in (3.22) and (3.24). The expressions for $\mathcal{E}^{\text{eqv}}[\begin{smallmatrix} j_1 & \cdots & j_l \\ k_1 & \cdots & k_l \end{smallmatrix}]$ obtained in section C.2 will then be used to demonstrate the equivalence between the two constructions of the modular forms $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ at modular depth three either as coefficients of $e_{k_i}^{(j_i)}$ from the generating series $\mathbb{J}^{\text{eqv}}(e_k; \tau)$, as discussed in section 3, or as coefficients of $(X_i - \tau Y_i)^{j_i} (X_i - \bar{\tau} Y_i)^{k_i - j_i - 2}$ from the equivariant integrals $M_{k_1,k_2,k_3}$, as investigated in section 5.

## C.1 (Anti-)meromorphic building blocks

We start by making the structure of the (anti-)meromorphic building blocks $\hat{\psi}^{\text{sv}}(\tilde{\mathbb{I}}_-(e_k))$ and $\mathbb{I}_+(e_k)$ of the generating series (3.22) more explicit up to modular depth three. For this purpose, we parametrise the change of alphabet $\hat{\psi}^{\text{sv}}$ of section 3.1.4 as well as the letters $e_{\Delta^{\pm}}$ of the cuspidal kernels in (3.7) and (3.8) via infinite families of real constants $\chi_k^j[\begin{smallmatrix} j_1 & \cdots \\ k_1 & \cdots \end{smallmatrix}]$, $\xi_{\Delta}^j[\begin{smallmatrix} j_1 & \cdots \\ k_1 & \cdots \end{smallmatrix}]$ and $\eta_{\Delta}^j[\begin{smallmatrix} j_1 & \cdots \\ k_1 & \cdots \end{smallmatrix}]$. With the conventions (3.16) for words $W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ in $e_k^{(j)}$, we parametrise the change of alphabet $\hat{\psi}^{\text{sv}}$ described in more detail in section 3.3 via
$$ \hat{\psi}^{\text{sv}}(W[\begin{smallmatrix} j \\ k \end{smallmatrix}]) = W[\begin{smallmatrix} j \\ k \end{smallmatrix}] + \sum_{k_1,k_2,j_1,j_2} \chi_k^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] W[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] + \sum_{k_1,k_2,k_3,j_1,j_2,j_3} \chi_k^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] + \dots, \quad (C.1) $$

where the sums $\sum_{k_1,k_2,j_1,j_2}$ and $\sum_{k_1,k_2,k_3,j_1,j_2,j_3}$ here and below are understood as covering the usual range of even $k_i \ge 4$ and $0 \le j_i \le k_i - 2$. The $\chi_k^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}]$ and $\chi_k^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ associated with words in up to three $e_k^{(j)}$ can be computed from the series $M^{\text{sv}}(\hat{z}_i)$ in zeta generators in (3.20), leading to rational multiples of $\zeta_{2n_1+1}$ or $\zeta_{2n_1+1}\zeta_{2n_2+1}$. However, we do not need the explicit form of $\chi_k^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}]$, $\chi_k^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \in \mathbb{R}$ for the main goals of this appendix. We also do not use any symmetry properties of the $\chi_k^j$ under permutations of columns such as $\chi_k^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] = -\chi_k^j[\begin{smallmatrix} j_2 & j_1 \\ k_2 & k_1 \end{smallmatrix}]$ which ensure that each order in (C.1) is Lie-algebra valued.

A similar parametrisation is used for the letters $e_{\Delta^{\pm}}$ associated with cuspidal integration kernels,

$$ W[\begin{smallmatrix} j \\ \Delta^+ \end{smallmatrix}] = \sum_{k_1,k_2,j_1,j_2} \xi_{\Delta}^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] W[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] + \sum_{k_1,k_2,k_3,j_1,j_2,j_3} \xi_{\Delta}^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] + \dots, \quad (C.2) $$

$$ W[\begin{smallmatrix} j \\ \Delta^- \end{smallmatrix}] = \sum_{k_1,k_2,j_1,j_2} \eta_{\Delta}^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] W[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] + \sum_{k_1,k_2,k_3,j_1,j_2,j_3} \eta_{\Delta}^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] + \dots, $$

where some of the $\xi_{\Delta}^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}]$, $\xi_{\Delta}^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$, $\eta_{\Delta}^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}]$, $\eta_{\Delta}^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \in \mathbb{R}$ can in principle be inferred from the expressions (3.73) for $e_{\Delta^{\pm}}$. By $\mathfrak{sl}_2$-invariance of $\hat{\psi}^{\text{sv}}$ and $e_{\Delta^{\pm}}^{(j)} = (\text{ad}_{e_0})^j e_{\Delta^{\pm}}$, the $j=0$ instances of $\chi_k^j[\begin{smallmatrix} j_1 & \cdots \\ k_1 & \cdots \end{smallmatrix}]$, $\xi_{\Delta_k}^j[\begin{smallmatrix} j_1 & \cdots \\ k_1 & \cdots \end{smallmatrix}]$ and $\eta_{\Delta_k}^j[\begin{smallmatrix} j_1 & \cdots \\ k_1 & \cdots \end{smallmatrix}]$ already determine those at non-zero $j$, but this appendix will not make any use of this property.

By virtue of the expansions (C.1) and (C.2), we can bring the coefficients of $W[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}]$ and $W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ in $\hat{\psi}^{\text{sv}}(\tilde{\mathbb{I}}_-(e_k; \tau))$ and $\mathbb{I}_+(e_k; \tau)$ into the following form:

$$ \begin{aligned} \mathbb{I}_+(e_k; \tau) = 1 &+ \sum_{k_1,j_1} W[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] \mathcal{E}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] + \sum_{k_1,k_2,j_1,j_2} W[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \left\{ \mathcal{E}[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] + \sum_{k,j} \sum_{\Delta_k \in S_k} \xi_{\Delta_k}^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \mathcal{E}[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}] \right\} \\ &+ \sum_{k_1,k_2,k_3,j_1,j_2,j_3} W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \left\{ \mathcal{E}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] + \sum_{k,j} \sum_{\Delta_k \in S_k} \left( \xi_{\Delta_k}^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \mathcal{E}[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}] + \xi_{\Delta_k}^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \mathcal{E}[\begin{smallmatrix} j_1 & j \\ k_1 & \Delta_k \end{smallmatrix}] \right) \right\} + \dots \end{aligned} \quad (C.3) $$
and

$$
\begin{align}
\hat{\psi}^{\text{sv}}(\tilde{\mathbb{I}}_-(e_k; \tau)) ={}& 1 + \sum_{k_1, j_1} W[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] + \sum_{k_1, k_2, j_1, j_2} W[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j_2 & j_1 \\ k_2 & k_1 \end{smallmatrix}] \nonumber \\
& + \sum_{k_1, k_2, j_1, j_2} W[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \sum_{k, j} \left\{ \chi_k^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ k \end{smallmatrix}] + \sum_{\Delta_k \in S_k} \eta_{\Delta_k}^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}] \right\} \nonumber \\
& + \sum_{k_1, k_2, k_3, j_1, j_2, j_3} W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \left\{ \tilde{\mathcal{E}}[\begin{smallmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{smallmatrix}] + \sum_{k, j} \chi_k^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ k \end{smallmatrix}] \right\} \nonumber \\
& + \sum_{k_1, k_2, k_3, j_1, j_2, j_3} W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \sum_{k, j} \left\{ \chi_k^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j_3 & j \\ k_3 & k \end{smallmatrix}] + \chi_k^j[\begin{smallmatrix} j_2 & j_3 \\ k_2 & k_3 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j & j_1 \\ k & k_1 \end{smallmatrix}] \right\} \nonumber \\
& + \sum_{k_1, k_2, k_3, j_1, j_2, j_3} W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \sum_{k, j} \sum_{\Delta_k \in S_k} \left( \eta_{\Delta_k}^j[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}] \right. \tag{C.4} \\
& \left. \qquad + \eta_{\Delta_k}^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j_3 & j \\ k_3 & \Delta_k \end{smallmatrix}] + \eta_{\Delta_k}^j[\begin{smallmatrix} j_2 & j_3 \\ k_2 & k_3 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j & j_1 \\ \Delta_k & k_1 \end{smallmatrix}] \right) \nonumber \\
& + \sum_{k_1, k_2, k_3, j_1, j_2, j_3} W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] \sum_{k, j} \sum_{\Delta_k \in S_k} \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}] \nonumber \\
& \qquad \times \sum_{k', j'} \left( \eta_{\Delta_k}^j[\begin{smallmatrix} j' & j_3 \\ k' & k_3 \end{smallmatrix}] \chi_{k'}^{j'}[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] + \eta_{\Delta_k}^j[\begin{smallmatrix} j_1 & j' \\ k_1 & k' \end{smallmatrix}] \chi_{k'}^{j'}[\begin{smallmatrix} j_2 & j_3 \\ k_2 & k_3 \end{smallmatrix}] \right) + \dots, \nonumber
\end{align}
$$

where we employ the following shorthand to absorb factors of $(-1)^{j_i}$

$$
\tilde{\mathcal{E}}\begin{bmatrix} j_1 & j_2 & \dots & j_l \\ k_1 & k_2 & \dots & k_l \end{bmatrix} = (-1)^{j_1+j_2+\dots+j_l} \overline{\mathcal{E}\begin{bmatrix} j_1 & j_2 & \dots & j_l \\ k_1 & k_2 & \dots & k_l \end{bmatrix}} \quad (C.5)
$$

which applies in identical form to cases with $k_i \to \Delta_k$.

