Proposition: Representation of and the onset at Σ

Let $\beta :={{\mathsf{E}}_0}^{\mathrm{A}}(\mathcal{P}\alpha )$, a smooth symmetric tensor on $M_T$ with $\mathrm{supp}\beta \subset {\mathcal{D}}_f$. Then

Viewed as a functional of ambient configurations through slab restriction, the degree-$(2,1)$ kernel $u$ of ${\mathcal{C}}_3$ is supported in $\{y_1=y_2=y\in {\mathcal{D}}_f\cap \{y^0\ge 0\}\}\times {\mathcal{D}}_f$ and has the form

where $j_3$ is the constant-coefficient bilinear local kernel of $J_3$ (a finite sum of products of derivatives of $\delta (y-y_1)$ and $\delta (y-y_2)$) and $\hat{\beta }$ is the $\varphi$-slot kernel of $\beta$. The factor $\theta (y^0)$ is forced: the frozen Cauchy prescription gives the second coefficient zero data on $\Sigma$, so ${{\mathsf{E}}_0}^{\mathrm{R}}$ truncates every source at $t=0$, and the Einstein–Hilbert density carries no $g$ and no ${\sigma }_{\epsilon }$ that could make $J_3[\gamma ]$ vanish near $\Sigma$.

Proof. ${\mathcal{C}}_3=\int \alpha \cdot u_3=\int \alpha \cdot \mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}{J}_3$. Trace reversal is self-adjoint under the $\eta$-contraction, $⟨\alpha ,\mathcal{P}r⟩=⟨\alpha ,r⟩-\frac{1}{2}\mathrm{tr}\alpha \mathrm{tr}r=⟨\mathcal{P}\alpha ,r⟩$, and transposition (applicable because $\mathrm{supp}\mathcal{P}\alpha \subset {\mathcal{D}}_f$ vanishes near $t=T$ and ${{\mathsf{E}}_0}^{\mathrm{R}}{J}_3$ has zero data) gives (eq:C3beta). The $y$-integral runs over the slab only, and $J_3[\gamma ](y)$ depends on the jet of $\gamma$ at $y$; this is (eq:ukernel). Compactness: $y\in {\mathcal{D}}_f$, and the $\varphi$-slot of $\alpha$ is supported in ${\mathcal{D}}_f$ by Lemma lem:alpha.