Lemma: Index bookkeeping of the displayed entries

In each of the six displayed entries $K^{I,J}(a,b,z)$ of sec:entries, the index $I$ is carried by the variable $a$ and the index $J$ by the variable $z$; the cross-line pullback $\pi (a,b,z,y)=(y-a,y-b)$ of XIV-J Jcross attaches the two cross-lines to $a$ and $b$. After contraction with the cross-lines, $J$ at $z$ is the only free tensor index, and $y$ is the only variable not consumed by an internal line.

Proof. Read off: in (eq:K5a), ${\mathsf{A}}^{\alpha \beta ,I}$ stands with ${\sigma }_{\epsilon }(a)$ and ${\partial }_a$, while $L^{\mu ,J}(a,z)={\sigma }_{\epsilon }(z){\mathsf{A}}^{\lambda \mu ,J}{\partial }_{z^{\lambda }}R(a-z)$ carries $J$ with ${\sigma }_{\epsilon }(z)$; in (eq:K5z) the roles of $(a,I)$ and $(z,J)$ are exchanged consistently; in (eq:K47), $L^{\mu ,J}(b,z)$ and $L^{\nu ,I}(b,a)$; in (eq:K6), XIV-I's defining display (eq:Dfrom) assigns $I$ to $h$ and $J$ to $G$, and ${\sigma }_{\epsilon }(a)$ accompanies $I$; in (eq:Q)–(eq:K9), the ordering $(\cdot ;a,z)$ of $Q^{L,I,J}(v;a,z)$ pairs $I$ with $a$ and $J$ with $z$. The statement about $\pi$ is the display (eq:cross) of the patch.