## C.2 Assembling $\mathcal{E}^{\text{eqv}}$

The next step is to combine the expansions (C.4) and (C.5) of the series in iterated integrals with the constant series

$$
\begin{align}
\mathbb{C}^{\text{sv}}(e_k) ={}& 1 + \sum_{k_1, j_1} W[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] c^{\text{sv}}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] + \sum_{k_1, k_2, j_1, j_2} W[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] c^{\text{sv}}[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \nonumber \\
& + \sum_{k_1, k_2, k_3, j_1, j_2, j_3} W[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] c^{\text{sv}}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}] + \dots . \tag{C.6}
\end{align}
$$

With these expansions at hand, one can obtain the following expressions for the coefficients $\mathcal{E}^{\text{eqv}}$ of $\mathbb{I}^{\text{eqv}}(e_k; \tau)$ by matching (3.22) with (3.24)

$$
\begin{align}
\mathcal{E}^{\text{eqv}}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] &= \tilde{\mathcal{E}}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] + c^{\text{sv}}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] + \mathcal{E}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] \tag{C.7} \\
\mathcal{E}^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] &= \tilde{\mathcal{E}}[\begin{smallmatrix} j_2 & j_1 \\ k_2 & k_1 \end{smallmatrix}] + c^{\text{sv}}[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] + \mathcal{E}[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \nonumber \\
&\quad + \tilde{\mathcal{E}}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] c^{\text{sv}}[\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}] + \tilde{\mathcal{E}}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] \mathcal{E}[\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}] + c^{\text{sv}}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] \mathcal{E}[\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}] \nonumber \\
&\quad + \sum_{k,j} \left\{ \chi_k^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ k \end{smallmatrix}] + \sum_{\Delta_k \in S_k} \left( \xi_{\Delta_k}^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \mathcal{E}[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}] + \eta_{\Delta_k}^j[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}] \right) \right\}, \nonumber
\end{align}
$$

where the case at modular depth two goes beyond the simpler classes of terms in (3.25). The double sum over $j, k$ collapses to finitely many terms bounded by $k \le k_1+k_2-2$ since all of
$\chi_k^j[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$, $\xi_{\Delta_k}^j[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$ and $\eta_{\Delta_k}^j[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$ vanish otherwise. The same procedure gives rise to the following more lengthy expression at modular depth three:

$$
\begin{equation}
\begin{aligned}
\mathcal{E}^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] ={}& \tilde{\mathcal{E}}[\begin{matrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{matrix}] + c^{\text{sv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] + \mathcal{E}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] + \tilde{\mathcal{E}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] c^{\text{sv}}[\begin{matrix} j_2 & j_3 & j_1 \\ k_2 & k_3 & k_1 \end{matrix}] \\
& + \tilde{\mathcal{E}}[\begin{matrix} j_2 & j_1 & j_3 \\ k_2 & k_1 & k_3 \end{matrix}] c^{\text{sv}}[\begin{matrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{matrix}] + \tilde{\mathcal{E}}[\begin{matrix} j_2 & j_1 & j_3 \\ k_2 & k_1 & k_3 \end{matrix}] \mathcal{E}[\begin{matrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{matrix}] \\
& + \tilde{\mathcal{E}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] \mathcal{E}[\begin{matrix} j_2 & j_3 & j_1 \\ k_2 & k_3 & k_1 \end{matrix}] + c^{\text{sv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] \mathcal{E}[\begin{matrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{matrix}] \\
& + \sum_{k,j} \left\{ \chi_k^j[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] c^{\text{sv}}[\begin{matrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{matrix}] \tilde{\mathcal{E}}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] + \chi_k^j[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] \mathcal{E}[\begin{matrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{matrix}] \tilde{\mathcal{E}}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] \right. \\
& \qquad \left. + \chi_k^j[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] \tilde{\mathcal{E}}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] \tilde{\mathcal{E}}[\begin{matrix} j_3 & j & j_1 \\ k_3 & k & k_2 \end{matrix}] + \chi_k^j[\begin{matrix} j_2 & j_3 & j_1 \\ k_2 & k_3 & k_1 \end{matrix}] \tilde{\mathcal{E}}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] \right\} \\
& + \sum_{k,j} \sum_{\Delta_k \in S_{\Delta_k}} \left\{ \eta_{\Delta_k}^j[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}] c^{\text{sv}}[\begin{matrix} j_3 & j_2 \\ k_3 & k_1 \end{matrix}] \tilde{\mathcal{E}}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] + \eta_{\Delta_k}^j[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}] \mathcal{E}[\begin{matrix} j_3 & j_2 \\ k_3 & k_1 \end{matrix}] \tilde{\mathcal{E}}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] \right. \\
& \qquad \left. + \eta_{\Delta_k}^j[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] \tilde{\mathcal{E}}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] + \eta_{\Delta_k}^j[\begin{matrix} j_1 & j_2 & j_3 \\ k_2 & k_3 & k_1 \end{matrix}] \tilde{\mathcal{E}}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] \right\} \\
& + \sum_{k,j} \sum_{\Delta_k \in S_{\Delta_k}} \left\{ \xi_{\Delta_k}^j[\begin{matrix} j_2 & j_3 \\ k_2 & k_3 \end{matrix}] c^{\text{sv}}[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}] \mathcal{E}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] + \xi_{\Delta_k}^j[\begin{matrix} j_2 & j_3 \\ k_2 & k_3 \end{matrix}] \tilde{\mathcal{E}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] \right. \\
& \qquad \left. + \xi_{\Delta_k}^j[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] \mathcal{E}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] + \xi_{\Delta_k}^j[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}] \mathcal{E}[\begin{matrix} j & j_1 & j_2 \\ k_1 & k_2 & k_3 \end{matrix}] + \xi_{\Delta_k}^j[\begin{matrix} j_2 & j_3 \\ k_2 & k_3 \end{matrix}] \mathcal{E}[\begin{matrix} j_1 & j & j_2 \\ k_1 & k_2 & k_3 \end{matrix}] \right\} \\
& + \sum_{k,j} \sum_{\Delta_k \in S_{\Delta_k}} \tilde{\mathcal{E}}[\begin{matrix} j & j_1 & j_2 \\ k & k_1 & k_2 \end{matrix}] \sum_{k',j'} \left\{ \eta_{\Delta_k}^j[\begin{matrix} j' & j_3 \\ k' & k_3 \end{matrix}] \chi_{k'}^{j'}[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}] + \eta_{\Delta_k}^j[\begin{matrix} j_1 & j' \\ k_1 & k' \end{matrix}] \chi_{k'}^{j'}[\begin{matrix} j_2 & j_3 \\ k_2 & k_3 \end{matrix}] \right\}.
\end{aligned}
\tag{C.8}
\end{equation}
$$

Again, all the sums over $k, j$ and $k', j'$ are finite in view of the vanishing of $\chi_k^j[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$, $\xi_{\Delta_k}^j[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$, $\eta_{\Delta_k}^j[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$ at $k > k_1+k_2-2$ as well as $\chi_k^j[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$, $\xi_{\Delta_k}^j[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$, $\eta_{\Delta_k}^j[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$ at $k > k_1+k_2+k_3-4$.

### C.3 Comparing $M_k$ and $M_{k_1,k_2}$ with $\mathcal{E}^{\text{eqv}}$

We shall next apply the expressions (C.7) for $\mathcal{E}^{\text{eqv}}$ to compare different representations of the equivariant integrals $M_k[X, Y; \tau]$ and $M_{k_1,k_2}[X_1, Y_1, X_2, Y_2; \tau]$ in section 5.1. The established $\beta^{\text{eqv}}$-representations in (5.5) and (5.18) [28] are equivalent to

$$
\begin{align}
M_k[X, Y; \tau] &= -\frac{1}{4}(k-1)! \sum_{j=0}^{k-2} (-1)^j \binom{k-2}{j} \left(\frac{Y}{2\pi i}\right)^j X^{k-2-j} \mathcal{E}^{\text{eqv}}[\begin{matrix} j \\ k \end{matrix}], \\
M_{k_1,k_2}[X_1, Y_1, X_2, Y_2; \tau] &= \frac{1}{16}(k_1-1)!(k_2-1)! \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} (-1)^{j_1+j_2} \binom{k_1-2}{j_1} \binom{k_2-2}{j_2} \nonumber \\
&\qquad \times \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} \mathcal{E}^{\text{eqv}}[\begin{matrix} j_2 & j_1 \\ k_2 & k_1 \end{matrix}]. \tag{C.9}
\end{align}
$$

By equating these expressions to the original construction of $M_k$ and $M_{k_1,k_2}$ through the (1,0)-form $\mathcal{G}_k[X, Y; \tau]$ in (5.3), we generate identities that facilitate the proof of the $\beta^{\text{eqv}}$ representation (5.31) of $M_{k_1,k_2,k_3}$ at modular depth three.
At modular depth one, matching the original expression in (5.5) with the first line of (C.9) (using the expression (C.7) for $\mathcal{E}^{\text{eqv}}[\begin{smallmatrix} j \\ k \end{smallmatrix}]$) yields

$$ M_k[X, Y; \tau] = -\frac{1}{2} \int_{\tau}^{i\infty} \underline{G}_k[X, Y; \tau_1] - \frac{1}{2} \int_{\bar{\tau}}^{-i\infty} \overline{G}_k[X, Y; \tau_1] - c_k[X, Y] \quad (C.10) \\ = -\frac{1}{4}(k-1)! \sum_{j=0}^{k-2} (-1)^j \begin{pmatrix} k-2 \\ j \end{pmatrix} \left(\frac{Y}{2\pi i}\right)^j X^{k-2-j} \left(\mathcal{E}[\begin{smallmatrix} j \\ k \end{smallmatrix}] + \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ k \end{smallmatrix}] + c^{\text{sv}}[\begin{smallmatrix} j \\ k \end{smallmatrix}] \right). $$

This can in fact be refined to three separate identities comparing terms with holomorphic, antiholomorphic or no iterated Eisenstein integrals:

$$ \begin{aligned} \int_{\tau}^{i\infty} \underline{G}_k[X, Y; \tau_1] &= \frac{1}{2}(k-1)! \sum_{j=0}^{k-2} (-1)^j \begin{pmatrix} k-2 \\ j \end{pmatrix} \left(\frac{Y}{2\pi i}\right)^j X^{k-2-j} \mathcal{E}[\begin{smallmatrix} j \\ k \end{smallmatrix}], \quad (C.11) \\ \int_{\bar{\tau}}^{-i\infty} \overline{G}_k[X, Y; \tau_1] &= \frac{1}{2}(k-1)! \sum_{j=0}^{k-2} (-1)^j \begin{pmatrix} k-2 \\ j \end{pmatrix} \left(\frac{Y}{2\pi i}\right)^j X^{k-2-j} \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ k \end{smallmatrix}], \\ c_k[X, Y] &= \frac{1}{4}(k-1)! \sum_{j=0}^{k-2} (-1)^j \begin{pmatrix} k-2 \\ j \end{pmatrix} \left(\frac{Y}{2\pi i}\right)^j X^{k-2-j} c^{\text{sv}}[\begin{smallmatrix} j \\ k \end{smallmatrix}]. \end{aligned} $$

