# Further details for the box

This file records statements about the box that support section 6 and appendix B of the paper. Formulas are in LaTeX notation, and equation labels refer to the paper source.

## Monodromy and the exact test.

 For two quadrilaterals that do not lie in a plane, the monodromy of $D_0$ contains elements with integer traces 4, 6, 18 and 20. For one quadrilateral in a plane we find traces between 6 and 486. A logarithmic form on the same curve gives unipotent monodromy around the same loops. In the exact test the module generated by the residue~\eqref{eq:boxresidue} has rank six on one family of quadrilaterals and rank four on a second family. In both cases it contains the module of rank two of the periods of the curve. A logarithmic form on the same curve generates a module of rank one. Along the first family we have also obtained the Picard--Fuchs operator and the $j$-invariant of the curve exactly as functions of the shape. The logarithmic singular points of the operator are the shapes at which the curve degenerates.

## The fifth-order operator.

 The symmetric square of the operator has order 14 rather than 15, which is the quadratic relation quoted in section~\ref{sec:boxk3}. The solutions of the operator are free of logarithms at infinite $y_{41}+y_{12}$. The same operator arises in two other ways. It is the operator of the surfaces along the line of site energies $X_v=1+t$, where the pole of site~1 lies at $y_{41}+y_{12}=-(1+t)$. It is also the fifth-order block of the reduction on that line. With the substitution $y_{41}+y_{12}\to1+t$ the three operators are identical.

## Coupling to the fifth-order block.

 With several legs per site, the leading term of the residue at large site energies generates the full block, whereas a form of the opposite parity generates none of it. For momenta that form a regular tetrahedron of unit side, the branch points $\sigma_c$ of the period block in $\sigma=y_{41}+y_{12}$ are the roots of $\sigma^4-4\sigma^2+12$. A function that contains the block inherits a term $(\sigma-\sigma_c)^{5/2}$. Its coefficient relative to the form that generates the block is fixed by the residue~\eqref{eq:boxresidue} at the point where two singular points of the surface collide. In the Taylor coefficients of the single-site discontinuity at one leg per site the measured exponent is $2.507$, and the coefficient agrees with the prediction to one part in $10^{4}$. The same test with several legs per site, where the coupling is known exactly, agrees to one part in $10^{7}$. A discontinuity with a single square root shows no singularity at $\sigma_c$. We find the same result for a less symmetric family of momenta. In the exact test the module generated by the integrand of the coefficient has dimension 36. The quotient by the remaining forms on the single-site pole has dimension five, and its operator is the fifth-order operator. Two forms on the same pole with different numerators are not contained in the module. The module of the integrand of the correlator has dimension 34.

## The systems on the line.

 The systems obtained by setting $\eps=0$ first, or by keeping only simple poles in $\eps$, do not contain the physical functions. The systems that account for the double poles have dimension 81 for the coefficient and 67 for the correlator. Of the coefficients of the double poles, 20 vanish for the correlator and 23 for the coefficient. Imposing that they vanish, together with their derivatives along the line, restricts the boundary constants to an invariant subspace of dimension 34 for the correlator and 44 for the coefficient. The coefficient of the double pole of each master integral equals $0$, $-\pi/8$ or $-\pi/16$. With these values the quotient systems have dimension 28 for the correlator and 34 for the coefficient. These numbers bound the order of the differential equations that the two functions satisfy. The point $t=0$ is a singular point of both systems, but the physical functions are analytic there. Each elliptic block occurs twice in the systems, once for each order of the pole in $\eps$, and only the copies attached to the simple pole contribute to the physical functions. The orbit of the fitted solutions under monodromy has a component of rank two in each of these blocks. It also has a component in the block that contains the fifth-order operator, on which the monodromy is not unipotent.

