^{\mu \nu }=\frac{1}{w_{\mu \nu }}\sum _{\beta }(-D{)}^{\beta }\frac{\partial \mathcal{L}}{\partial ({\partial }^{\beta }{h}_{\mu \nu })}.$$

4. External data: ${\sigma }_{\epsilon }$ and ${\chi }_{\mathrm{I}}$ are fixed under field variation, but their derivatives are kept in every integration by parts.

5. Order: $h$-linear collapsed tensors; quadratic functional ansatz modulo field-independent terms.

6. Helmholtz test: $\int r\cdot (D{E}_C[h]\cdot t)=\int t\cdot (D{E}_C[h]\cdot r)$; for homogeneous linear $E_C$ this is $\int ⟨r,E_C[t]⟩=\int ⟨t,E_C[r]⟩$.