The chord-diagram summation identity for Dodgson polynomials

Let $D_0\in \mathcal{D}^n_0$ with $n\in \mathbb{N}^{\ell}$ and $N=\sum n_i$. For the chord diagrams $D$ in $\mathcal{D}^n_N$, let $s(D)$ be the number of chords and let $E_D^0$ denote the set of uncoloured edges, with $\chi_{\Gamma}^{(u|v)}$ the associated contraction of Dirac matrices. Define

$$Z_{ij}(D_0)=\sum_{\mathcal{E}\in \mathcal{P}(E_{D_0}^i)}(-\Psi_{\Gamma})^{N-|\mathcal{E}|}(|\mathcal{E}|+1)!Y(\mathcal{E},E_{D_0}^j).$$

Here $\mathcal{P}(E_{D_0}^i)$ is the set of partitions of the $i$-coloured base edges, and $Y(\mathcal{E},E_{D_0}^j)$ is a sum of products of $|\mathcal{E}|$ Dodgson polynomials $\Psi_{\Gamma,K}^{I,J}$ with $|I|=|E_k|=|J|$ and $K=\emptyset$. The chord-diagram summation identity. One has

$$\sum_{D\in \mathcal{D}^n_N}(-2)^{s(D)}\prod_{(u,v)\in E_D^0}\chi_{\Gamma}^{(u|v)}=(-2)^\ell\bigl(Z_{12}(D_0)+Z_{21}(D_0)\bigr).$$

This is presented as a corollary-level identity following the preceding iterative chord-addition theorem; the supplied text gives no evidence that it is conjectural or unresolved.