In fact $d_2$ and $f_2$ integral can be directly computed by the package hyperlogprocedures [39, 54] (but not for $f$ integral), which proves to be extremely useful for these computations.

This is a special case of Coulomb branch amplitude where $y_i^2=0$.

Slingshot rule can only generate part of the ladder-type identities. We need to consider relation like ${\tilde{\mathcal{I}}}^{(\ell )}={\tilde{\mathcal{I}}}^{({\ell }^{\prime })}{\tilde{\mathcal{I}}}^{(\ell -{\ell }^{\prime })}$, which has the slingshot rule as its special case when ${\ell }^{\prime }=1$.

In fact, $M_{a,b}^{(\ell >ab)}$ vanishes exactly at the integrated level, and we found that they vanish at the integrand level up to $\ell =11$.

The ladder-equivalent integrals contribute $36F_6$, that is why the LHS. of (50) is nonzero.

Here $I^{(\ell )}$ denotes the integrand of DCI integrals defined in the appendix. Note that we always suppress a factor $1/(x_{13}^2{x}_{24}^2)$ in the definition. It should be distinguished from $I_{n,l}^{(l)}$ in the octagons, which always has two indices in the subscript.

This counting of ${\mathcal{L}}^{w=n}$ in Tab. 3 at weight $n$ does not include terms such as ${\zeta }_m{\mathcal{L}}^{w=n-m},m>0$.

In the definition of ${\mathcal{I}}^{d2}$ and ${\mathcal{I}}^{f2}$ of [27], the external legs are ordered as 1, 3, 2, 4. Therefore, in the expressions for ${\mathcal{I}}^{d2}$ and ${\mathcal{I}}^f$, $z$ and $\overline{z}$ should be replaced by $1/z$ and $1/\overline{z}$, and an overall factor $z\overline{z}$ should be factored out to get the expressions of order 1, 2, 3, 4.

For more details on the function’s usage and the meaning of every argument, refer to the documentation of hyperlogprocedures.

${\mathcal{I}}^{f2}$ is a fishnet, and its result is pre-implemented in hyperlogprocedures [54]. Thus, it is not explicitly calculated. In this sense, our result for ${\mathcal{I}}^f$ can also serve as a seed for hyperlogprocedures.

For those integrals that are not of uniform weight, the leading singularity analysis only accounts for the coefficients of the weight-10 part.

In the inverse boxing of ${\mathcal{I}}^f$, $1/{\mathrm{\Delta }}^2$ will result in the single-valued integration of the following type: $\int \frac{\mathrm{d}z\mathrm{d}\overline{z}}{z\overline{z}(1-z)(1-\overline{z})(z-\overline{z})}\times \text{pure functions}$, the measure of which is not of $\mathrm{d}\log$ type for $z,\overline{z}$. Thus, integration will result in a nonuniform-weight part.