Lemma: Chart form of the response integrals

Under (P1), as distributions on ${\mathbb{R}}^{12}$,

where $R(b-a){\delta }^{(4)}(a-z)$ denotes the pullback of $R\otimes \delta$ by $(a,b,z)\mapsto (b-a,a-z)$. On $V_{+}$ the bracketed distributions are $\frac{1}{2\pi }{q}_{\alpha \beta }(u){\mathsf{E}}_0(1)$ and $\frac{1}{4\pi }{q}_{\alpha \beta }(u)d_{\kappa }{\mathsf{E}}_0(1)$, elements of ${{\mathcal{J}}^0}_{\infty }$.

Proof. By (P1), (P2), $⟨{\mathcal{T}}_0,\varphi ⟩=\sum \int q_{\alpha \beta }(v)({{\partial }_z}^{\alpha }{{\partial }_a}^{\beta }{{\Phi }^R}_{\varphi })(v,v,v)\mathrm{d}v$, and $({{\partial }_z}^{\alpha }{{\partial }_a}^{\beta }{{\Phi }^R}_{\varphi })(v,v,v)=⟨R,({{\partial }_z}^{\alpha }{{\partial }_a}^{\beta }\varphi )(v,v+\cdot ,v)⟩$. The right side equals $\sum ⟨q_{\alpha \beta }(a)R(b-a)\delta (a-z),{{\partial }_z}^{\alpha }{{\partial }_a}^{\beta }\varphi ⟩$, which is the pairing of the right side of (eq:T0chart) with $\varphi$. For (eq:TBchart) use Corollary cor:K8supp's identification of ${\partial }_{b^{\kappa }}{\mathcal{T}}_{\mathcal{B}}$ as the same construction with $\frac{1}{2}{x}_{\kappa }R$. The chart forms are (P4) and Lemma lem:dB; $\delta (q)\otimes \delta (r)={\mathsf{E}}_0(1)$ along $S^{+}$.