The analogous comparison at modular depth two reads (see (5.10) and (5.18)):

$$ \begin{aligned} M_{k_1, k_2}[\tau] &= \frac{1}{4} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \int_{\tau_1}^{i\infty} \underline{G}_{k_2}[\tau_2] + \frac{1}{4} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \times \int_{\bar{\tau}}^{-i\infty} \overline{G_{k_2}[\tau_2]} \quad (C.12) \\ &\quad + \frac{1}{4} \int_{\bar{\tau}}^{-i\infty} \overline{G_{k_2}[\tau_2]} \int_{\tau_2}^{-i\infty} \overline{G_{k_1}[\tau_1]} + \frac{1}{2} c_{k_2} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] + \frac{1}{2} c_{k_1} \int_{\bar{\tau}}^{-i\infty} \overline{G_{k_2}[\tau_2]} \\ &\quad - c_{k_1, k_2} - \frac{1}{2} \int_{\bar{\tau}}^{-i\infty} \underline{G}_{k_1, k_2}[\tau_1] - \frac{1}{2} \int_{\tau}^{i\infty} \underline{f}_{k_1, k_2}[\tau_1] \\ &= \frac{1}{16}(k_1-1)!(k_2-1)! \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} (-1)^{j_1+j_2} \begin{pmatrix} k_1-2 \\ j_1 \end{pmatrix} \begin{pmatrix} k_2-2 \\ j_2 \end{pmatrix} \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} \\ &\quad \times \left\{ \mathcal{E}[\begin{smallmatrix} j_2 & j_1 \\ k_2 & k_1 \end{smallmatrix}] + \tilde{\mathcal{E}}[\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}] \mathcal{E}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] + \tilde{\mathcal{E}}[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}] + c^{\text{sv}}[\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}] \mathcal{E}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] + \tilde{\mathcal{E}}[\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}] c^{\text{sv}}[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}] + c^{\text{sv}}[\begin{smallmatrix} j_2 & j_1 \\ k_2 & k_1 \end{smallmatrix}] \right\} \\ &\quad + \sum_{k,j} \left[ \chi_k^j[\begin{smallmatrix} j_2 & j_1 \\ k_2 & k_1 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ k \end{smallmatrix}] + \sum_{\Delta_k \in S_k} \left( \eta_{\Delta_k}^j[\begin{smallmatrix} j_2 & j_1 \\ k_2 & k_1 \end{smallmatrix}] \tilde{\mathcal{E}}[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}] + \xi_{\Delta_k}^j[\begin{smallmatrix} j_2 & j_1 \\ k_2 & k_1 \end{smallmatrix}] \mathcal{E}[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}] \right) \right\}, \end{aligned} $$

where we stopped displaying the dependence on $X_i, Y_i$ to avoid cluttering (see the comments below (5.11) for our conventions). The first five terms on the two sides are readily seen to match individually by (C.11) or immediate consequences such as

$$ \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \int_{\tau_1}^{i\infty} \underline{G}_{k_2}[\tau_2] = \frac{1}{4}(k_1-1)!(k_2-1)! \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} (-1)^{j_1+j_2} \begin{pmatrix} k_1-2 \\ j_1 \end{pmatrix} \begin{pmatrix} k_2-2 \\ j_2 \end{pmatrix} \times \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} \mathcal{E}[\begin{smallmatrix} j_2 & j_1 \\ k_2 & k_1 \end{smallmatrix}]. \quad (C.13) $$

The sixth terms $\sim c_{k_1, k_2}$ and $c^{\text{sv}}[\begin{smallmatrix} j_2 & j_1 \\ k_2 & k_1 \end{smallmatrix}]$ on both sides of (C.12) match by (5.20). Then, by separately equating the leftover terms involving holomorphic and antiholomorphic integrals,
we deduce two new identities from (C.12):

$$
\begin{align}
\int_{\bar{\tau}}^{-i\infty} \overline{\mathcal{G}_{k_1, k_2}[\tau_1]} &= -\frac{1}{8}(k_1-1)!(k_2-1)!
\sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} (-1)^{j_1+j_2} \binom{k_1-2}{j_1} \binom{k_2-2}{j_2} \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \nonumber \\
&\quad \times \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} \sum_{k,j} \left\{ \chi_k^j \genfrac{[}{]}{0pt}{}{j_2}{k_2} \chi_1^j \genfrac{[}{]}{0pt}{}{j_1}{k_1} \tilde{\mathcal{E}} \genfrac{[}{]}{0pt}{}{j}{k} + \sum_{\Delta_k \in S_k} \eta_{\Delta_k}^j \genfrac{[}{]}{0pt}{}{j}{k_2} \chi_1^j \genfrac{[}{]}{0pt}{}{j_1}{k_1} \tilde{\mathcal{E}} \genfrac{[}{]}{0pt}{}{j}{\Delta_k} \right\}, \nonumber \\
\int_{\tau}^{i\infty} \mathfrak{f}_{k_1, k_2}[\tau_1] &= -\frac{1}{8}(k_1-1)!(k_2-1)!
\sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} (-1)^{j_1+j_2} \binom{k_1-2}{j_1} \binom{k_2-2}{j_2} \tag{C.14} \\
&\quad \times \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} \sum_{k,j} \sum_{\Delta_k \in S_k} \xi_{\Delta_k}^j \genfrac{[}{]}{0pt}{}{j_2}{k_2} \chi_1^j \genfrac{[}{]}{0pt}{}{j_1}{k_1} \mathcal{E} \genfrac{[}{]}{0pt}{}{j}{\Delta_k} \nonumber
\end{align}
$$

The sums over $k, j$ in the second and fourth line are both bounded by $k \le k_1 + k_2 - 2$. Note that the two types of contributions $\chi_k^j \genfrac{[}{]}{0pt}{}{j_2}{k_2} \chi_1^j \genfrac{[}{]}{0pt}{}{j_1}{k_1} \tilde{\mathcal{E}} \genfrac{[}{]}{0pt}{}{j}{k}$ and $\eta_{\Delta_k}^j \genfrac{[}{]}{0pt}{}{j}{k_2} \chi_1^j \genfrac{[}{]}{0pt}{}{j_1}{k_1} \tilde{\mathcal{E}} \genfrac{[}{]}{0pt}{}{j}{\Delta_k}$ to the integral over $\overline{\mathcal{G}_{k_1, k_2}[\tau_1]}$ correspond to the Eisenstein part and the cuspidal part of the underlying antiholomorphic modular form.

### C.4 Comparing $M_{k_1,k_2,k_3}$ with $\mathcal{E}^{\text{eqv}}$

At modular depth three, the $\beta^{\text{eqv}}$-representation of $M_{k_1,k_2,k_3}$ in (5.31) that we wish to prove is equivalent to the following generalisation of (C.9):

$$
M_{k_1,k_2,k_3}[\tau] = -\frac{1}{64}(k_1-1)!(k_2-1)!(k_3-1)!
\sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \sum_{j_3=0}^{k_3-2} (-1)^{j_1+j_2+j_3} \binom{k_1-2}{j_1} \binom{k_2-2}{j_2} \binom{k_3-2}{j_3} \nonumber \\
\times \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} \left(\frac{Y_3}{2\pi i}\right)^{j_3} X_3^{k_3-2-j_3} \mathcal{E}^{\text{eqv}} \genfrac{[}{]}{0pt}{}{j_3}{k_3} \genfrac{[}{]}{0pt}{}{j_2}{k_2} \genfrac{[}{]}{0pt}{}{j_1}{k_1} . \quad (C.15)
$$

The key idea of the proof is to insert the representation (C.8) of $\mathcal{E}^{\text{eqv}} \genfrac{[}{]}{0pt}{}{j_3}{k_3} \genfrac{[}{]}{0pt}{}{j_2}{k_2} \genfrac{[}{]}{0pt}{}{j_1}{k_1}$ and to compare term by term with the original definition of $M_{k_1,k_2,k_3}$ via (5.29) with $K_{k_1,k_2,k_3}$ given by (5.23). Based on (C.11), (C.14) and their corollaries for products and double or triple integrals, we can make 21 out of the 27 terms in the expression (C.8) for $\mathcal{E}^{\text{eqv}} \genfrac{[}{]}{0pt}{}{j_3}{k_3} \genfrac{[}{]}{0pt}{}{j_2}{k_2} \genfrac{[}{]}{0pt}{}{j_1}{k_1}$ match with $K_{k_1,k_2,k_3}$ in (5.23). After setting these matching terms aside, the leftover task in proving (5.31) is to show that one can find solutions for $c_{k_1,k_2,k_3}$, $\overline{\mathcal{G}_{k_1,k_2,k_3}[\tau_1]}$ and $\mathfrak{f}_{k_1,k_2,k_3}[\tau_1]$ such that

$$
\begin{align}
M_{k_1,k_2,k_3}[\tau] - K_{k_1,k_2,k_3}[\tau] &= -c_{k_1,k_2,k_3} - \frac{1}{2} \int_{\bar{\tau}}^{-i\infty} \overline{\mathcal{G}_{k_1,k_2,k_3}[\tau_1]} - \frac{1}{2} \int_{\tau}^{i\infty} \mathfrak{f}_{k_1,k_2,k_3}[\tau_1] \tag{C.16} \\
&= -\frac{1}{64}(k_1-1)!(k_2-1)!(k_3-1)!
\sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \sum_{j_3=0}^{k_3-2} (-1)^{j_1+j_2+j_3} \binom{k_1-2}{j_1} \binom{k_2-2}{j_2} \binom{k_3-2}{j_3} \nonumber \\
&\quad \times \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} \left(\frac{Y_3}{2\pi i}\right)^{j_3} X_3^{k_3-2-j_3} \nonumber \\
&\quad + \sum_{k,j} \left[ \chi_k^j \genfrac{[}{]}{0pt}{}{j_3}{k_3} \genfrac{[}{]}{0pt}{}{j_2}{k_2} \genfrac{[}{]}{0pt}{}{j_1}{k_1} \tilde{\mathcal{E}} \genfrac{[}{]}{0pt}{}{j}{k} + \sum_{\Delta_k \in S_k} \left( \xi_{\Delta_k}^j \genfrac{[}{]}{0pt}{}{j_3}{k_3} \genfrac{[}{]}{0pt}{}{j_2}{k_2} \genfrac{[}{]}{0pt}{}{j_1}{k_1} \mathcal{E} \genfrac{[}{]}{0pt}{}{j}{\Delta_k} + \eta_{\Delta_k}^j \genfrac{[}{]}{0pt}{}{j_3}{k_3} \genfrac{[}{]}{0pt}{}{j_2}{k_2} \genfrac{[}{]}{0pt}{}{j_1}{k_1} \tilde{\mathcal{E}} \genfrac{[}{]}{0pt}{}{j}{\Delta_k} \right) \nonumber \\
&\quad + \sum_{\Delta_k \in S_k} \tilde{\mathcal{E}} \genfrac{[}{]}{0pt}{}{j}{\Delta_k} \sum_{k',j'} \left( \eta_{\Delta_k}^j \genfrac{[}{]}{0pt}{}{j'}{k'} \genfrac{[}{]}{0pt}{}{j_3}{k_3} \genfrac{[}{]}{0pt}{}{j_2}{k_2} \genfrac{[}{]}{0pt}{}{j_1}{k_1} + \eta_{\Delta_k}^j \genfrac{[}{]}{0pt}{}{j}{k'} \chi_{k'}^{j'} \genfrac{[}{]}{0pt}{}{j_3}{k_3} \genfrac{[}{]}{0pt}{}{j_2}{k_2} \genfrac{[}{]}{0pt}{}{j_1}{k_1} \right) \right\}. \nonumber
\end{align}
$$
It is a non-trivial confirmation that all iterated integrals beyond depth one in the expression (C.8) for $\mathcal{E}^{\text{eqv}}\begin{bmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{bmatrix}$ have been successfully lined up with the expression (5.23) for $K_{k_1,k_2,k_3}$. It is always possible to separately match the depth-one integrals $\mathcal{E}$, their complex conjugates $\pm \tilde{\mathcal{E}}$ and the $\tau$-independent terms in (C.16) — this process will in fact determine the quantities $\mathfrak{f}_{k_1,k_2,k_3}[\tau_1]$, $\mathfrak{g}_{k_1,k_2,k_3}[\tau_1]$ and $c_{k_1,k_2,k_3}$. The $\tau$-independent terms in (C.16) imply the identity (5.33) between $c_{k_1,k_2,k_3}$ and $c^{\text{sv}}\begin{bmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{bmatrix}$ noted in the main text, and the equality of the (anti-)holomorphic integral amounts to

$$ \int_{\tau}^{-i\infty} \overline{\mathfrak{g}_{k_1,k_2,k_3}[\tau_1]} = \frac{(k_1-1)!(k_2-1)!(k_3-1)!}{32} \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \sum_{j_3=0}^{k_3-2} (-1)^{j_1+j_2+j_3} \begin{pmatrix} k_1-2 \\ j_1 \end{pmatrix} \begin{pmatrix} k_2-2 \\ j_2 \end{pmatrix} \begin{pmatrix} k_3-2 \\ j_3 \end{pmatrix} \\ \times \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} \left(\frac{Y_3}{2\pi i}\right)^{j_3} X_3^{k_3-2-j_3} \sum_{k,j} \left\{ \chi_k^j \begin{bmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{bmatrix} \tilde{\mathcal{E}}\begin{bmatrix} j \\ k \end{bmatrix} + \sum_{\Delta_k \in S_k} \tilde{\mathcal{E}}\begin{bmatrix} j \\ \Delta_k \end{bmatrix} \left[ \eta_{\Delta_k}^j \begin{bmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{bmatrix} + \sum_{k',j'} \left( \eta_{\Delta_k}^j \begin{bmatrix} j' & j_1 \\ k' & k_1 \end{bmatrix} \chi_{k'}^{j'} \begin{bmatrix} j_3 & j_2 \\ k_3 & k_2 \end{bmatrix} + \eta_{\Delta_k}^j \begin{bmatrix} j_3 & j' \\ k_3 & k' \end{bmatrix} \chi_{k'}^{j'} \begin{bmatrix} j_2 & j_1 \\ k_2 & k_1 \end{bmatrix} \right) \right\}, \quad (C.17) $$

as well as

$$ \int_{\tau}^{i\infty} \mathfrak{f}_{k_1,k_2,k_3}[\tau_1] = \frac{(k_1-1)!(k_2-1)!(k_3-1)!}{32} \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \sum_{j_3=0}^{k_3-2} (-1)^{j_1+j_2+j_3} \begin{pmatrix} k_1-2 \\ j_1 \end{pmatrix} \begin{pmatrix} k_2-2 \\ j_2 \end{pmatrix} \begin{pmatrix} k_3-2 \\ j_3 \end{pmatrix} \\ \times \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} \left(\frac{Y_3}{2\pi i}\right)^{j_3} X_3^{k_3-2-j_3} \sum_{k,j} \sum_{\Delta_k \in S_k} \xi_{\Delta_k}^j \begin{bmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{bmatrix} \mathcal{E}\begin{bmatrix} j \\ \Delta_k \end{bmatrix}. \quad (C.18) $$

The sums over $k, j$ and $k', j'$ are again finite by the vanishing conditions stated below (C.8). In conclusion, the expansion (5.31) of $M_{k_1,k_2,k_3}$ in terms of $\beta^{\text{eqv}}\begin{bmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{bmatrix}$ obtained from the generating series $\mathcal{J}^{\text{eqv}}$ has been shown to agree with the original construction in section 5.2 through the quantity $K_{k_1,k_2,k_3}$ in (5.23). As a side effect, we can explicitly determine the quantities $\mathfrak{g}_{k_1,k_2,k_3}[\tau_1]$ and $\mathfrak{f}_{k_1,k_2,k_3}[\tau_1]$ via (C.17) and (C.18) in terms of information $\hat{\psi}^{\text{sv}}$ and $e_{\Delta^{\pm}}$ from the generating series $\mathcal{J}^{\text{eqv}}$.

## D Proof of Lemma 2

In this appendix, we shall prove that the T-cocycle at infinity defined by (4.9) can be brought into the compact form (4.13). Given the definition of $\hat{N}$ and $\hat{N}_+$ in (4.12), we see that the expansion (4.11) can be decomposed according to the double grading $r = \#\hat{N}_+$ and $J = \#e_0$. Let us then consider the particular contribution with fixed gradings $r, J$:

$$ \mathrm{P-exp} \left( \int_{i\infty}^{i\infty-1} \mathbb{A}_+(\tau_1) \right) \Big|_{\substack{r=\#\hat{N}_+ \\ J=\#e_0}} = \sum_{j_1,j_2,\dots,j_r=0}^{\infty} \delta(j_1+\dots+j_r-J) (2\pi i)^{J+r} \quad (D.1) \\ \times \left( \prod_{i=1}^{r} \frac{1}{j_i! \sum_{m=i}^{r} (j_m+1)} \right) \hat{N}_+^{(j_1)} \hat{N}_+^{(j_2)} \dots \hat{N}_+^{(j_r)}. $$
We can now use the simple fact that the repeated adjoint action $X^{(j)} = \text{ad}_{e_0}^j X$ of $e_0$ on any algebra element $X$ can be rewritten as

$$X^{(j)} = \sum_{p=0}^{j} \binom{j}{p} (-1)^{j-p} e_0^p X e_0^{j-p} = \sum_{a_1, a_2=0}^{\infty} \delta(a_1+a_2-j) \binom{a_1+a_2}{a_2} (-1)^{a_2} e_0^{a_1} X e_0^{a_2}, \quad (D.2)$$

leading to

$$\hat{N}_+^{(j_1)} \hat{N}_+^{(j_2)} \dots \hat{N}_+^{(j_r)} = \sum_{a_1, \dots, a_{2r}=0}^{\infty} \prod_{i=1}^{r} \left[ \delta(a_{2i-1}+a_{2i}-j_i) \binom{a_{2i-1}+a_{2i}}{a_{2i}} (-1)^{a_{2i}} \right] \times e_0^{a_1} \hat{N}_+ e_0^{a_2+a_3} \hat{N}_+ e_0^{a_4+a_5} \dots \hat{N}_+ e_0^{a_{2r-2}+a_{2r-1}} \hat{N}_+ e_0^{a_{2r}}. \quad (D.3)$$

We now change summation variables and define

$$b_i := a_{2i-1} + a_{2i-2}, \quad i = 1, 2, \dots, r+1, \quad (D.4)$$

with the convention $a_0 = a_{2r+1} = 0$ such that $b_1 = a_1$ and $b_{r+1} = a_{2r}$. We can eliminate the $j_i$ variables using the Kronecker delta $\delta(j_1+\dots+j_r-J)$ which, when combined with $\prod_i \delta(a_{2i-1}+a_{2i}-j_i)$, reduces to $\delta(b_1+\dots+b_{r+1}-J)$.

Furthermore, we can also get rid of the variables $a_{2i-1}$ for $i = 2, \dots, r$ in favour of the new variables $b_i \ge 0$ with $i = 1, \dots, r+1$. We still have to perform the sum over the remaining variables $a_{2i-2}$ with $i = 2, \dots, r$ over the range $0 \le a_{2i-2} \le b_i$ as we can see from (D.4).

It is convenient to focus on the combinatorial factors appearing in (D.1) and express them first in terms of the variables $a_i$:

$$\left( \prod_{i=1}^{r} \frac{1}{j_i! \sum_{m=i}^{r} (j_m+1)} \right) = \prod_{i=1}^{r} \frac{1}{(a_{2i-1}+a_{2i})! (r+1-i+\sum_{m=2i-1}^{2r} a_m)} \quad (D.5)$$

When we combine this expression with the additional factors coming from (D.3) we obtain

$$\prod_{i=1}^{r} \left[ \frac{1}{(a_{2i-1}+a_{2i})!(r+1-i+\sum_{m=2i-1}^{2r} a_m)} \binom{a_{2i-1}+a_{2i}}{a_{2i}} (-1)^{a_{2i}} \right] = (-1)^{\sum_{i=1}^{r} a_{2i}} \prod_{i=1}^{r} \frac{1}{(a_{2i-1})! (a_{2i})!} \prod_{i=1}^{r} \frac{1}{(r+1-i+\sum_{m=2i-1}^{2r} a_m)} \quad (D.6)$$

The next step is to substitute $a_{2i+1} = b_{i+1} - a_{2i}$ for $i = 1, \dots, r-1$, remembering the conditions $a_1 = b_1$ and $a_{2r} = b_{r+1}$. After a shift in the label $i$ and a rearrangement of the terms in the first product, we arrive at

$$\frac{1}{b_1!} \left[ \prod_{i=1}^{r-1} \frac{(-1)^{a_{2i}}}{(b_{i+1}-a_{2i})! (a_{2i})!} \right] \frac{(-1)^{b_{r+1}}}{b_{r+1}!} \prod_{i=1}^{r} \frac{1}{(r+1-i-a_{2i-2}+\sum_{m=i}^{r+1} b_m)} \quad (D.7)$$

Note that for $i = 1$ the last fraction produces

$$\frac{1}{(r-a_0+\sum_{m=1}^{r+1} b_m)} = \frac{1}{(J+r)}, \quad (D.8)$$

since $a_0 = 0$ and the $b$ variables are constrained by $\delta(b_1+\dots+b_{r+1}-J)$.
We can now consider one by one the remaining summations over $0 \le a_{2i} \le b_{i+1}$ with $i = 1, \dots, r-1$:

$$ \sum_{a_{2i}=0}^{b_{i+1}} \frac{(-1)^{a_{2i}}}{(b_{i+1}-a_{2i})!(a_{2i})!} \frac{1}{(r-i-a_{2i}+\sum_{m=i+1}^{r+1} b_m)} = (-1)^{b_{i+1}} \frac{\Gamma(r-i+\sum_{m=i+2}^{r+1} b_m)}{\Gamma(r+1-i+\sum_{m=i+1}^{r+1} b_m)}. \quad (D.9) $$

When we combine all these factors together we see that summing (D.7) over all $a_{2i}$ with $i = 1, \dots, r-1$ yields the telescopic product

$$ \begin{aligned} & \frac{(-1)^{b_{r+1}}}{(b_1)!(b_{r+1})!} \frac{1}{(J+r)} \prod_{i=1}^{r-1} (-1)^{b_{i+1}} \frac{\Gamma(r-i+\sum_{m=i+2}^{r+1} b_m)}{\Gamma(r+1-i+\sum_{m=i+1}^{r+1} b_m)} = \frac{1}{(b_1)!(b_{r+1})!} \frac{1}{(J+r)} (-1)^{\sum_{i=2}^{r+1} b_i} \\ & \qquad \times \frac{\Gamma(r-1+\sum_{m=3}^{r+1} b_m)}{\Gamma(r+\sum_{m=2}^{r+1} b_m)} \frac{\Gamma(r-2+\sum_{m=4}^{r+1} b_m)}{\Gamma(r-1+\sum_{m=3}^{r+1} b_m)} \cdots \frac{\Gamma(r-(r-1)+\sum_{m=(r-1)+2}^{r+1} b_m)}{\Gamma(r+1-(r-1)+\sum_{m=(r-1)+1}^{r+1} b_m)} \quad (D.10) \\ & = \frac{1}{(b_1)!(b_{r+1})!} \frac{1}{(J+r)} (-1)^{\sum_{i=2}^{r+1} b_i} \frac{\Gamma(r-(r-1)+\sum_{m=(r-1)+2}^{r+1} b_m)}{\Gamma(r+\sum_{m=2}^{r+1} b_m)} = \frac{(-1)^{J-b_1}}{(b_1)!(J+r-b_1-1)!(J+r)}, \end{aligned} $$

where again we used $\delta(b_1+\cdots+b_{r+1}-J)$.

We finally rewrite (D.1) in these new variables

$$ \begin{aligned} & \mathrm{P-exp} \left( \int_{i\infty}^{i\infty-1} \mathrm{A}_+(\tau_1) \right) \Big|_{\substack{r=\#\hat{N}_+ \\ J=\#e_0}} && \quad (D.11) \\ & = \sum_{b_1, \dots, b_{r+1}=0}^{\infty} \delta(b_1+\cdots+b_{r+1}-J) \frac{(2\pi i)^{J+r} (-1)^{J-b_1}}{(b_1)!(J+r-b_1-1)!(J+r)} e_0^{b_1} \hat{N}_+ e_0^{b_2} \hat{N}_+ \cdots \hat{N}_+ e_0^{b_{r+1}}. \end{aligned} $$

It is much easier to compute

$$ \begin{aligned} & e^{2\pi i e_0} e^{2\pi i (\hat{N}_+ - e_0)} \Big|_{\substack{r=\#\hat{N}_+ \\ J=\#e_0}} = \sum_{k_1, k_2=0}^{\infty} (2\pi i)^{k_1+k_2} [e_0^{k_1} (\hat{N}_+ - e_0)^{k_2}] \Big|_{\substack{r=\#\hat{N}_+ \\ J=\#e_0}} && \quad (D.12) \\ & = \sum_{k_2=r}^{J+r} \frac{(2\pi i)^{J+r}}{(J-(k_2-r))!k_2!} e_0^{J-(k_2-r)} \left[ (\hat{N}_+ - e_0)^{k_2} \Big|_{\substack{r=\#\hat{N}_+}} \right] \\ & = \sum_{\ell_0, \dots, \ell_{r+1}=0}^{J} \delta(\ell_0+\ell_1+\cdots+\ell_{r+1}-J) \frac{(2\pi i)^{J+r}}{(J+r-\ell_0)!\ell_0!} (-1)^{J-\ell_0} e_0^{\ell_0+\ell_1} \hat{N}_+ e_0^{\ell_2} \hat{N}_+ \cdots \hat{N}_+ e_0^{\ell_{r+1}}, \end{aligned} $$

where we used the fact that $k_2$ must be at least $r$ as to obtain enough powers of $(\hat{N}_+-e_0)$ such that $r = \#\hat{N}_+$ can be satisfied, and it must be at most $J+r$ otherwise we would get too many $e_0$ to satisfy $J = \#e_0$. In the last line we simply changed variables $k_2 \to \ell_0 = J-(k_2-r) \in \{0, \dots, J\}$.

The final step is to change summation variables $b_1 := \ell_0+\ell_1$ while $b_i = \ell_i$ for $i = 2, \dots, r+1$. If we substitute for $\ell_1 = b_1-\ell_0$ and perform the sum over $\ell_0 \in \{0, \dots, b_1\}$ we simply have:

$$ \sum_{\ell_0=0}^{b_1} \frac{(-1)^{J-\ell_0}}{(J+r-\ell_0)!\ell_0!} = \frac{(-1)^{J-b_1}}{b_1!(J+r-b_1-1)!(J+r)}, \quad (D.13) $$

so that our expression (D.12) identically reproduces (D.1).
## E Zeta generators at modular depth three

In this appendix, we add explicit examples at modular depth three to the discussion of zeta generators in section 2.4.2. The subsequent expressions apply to the uplift $\hat{\sigma}_w$ adapted to the free-algebra generators $e_k$ and can be straightforwardly specialised to the zeta generators $\sigma_w$ associated with the derivations $\epsilon_k$ on Lie[a, b].

### E.1 [$\hat{z}_w$, $e_k$] beyond modular depth two

For the arithmetic terms $\hat{z}_w$ of $\hat{\sigma}_w$, the modular-depth-two contributions to their commutators with $e_k$ are known in a simple closed form (2.81) in terms of the $t^d$ operation in (2.60). We shall here give the terms of modular depth three in [$\hat{z}_w$, $e_k$] for all degrees $2w+k \le 18$ and refer to tentative extra contributions of modular depth four via ellipses.

* $w=3$

$$ [\hat{z}_3, e_4] = \frac{\mathrm{BF}_6}{\mathrm{BF}_4} t^4(e_4, e_6), \qquad (\mathrm{E.1}) $$

$$ [\hat{z}_3, e_6] = \frac{\mathrm{BF}_8}{\mathrm{BF}_6} t^4(e_4, e_8) - \frac{9\mathrm{BF}_4^2}{10\mathrm{BF}_6} t^2(e_4, t^3(e_4, e_4)), $$

$$ [\hat{z}_3, e_8] = \frac{\mathrm{BF}_{10}}{\mathrm{BF}_8} t^4(e_4, e_{10}) + \frac{\mathrm{BF}_4\mathrm{BF}_6}{\mathrm{BF}_8} \left\{ -\frac{15}{7} t^2(e_4, t^3(e_4, e_6)) + \frac{9}{4} t^3(e_4, t^2(e_4, e_6)) \right\}, $$

$$ [\hat{z}_3, e_{10}] = \frac{\mathrm{BF}_{12}}{\mathrm{BF}_{10}} t^4(e_4, e_{12}) + \frac{\mathrm{BF}_4\mathrm{BF}_8}{\mathrm{BF}_{10}} \left\{ -\frac{7}{3} t^2(e_4, t^3(e_4, e_8)) + \frac{12}{5} t^3(e_4, t^2(e_4, e_8)) \right\} \\ + \frac{\mathrm{BF}_6^2}{\mathrm{BF}_{10}} \left\{ -\frac{25}{27} t^2(e_6, t^3(e_4, e_6)) + \frac{2}{9} t^3(e_6, t^2(e_4, e_6)) \right\}, $$

$$ [\hat{z}_3, e_{12}] = \frac{\mathrm{BF}_{14}}{\mathrm{BF}_{12}} t^4(e_4, e_{14}) + \frac{\mathrm{BF}_4\mathrm{BF}_{10}}{\mathrm{BF}_{12}} \left\{ -\frac{27}{11} t^2(e_4, t^3(e_4, e_{10})) + \frac{5}{2} t^3(e_4, t^2(e_4, e_{10})) \right\} \\ + \frac{\mathrm{BF}_6\mathrm{BF}_8}{\mathrm{BF}_{12}} \left\{ -\frac{35}{44} t^2(e_6, t^3(e_4, e_8)) + \frac{5}{33} t^3(e_6, t^2(e_4, e_8)) \right. \\ \left. - \frac{35}{33} t^2(e_8, t^3(e_4, e_6)) + \frac{7}{22} t^3(e_8, t^2(e_4, e_6)) \right\} + \dots $$

* $w=5$

$$ [\hat{z}_5, e_4] = \frac{\mathrm{BF}_8}{\mathrm{BF}_4} t^6(e_6, e_8) + \frac{\mathrm{BF}_6\mathrm{BF}_2^3}{2\mathrm{BF}_4^2} t^4(e_6, t^3(e_4, e_4)) \qquad (\mathrm{E.2}) \\ - \mathrm{BF}_4 \left\{ \frac{9}{10} t^3(e_4, t^4(e_4, e_6)) + \frac{1}{5} t^4(e_4, t^3(e_4, e_6)) \right\}, $$

$$ [\hat{z}_5, e_6] = \frac{\mathrm{BF}_{10}}{\mathrm{BF}_6} t^6(e_6, e_{10}) + \frac{\mathrm{BF}_8\mathrm{BF}_2^3}{2\mathrm{BF}_4\mathrm{BF}_6} t^4(e_8, t^3(e_4, e_4)) \\ - \mathrm{BF}_4 \left\{ 2t^3(e_6, t^4(e_4, e_6)) + \frac{20}{7} t^4(e_6, t^3(e_4, e_6)) + \frac{5}{2} t^5(e_6, t^2(e_4, e_6)) \right\} + \dots, $$
$$ [\hat{z}_5, e_8] = \frac{\mathrm{BF}_{12}}{\mathrm{BF}_8} t^6(e_6, e_{12}) + \frac{\mathrm{BF}_{10} \mathrm{BF}_2^3}{2\mathrm{BF}_4 \mathrm{BF}_8} t^4(e_{10}, t^3(e_4, e_4)) - \frac{125 \mathrm{BF}_6^2}{84 \mathrm{BF}_8} t^2(e_6, t^5(e_6, e_6)) \\ + \mathrm{BF}_4 \left\{ -\frac{15}{14} t^2(e_4, t^5(e_6, e_8)) + \frac{3}{8} t^3(e_4, t^4(e_6, e_8)) - \frac{1}{30} t^4(e_4, t^3(e_6, e_8)) \right. \\ \left. - \frac{27}{20} t^3(e_8, t^4(e_4, e_6)) - \frac{5}{6} t^4(e_8, t^3(e_4, e_6)) - \frac{1}{6} t^5(e_8, t^2(e_4, e_6)) \right\} + \dots $$

$$ \bullet \quad w = 7 $$

$$ [\hat{z}_7, e_4] = \frac{\mathrm{BF}_{10}}{\mathrm{BF}_4} t^8(e_8, e_{10}) + \frac{\mathrm{BF}_8 \mathrm{BF}_2^2}{\mathrm{BF}_6} t^6(e_8, t^3(e_4, e_6)) + \frac{\mathrm{BF}_6 \mathrm{BF}_2^2}{2\mathrm{BF}_4} t^4(e_6, t^5(e_6, e_6)) \\ - \mathrm{BF}_6 \left\{ \frac{15}{14} t^3(e_4, t^6(e_6, e_8)) + \frac{5}{14} t^4(e_4, t^5(e_6, e_8)) \right. \\ \left. + \frac{5}{7} t^5(e_4, t^4(e_6, e_8)) + \frac{3}{28} t^6(e_4, t^3(e_6, e_8)) \right\} + \dots \quad (\mathrm{E.3}) $$

These expressions are derived from the commutation relation (2.80) with $\hat{N}$ based on the terms of modular depth two in (2.83), see [34] for further details and generalisations.

## E.2 Expansion of zeta generators

We shall next supplement the closed formula (2.83) for the expansion of zeta generators up to modular depth two by contributions at modular depth three. The following examples are given up to and including degree 14

$$ \hat{\sigma}_3 = -\frac{1}{2}e_4^{(2)} + \hat{z}_3 + \frac{s_{4,4}^5}{1440} - \frac{[e_4, e_6^{(1)}]}{120960} + \frac{[e_4^{(1)}, e_6]}{30240} + \frac{[e_4, e_8^{(1)}]}{7257600} - \frac{[e_4^{(1)}, e_8]}{1209600} \quad (\mathrm{E.4}) \\ - \frac{[e_4, e_{10}^{(1)}]}{383201280} + \frac{[e_4^{(1)}, e_{10}]}{47900160} - \frac{[e_4, [e_4, e_6]]}{58060800} + \dots, $$

$$ \hat{\sigma}_5 = -\frac{1}{4!}e_6^{(4)} + \frac{5s_{4,4}^3}{24} + \hat{z}_5 + \frac{s_{4,6}^5}{720} - \frac{s_{6,6}^7}{60480} + \frac{1}{6912}([e_4^{(1)}, [e_4^{(1)}, e_4^{(0)}]] + 2[e_4^{(0)}, [e_4^{(0)}, e_4^{(2)}]]) \\ + \frac{[e_6, e_8^{(3)}]}{145152000} - \frac{[e_6^{(1)}, e_8^{(2)}]}{36288000} + \frac{[e_6^{(2)}, e_8^{(1)}]}{14515200} - \frac{[e_6^{(3)}, e_8]}{7257600} \\ - \frac{[e_4, [e_4, e_6^{(2)}]]}{2073600} - \frac{23[e_4, [e_4^{(1)}, e_6^{(1)}]]}{24192000} + \frac{289[e_4, [e_4^{(2)}, e_6]]}{48384000} \\ + \frac{139[e_4^{(1)}, [e_4, e_6^{(1)}]]}{72576000} - \frac{[e_4^{(1)}, [e_4^{(1)}, e_6]]}{4147200} - \frac{1007[e_4^{(2)}, [e_4, e_6]]}{145152000} + \dots, $$

$$ \hat{\sigma}_7 = -\frac{1}{6!}e_8^{(6)} + \frac{7s_{4,6}^3}{24} + \frac{s_{4,8}^5}{720} - \frac{s_{6,6}^5}{288} - \frac{661s^3(e_4, t^3(e_4, e_4))}{14400} \\ + \hat{z}_7 - \frac{s_{6,8}^7}{30240} + \frac{7s^5(e_4, t^3(e_4, e_6))}{17280} + \dots, $$
where $s_{p,q}^d = s^d(e_p, e_q)$ are defined by (2.61). The next examples are displayed up to and including degree 16 (omitting tentative contributions of modular depth four and degree 16):

$$ \hat{\sigma}_9 = -\frac{1}{8!}e_{10}^{(8)} + \frac{5s_{4,8}^3}{18} + \frac{7s_{6,6}^3}{72} + \frac{s_{4,10}^5}{720} - \frac{7s_{6,8}^5}{1440} \qquad (E.5) \\ + \frac{34921s^2(e_4, t^4(e_4, e_6))}{1134000} + \frac{2587s^3(e_4, t^3(e_4, e_6))}{37800} - \frac{529s^4(e_4, t^2(e_4, e_6))}{14400} \\ - \frac{s_{6,10}^7}{30240} + \frac{s_{8,8}^7}{12096} + \frac{s^5(e_4, t^3(e_4, e_8))}{2592} + \frac{7s^5(e_4, t^3(e_6, e_6))}{51840} \\ - \frac{34921s^4(e_6, t^4(e_6, e_4))}{47628000} - \frac{2587s^5(e_6, t^3(e_6, e_4))}{1587600} + \frac{529s^6(e_6, t^2(e_6, e_4))}{604800} + \dots $$

$$ \hat{\sigma}_{11} = -\frac{1}{10!}e_{12}^{(10)} + \frac{11s_{4,10}^3}{40} + \frac{11s_{6,8}^3}{60} + \frac{242407}{14735232}s^2(e_4, t^2(e_4, e_6)) + \frac{s_{4,12}^5}{720} - \frac{s_{6,10}^5}{216} - \frac{7s_{8,8}^5}{4320} \\ + \frac{11090423s^2(e_4, t^4(e_4, e_8))}{309439872} + \frac{3197s^3(e_4, t^3(e_4, e_8))}{57600} - \frac{2983s^4(e_4, t^2(e_4, e_8))}{86400} \\ + \frac{148753s^3(e_4, t^3(e_6, e_6))}{7367616} + \frac{490853s^3(e_6, t^3(e_6, e_4))}{17191104} + \frac{156805s^4(e_6, t^2(e_6, e_4))}{14735232} + \dots $$

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## References

[1] M.B. Green and P. Vanhove, *The Low-energy expansion of the one loop type II superstring amplitude*, Phys. Rev. D 61 (2000) 104011 [hep-th/9910056] [INSPIRE].

[2] M.B. Green, J.G. Russo and P. Vanhove, *Low energy expansion of the four-particle genus-one amplitude in type II superstring theory*, JHEP 02 (2008) 020 [arXiv:0801.0322] [INSPIRE].

[3] E. D'Hoker, M.B. Green and P. Vanhove, *On the modular structure of the genus-one Type II superstring low energy expansion*, JHEP 08 (2015) 041 [arXiv:1502.06698] [INSPIRE].

[4] J.E. Gerken, A. Kleinschmidt and O. Schlotterer, *Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings*, JHEP 01 (2019) 052 [arXiv:1811.02548] [INSPIRE].

[5] E. D'Hoker, M.B. Green, Ö. Gürdogan and P. Vanhove, *Modular graph functions*, Commun. Num. Theor. Phys. 11 (2017) 165 [arXiv:1512.06779] [INSPIRE].

[6] E. D'Hoker and M.B. Green, *Identities between modular graph forms*, J. Number Theor. 189 (2018) 25 [arXiv:1603.00839] [INSPIRE].

[7] E. D'Hoker and M.B. Green, *Exploring transcendentality in superstring amplitudes*, JHEP 07 (2019) 149 [arXiv:1906.01652] [INSPIRE].

[8] D. Dorigoni, M.B. Green and C. Wen, *The SAGEX review on scattering amplitudes Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their supersymmetric Yang-Mills duals*, J. Phys. A 55 (2022) 443011 [arXiv:2203.13021] [INSPIRE].

[9] J. Broedel, O. Schlotterer and F. Zerbini, *From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop*, JHEP 01 (2019) 155 [arXiv:1803.00527] [INSPIRE].
[10] J.E. Gerken et al., Towards closed strings as single-valued open strings at genus one, J. Phys. A 55 (2022) 025401 [arXiv:2010.10558] [INSPIRE].

[11] F. Brown, Multiple modular values and the relative completion of the fundamental group of $M_{1,1}$, arXiv:1407.5167 [INSPIRE].

[12] F. Zerbini, Single-valued multiple zeta values in genus 1 superstring amplitudes, Commun. Num. Theor. Phys. 10 (2016) 703 [arXiv:1512.05689] [INSPIRE].

[13] F. Brown, A class of non-holomorphic modular forms I, Res. Math. Sci. 5 (2018) 5:7 [arXiv:1707.01230] [INSPIRE].

[14] F. Brown, A class of non-holomorphic modular forms II: equivariant iterated Eisenstein integrals, Forum Math. Sigma 8 (2020) e31 [arXiv:1708.03354] [INSPIRE].

[15] D. Zagier and F. Zerbini, Genus-zero and genus-one string amplitudes and special multiple zeta values, Commun. Num. Theor. Phys. 14 (2020) 413 [arXiv:1906.12339] [INSPIRE].

[16] J. Drewitt, Laplace-eigenvalue equations for length three modular iterated integrals, J. Number Theory 239 (2022) 78 [arXiv:2104.09916].

[17] E. D'Hoker and W. Duke, Fourier series of modular graph functions, J. Number Theor. 192 (2018) 1 [arXiv:1708.07998] [INSPIRE].

[18] E. D'Hoker and M.B. Green, Absence of irreducible multiple zeta-values in melon modular graph functions, Commun. Num. Theor. Phys. 14 (2020) 315 [arXiv:1904.06603] [INSPIRE].

[19] P. Vanhove and F. Zerbini, Building blocks of closed and open string amplitudes, PoS MA2019 (2022) 022 [arXiv:2007.08981] [INSPIRE].

[20] O. Schnetz, Graphical functions and single-valued multiple polylogarithms, Commun. Num. Theor. Phys. 08 (2014) 589 [arXiv:1302.6445] [INSPIRE].

[21] F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA 2 (2014) e25 [arXiv:1309.5309] [INSPIRE].

[22] O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].

[23] S. Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A 47 (2014) 155401 [arXiv:1310.3259] [INSPIRE].

[24] S. Stieberger and T.R. Taylor, Closed String Amplitudes as Single-Valued Open String Amplitudes, Nucl. Phys. B 881 (2014) 269 [arXiv:1401.1218] [INSPIRE].

[25] O. Schlotterer and O. Schnetz, Closed strings as single-valued open strings: A genus-zero derivation, J. Phys. A 52 (2019) 045401 [arXiv:1808.00713] [INSPIRE].

[26] P. Vanhove and F. Zerbini, Single-valued hyperlogarithms, correlation functions and closed string amplitudes, Adv. Theor. Math. Phys. 26 (2022) 455 [arXiv:1812.03018] [INSPIRE].

[27] F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, Commun. Math. Phys. 382 (2021) 815 [arXiv:1910.01107] [INSPIRE].

[28] D. Dorigoni et al., Modular graph forms from equivariant iterated Eisenstein integrals, JHEP 12 (2022) 162 [arXiv:2209.06772] [INSPIRE].

[29] J.E. Gerken, A. Kleinschmidt and O. Schlotterer, All-order differential equations for one-loop closed-string integrals and modular graph forms, JHEP 01 (2020) 064 [arXiv:1911.03476] [INSPIRE].
[30] J.E. Gerken, A. Kleinschmidt and O. Schlotterer, *Generating series of all modular graph forms from iterated Eisenstein integrals*, JHEP 07 (2020) 190 [arXiv:2004.05156] [INSPIRE].

[31] A.B. Goncharov, *Galois symmetries of fundamental groupoids and noncommutative geometry*, Duke Math. J. 128 (2005) 209 [math/0208144] [INSPIRE].

[32] F. Brown, *Mixed Tate motives over $Z$*, Annals Math. 175 (2012) 949 [arXiv:1102.1312].

[33] F. Brown, *On the decomposition of motivic multiple zeta values*, in Galois-Teichmüller theory and arithmetic geometry, Adv. Stud. Pure Math. 63 (2012) 31 [arXiv:1102.1310] [INSPIRE].

[34] D. Dorigoni et al., *Canonicalizing zeta generators: genus zero and genus one*, arXiv:2406.05099 [INSPIRE].

[35] H. Tsunogai, *On Some Derivations of Lie Algebras Related to Galois Representations*, Publ. Res. Inst. Math. Sci. 31 (1995) 113.

[36] A. Pollack, *Relations between derivations arising from modular forms*, Undergraduate thesis, Duke University, U.S.A., https://dukespace.lib.duke.edu/dspace/handle/10161/1281, (2009).

[37] D. Dorigoni, A. Kleinschmidt and O. Schlotterer, *Poincaré series for modular graph forms at depth two. Part II. Iterated integrals of cusp forms*, JHEP 01 (2022) 134 [arXiv:2109.05018] [INSPIRE].

[38] F. Brown, *From the Deligne-Ihara conjecture to multiple modular values*, arXiv:1904.00179.

[39] P.P. Deligne, *Le groupe fondamental de la droite projective moins trois points*, in Galois Groups over $Q$, Y. Ihara, K. Ribet and J.-P. Serre, eds., Springer US, New York, NY, U.S.A. (1989), pp. 79–297.

[40] Y. Ihara, *Braids, Galois groups, and some arithmetic functions*, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, Japan (1991), pp. 99–120.

[41] Y. Ihara and N. Takao, *A seminar talk*, (1993).

[42] A.B. Goncharov, *Multiple $\zeta$-values*, Galois groups, and geometry of modular varieties, 3rd European Congress of Mathematics: Shaping the 21st Century 201 (2001) 361 [INSPIRE].

[43] H. Gangl, M. Kaneko and D. Zagier, *Double zeta values and modular forms*, in Automorphic forms and zeta functions, pp. 71–106. World Sci. Publ., Hackensack, NJ, U.S.A. (2006) [DOI:10.1142/9789812774415_0004].

[44] L. Schneps, *On the Poisson bracket on the free Lie algebra in two generators*, J. Lie Theory 16 (2006) 19.

[45] F. Brown, *Anatomy of the motivic Lie algebra*, talk given at the program Grothendieck-Teichmüller Groups, Deformation and Operads, Newton Institute, Cambridge, U.K. (2013) https://sms.cam.ac.uk/media/1459610.

[46] F. Brown, *Zeta Elements in Depth 3 and the Fundamental Lie Algebra of The infinitesimal Tate Curve*, Forum Math. Sigma 5 (2017) e1 [arXiv:1504.04737] [INSPIRE].

[47] S. Baumard and L. Schneps, *On the derivation representation of the fundamental Lie algebra of mixed elliptic motives*, Ann. Math. Qué. 41 (2016) 43 [arXiv:1510.05549].

[48] R. Hain and M. Matsumoto, *Universal mixed elliptic motives*, J. Inst. Math. Jussieu 19 (2018) 663 [arXiv:1512.03975].

[49] C.R. Mafra and O. Schlotterer, *Tree-level amplitudes from the pure spinor superstring*, Phys. Rept. 1020 (2023) 1 [arXiv:2210.14241] [INSPIRE].
[50] H. Frost et al., *Motivic coaction and single-valued map of polylogarithms from zeta generators*, *J. Phys. A* **57** (2024) 31LT01 [arXiv:2312.00697] [INSPIRE].

[51] F. Brown, *Polylogarithmes multiples uniformes en une variable*, *Compt. Rend. Math.* **338** (2004) 527 [INSPIRE].

[52] Y. Ihara, *On the stable derivation algebra associated with some braid groups*, *Isr. J. Math.* **80** (1992) 35-153.

[53] H. Furusho, *The multiple zeta value algebra and the stable derivation algebra*, *Publ. Res. Inst. Math. Sci.* **39** (2003) 695 http://projecteuclid.org/euclid.prims/1145476044 [math/0011261].

[54] J.I. Manin, *Periods of cusp forms, and p-adic Hecke series*, *Mat. Sb. (N.S.)* **92**(134) (1973) 378-401, 503.

[55] J.L. Bourjaily et al., *Functions Beyond Multiple Polylogarithms for Precision Collider Physics*, in the proceedings of the *Snowmass 2021*, Seattle, U.S.A., July 17-26 (2022) [arXiv:2203.07088] [INSPIRE].

[56] S. Abreu, R. Britto and C. Duhr, *The SAGEX review on scattering amplitudes Chapter 3: Mathematical structures in Feynman integrals*, *J. Phys. A* **55** (2022) 443004 [arXiv:2203.13014] [INSPIRE].

[57] J.E. Gerken, *Modular Graph Forms and Scattering Amplitudes in String Theory*, Ph.D. thesis, Humboldt University, Berlin, Germany (2020) [arXiv:2011.08647] [INSPIRE].

[58] N. Berkovits et al., *Snowmass White Paper: String Perturbation Theory*, in the proceedings of the *Snowmass 2021*, Seattle, U.S.A., July 17-26 (2022) [arXiv:2203.09099] [INSPIRE].

[59] E. D'Hoker and J. Kaidi, *Lectures on modular forms and strings*, arXiv:2208.07242 [INSPIRE].

[60] D. Zagier, *Periods of modular forms and Jacobi theta functions*, *Invent. Math.* **104** (1991) 449.

[61] H. Maass, *Lectures on modular functions of one complex variable*, vol. 29 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tata Institute of Fundamental Research, Bombay, second ed., (1983).

[62] J.E. Gerken and J. Kaidi, *Holomorphic subgraph reduction of higher-point modular graph forms*, *JHEP* **01** (2019) 131 [arXiv:1809.05122] [INSPIRE].

[63] A. Basu, *Poisson equation for the Mercedes diagram in string theory at genus one*, *Class. Quant. Grav.* **33** (2016) 055005 [arXiv:1511.07455] [INSPIRE].

[64] A. Kleinschmidt and V. Verschinin, *Tetrahedral modular graph functions*, *JHEP* **09** (2017) 155 [arXiv:1706.01889] [INSPIRE].

[65] E. D'Hoker, M.B. Green and P. Vanhove, *Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus*, *J. Number Theor.* **196** (2019) 381 [arXiv:1509.00363] [INSPIRE].

[66] A. Basu, *Proving relations between modular graph functions*, *Class. Quant. Grav.* **33** (2016) 235011 [arXiv:1606.07084] [INSPIRE].

[67] E. D'Hoker and J. Kaidi, *Hierarchy of Modular Graph Identities*, *JHEP* **11** (2016) 051 [arXiv:1608.04393] [INSPIRE].

[68] D.M. Richards, *The One-Loop Five-Graviton Amplitude and the Effective Action*, *JHEP* **10** (2008) 042 [arXiv:0807.2421] [INSPIRE].

[69] M.B. Green, C.R. Mafra and O. Schlotterer, *Multiparticle one-loop amplitudes and S-duality in closed superstring theory*, *JHEP* **10** (2013) 188 [arXiv:1307.3534] [INSPIRE].
[70] A. Basu, Simplifying the one loop five graviton amplitude in type IIB string theory, Int. J. Mod. Phys. A 32 (2017) 1750074 [arXiv:1608.02056] [INSPIRE].

[71] J.E. Gerken, Basis Decompositions and a Mathematica Package for Modular Graph Forms, J. Phys. A 54 (2021) 195401 [arXiv:2007.05476] [INSPIRE].

[72] D. Zagier and H. Gangl, Classical and elliptic polylogarithms and special values of L-series, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), vol. 548 of NATO Sci. Ser. C Math. Phys. Sci., pp. 561–615. Kluwer Acad. Publ., Dordrecht, The Netherlands (2000).

[73] N. Matthes, On the algebraic structure of iterated integrals of quasimodular forms, Alg. Numb. Theor. 11 (2017) 2113 [arXiv:1708.04561].

[74] D. Dorigoni, A. Kleinschmidt and O. Schlotterer, Poincaré series for modular graph forms at depth two. Part I. Seeds and Laplace systems, JHEP 01 (2022) 133 [arXiv:2109.05017] [INSPIRE].

[75] J. Broedel, N. Matthes and O. Schlotterer, Relations between elliptic multiple zeta values and a special derivation algebra, J. Phys. A 49 (2016) 155203 [arXiv:1507.02254] [INSPIRE].

[76] J.-G. Luque, J.-C. Novelli and J.-Y. Thibon, Period polynomials and Ihara brackets, math/0606301.

[77] R. Hain, Notes on the universal elliptic KZB connection, Pure Appl. Math. Quart. 16 (2020) 229 [arXiv:1309.0580] [INSPIRE].

[78] J. Broedel, N. Matthes and O. Schlotterer. https://tools.aei.mpg.de/emzv.

[79] D. Calaque, B. Enriquez and P. Etingof, Universal KZB equations: the elliptic case, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math. 269 (2009) 165, Birkhäuser Boston, Boston, MA, U.S.A. (2009) [math/0702670].

[80] B. Enriquez, Elliptic associators, Selecta Math. 20 (2013) 491 [arXiv:1003.1012].

[81] L. Schneps, Elliptic double shuffle, Grothendieck-Teichmüller and mould theory, Ann. Math. Qué. 44 (2020) 261 [arXiv:1506.09050].

[82] F. Brown, A class of non-holomorphic modular forms III: real analytic cusp forms for SL₂(Z), Res. Math. Sci. 5 (2018) 36 [arXiv:1710.07912].

[83] S. Abreu, R. Britto, C. Duhr and E. Gardi, Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction, Phys. Rev. Lett. 119 (2017) 051601 [arXiv:1703.05064] [INSPIRE].

[84] S. Abreu, R. Britto, C. Duhr and E. Gardi, Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case, JHEP 12 (2017) 090 [arXiv:1704.07931] [INSPIRE].

[85] R. Britto, S. Mizera, C. Rodriguez and O. Schlotterer, Coaction and double-copy properties of configuration-space integrals at genus zero, JHEP 05 (2021) 053 [arXiv:2102.06206] [INSPIRE].

[86] A. Saad, Multiple zeta values and iterated Eisenstein integrals, arXiv:2009.09885.

[87] P. Lochak, N. Matthes and L. Schneps, Elliptic Multizetas and the Elliptic Double Shuffle Relations, Int. Math. Res. Not. 2021 (2020) 695 [arXiv:1703.09410].

[88] P. Fleig, H.P.A. Gustafsson, A. Kleinschmidt and D. Persson, Eisenstein series and automorphic representations, Cambridge University Press (2018) [DOI:10.1017/9781316995860] [INSPIRE].

[89] K. Klinger-Logan, Differential equations in automorphic forms, Commun. Num. Theor. Phys. 12 (2018) 767 [INSPIRE].
[90] M.B. Green, S.D. Miller and P. Vanhove, *SL(2, Z)-invariance and D-instanton contributions to the D⁶R⁴ interaction*, Commun. Num. Theor. Phys. **09** (2015) 307 [arXiv:1404.2192] [INSPIRE].

[91] K. Fedosova, K. Klinger-Logan and D. Radchenko, *Convolution identities for divisor sums and modular forms*, arXiv:2312.00722 [INSPIRE].

[92] L.F. Alday et al., *Relations between integrated correlators in $N=4$ supersymmetric Yang-Mills theory*, JHEP **05** (2024) 044 [arXiv:2310.12322] [INSPIRE].

[93] D. Dorigoni, A. Kleinschmidt and R. Treilis, *To the cusp and back: resurgent analysis for modular graph functions*, JHEP **11** (2022) 048 [arXiv:2208.14087] [INSPIRE].

[94] D. Dorigoni and R. Treilis, *Two string theory flavours of generalised Eisenstein series*, JHEP **11** (2023) 102 [arXiv:2307.07491] [INSPIRE].

[95] M. Eichler, *Eine Verallgemeinerung der Abelschen Integrale*, Math. Z. **67** (1957) 267.

[96] P. Goro Shimura, *Sur les intégrales attachées aux formes automorphes*, J. Math. Soc. Jap. **11** (1959) 291.

[97] E. D'Hoker, M.B. Green and B. Pioline, *Asymptotics of the D⁸R⁴ genus-two string invariant*, Commun. Num. Theor. Phys. **13** (2019) 351 [arXiv:1806.02691] [INSPIRE].

[98] E. D'Hoker, A. Kleinschmidt and O. Schlotterer, *Elliptic modular graph forms. Part I. Identities and generating series*, JHEP **03** (2021) 151 [arXiv:2012.09198] [INSPIRE].

[99] M. Hidding, O. Schlotterer and B. Verbeek, *Elliptic modular graph forms II: Iterated integrals*, arXiv:2208.11116 [INSPIRE].

[100] E. D'Hoker, M.B. Green and B. Pioline, *Higher genus modular graph functions, string invariants, and their exact asymptotics*, Commun. Math. Phys. **366** (2019) 927 [arXiv:1712.06135] [INSPIRE].

[101] E. D'Hoker and O. Schlotterer, *Identities among higher genus modular graph tensors*, Commun. Num. Theor. Phys. **16** (2022) 35 [arXiv:2010.00924] [INSPIRE].

[102] B. Pioline, *A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces*, J. Number Theor. **163** (2016) 520 [arXiv:1504.04182] [INSPIRE].

[103] B. Enriquez, *Analogues elliptiques des nombres multizétas*, Bull. Soc. Math. Fr. **144** (2016) 395 [arXiv:1301.3042].

[104] C.R. Mafra and O. Schlotterer, *All Order α' Expansion of One-Loop Open-String Integrals*, Phys. Rev. Lett. **124** (2020) 101603 [arXiv:1908.09848] [INSPIRE].

[105] C.R. Mafra and O. Schlotterer, *One-loop open-string integrals from differential equations: all-order α'-expansions at n points*, JHEP **03** (2020) 007 [arXiv:1908.10830] [INSPIRE].

[106] S. Stieberger, *A Relation between One-Loop Amplitudes of Closed and Open Strings (One-Loop KLT Relation)*, arXiv:2212.06816 [INSPIRE].

[107] S. Stieberger, *One-Loop Double Copy Relation in String Theory*, Phys. Rev. Lett. **132** (2024) 191602 [arXiv:2310.07755] [INSPIRE].

[108] R. Bhardwaj, A. Pokraka, L. Ren and C. Rodriguez, *A double copy from twisted (co)homology at genus one*, JHEP **07** (2024) 040 [arXiv:2312.02148] [INSPIRE].

[109] P. Mazloumi and S. Stieberger, *One-loop Double Copy Relation from Twisted (Co)homology*, arXiv:2403.05208 [INSPIRE].
[110] E. D'Hoker, *Integral of two-loop modular graph functions*, JHEP 06 (2019) 092
[arXiv:1905.06217] [INSPIRE].

[111] E. D'Hoker and N. Geiser, *Integrating three-loop modular graph functions and transcendentality of string amplitudes*, JHEP 02 (2022) 019 [arXiv:2110.06237] [INSPIRE].

[112] M. Doroudiani, *Integral of depth zero to three basis of Modular Graph Functions*, JHEP 07 (2024) 029 [arXiv:2311.07287] [INSPIRE].

[113] D. Zagier, *The rankin-selberg method for automorphic functions which are not of rapid decay*, J. Fac. Sci. Tokyo (Sect. 1A Math.) 28 (1982) 415
https://apisemanticscholar.org/CorpusID:122093412.

[114] C. Angelantonj, I. Florakis and B. Pioline, *A new look at one-loop integrals in string theory*, Commun. Num. Theor. Phys. 6 (2012) 159 [arXiv:1110.5318] [INSPIRE].

[115] C. Angelantonj, I. Florakis and B. Pioline, *One-Loop BPS amplitudes as BPS-state sums*, JHEP 06 (2012) 070 [arXiv:1203.0566] [INSPIRE].

[116] A. Basu, *Poisson equation for the three loop ladder diagram in string theory at genus one*, Int. J. Mod. Phys. A 31 (2016) 1650169 [arXiv:1606.02203] [INSPIRE].

[117] N. Diamantis, *Modular iterated integrals associated with cusp forms*, Forum Math. 34 (2021) 157 [arXiv:2009.07128].

[118] J. Blümlein, D.J. Broadhurst and J.A.M. Vermaseren, *The Multiple Zeta Value Data Mine*, Comput. Phys. Commun. 181 (2010) 582 [arXiv:0907.2557] [INSPIRE].

[119] F. Brown, *Motivic periods and the projective line minus three points*, arXiv:1407.5165.

[120] F. Brown, *Notes on Motivic Periods*, Commun. Num. Theor. Phys. 11 (2017) 557
[arXiv:1512.06410] [INSPIRE].

[121] A.B. Goncharov, *Multiple polylogarithms and mixed Tate motives*, math/0103059 [INSPIRE].

[122] D.J. Broadhurst and D. Kreimer, *Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops*, Phys. Lett. B 393 (1997) 403 [hep-th/9609128] [INSPIRE].

[123] J. Zhao, *Multiple zeta functions, multiple polylogarithms, and their special values*, World Scientific, New Jersey, U.S.A. (2016).

[124] J.I.B. Gil and J. Fresan, *Multiple zeta values: from numbers to motives*, Clay Mathematics Proceedings, to appear. http://javier.fresan.perso.math.cnrs.fr/mzv.pdf.

[125] J. Broedel, M. Sprenger and A. Torres Orjuela, *Towards single-valued polylogarithms in two variables for the seven-point remainder function in multi-Regge kinematics*, Nucl. Phys. B 915 (2017) 394 [arXiv:1606.08411] [INSPIRE].

[126] V. Del Duca et al., *Multi-Regge kinematics and the moduli space of Riemann spheres with marked points*, JHEP 08 (2016) 152 [arXiv:1606.08807] [INSPIRE].