## The elliptic curves on the line.

 The two subgraphs whose energies vanish are two adjacent sites, two opposite sites, or one site and the pair of adjacent sites that contains it. We denote the curves by $E_{\rm adj}$, $E_{\rm opp}$ and $E_{\rm nest}$. The second-order operator $\mathcal L_E$ of each curve satisfies an exact identity. With $J=1728/j(t)$, and with both sides normalized to unit leading coefficient in $\dd/\dd t$,
\begin{equation}
\mathcal L_E \;=\; c_4^{-1/4}\,\Bigl[\,J(1-J)\,\frac{\dd^2}{\dd J^2}+\Bigl(1-\frac{3J}{2}\Bigr)\frac{\dd}{\dd J}-\frac{5}{144}\,\Bigr]\,c_4^{1/4}\,,
\label{eq:boxhyp}
\end{equation}
where the operator in brackets annihilates ${}_2F_1(\tfrac1{12},\tfrac5{12};1;J)$ and is rewritten in the variable $t$. A basis of solutions of $\mathcal L_E$ is therefore $c_4^{-1/4}\,{}_2F_1(\tfrac1{12},\tfrac5{12};1;J)$ and its partner with a logarithm at $J=0$. The point $J=0$ is $j=\infty$, which covers every degeneration of the three curves. These include $t\to0$ for all three and the threshold $t=-1+\sqrt{3/2}$ of $E_{\rm opp}$. Here $c_4$ is a polynomial of degree eight in $t$, whose cube is the numerator of $j$ up to a constant factor. The three $j$-invariants have degree 24 in $t$, and we give them in the ancillary files. In none of the three cases is $t$ a modular function for the curve. Each map $t\mapsto j$ has critical points away from $j=0$, $1728$ and $\infty$, located at the apparent singularities of $\mathcal L_E$. The periods are therefore not modular forms in $t$.

## The functions $u$ and $v$.

 In the system of dimension 28, the solutions that are single-valued around $t=1$, $-1+\sqrt{3/2}$, $0$, $-1+1/\sqrt2$, $-1+2/\sqrt{13}$ and $-\tfrac12$ form the space of dimension four of section~\ref{sec:boxclosed}. Those that are in addition single-valued around the logarithmic point $t=-\tfrac23$ form a space of dimension three. For the coefficient the corresponding dimensions are eight and seven. These counts use no value of either function, and a fit restricted to these spaces reproduces the numerical values of both functions. The expansion of $v$ is even in $1/X$ and has rational coefficients,
\begin{equation}
v\;=\;X^{-4}+\frac{29}{80}\,X^{-6}+\frac{923}{12800}\,X^{-8}-\frac{1531}{204800}\,X^{-10}+O\bigl(X^{-12}\bigr)\,.
\label{eq:boxv}
\end{equation}
The expansion~\eqref{eq:boxlargeX} has no higher power of the logarithm, and $\beta_n=0$ for every even $n$. Both properties follow from the expansion by regions, in which the loop momentum is either of order $X$ or of order one. In terms of eq.~\eqref{eq:boxv} the coefficients are $\alpha_n=a_n\Omega+u_n$, with $a_n$ the coefficients of $v$, and all logarithms belong to $u$. Beyond the coefficients in eq.~\eqref{eq:boxlargeX}, the expansion by regions gives $\alpha_5=\tfrac{\pi}{12}\,(22+291\log3-108\log2)$, $\alpha_6=-\tfrac{95}{3}\pi+8\,\widetilde\Omega$ and $\alpha_7=\pi\bigl(\tfrac{22519}{360}+\tfrac{423}{8}\log3-16\log2\bigr)$. Here $\widetilde\Omega=-3.3142572208468856411918120617$ is the dimensionally regularized value of the integral in eq.~\eqref{eq:boxclosed} with $y_{12}^{-1}$ replaced by $y_{12}$. From the numerical values we identify in addition $\beta_9=-\tfrac{627}{10}\pi$ and $\beta_{11}=\tfrac{10247}{840}\pi$. The series converges for $|X|>2$, beyond the outermost singular point of the system. With the coefficients through $X^{-44}$, which we give in the ancillary files, it is the explicit form of $C_4$ for $X\ge14$. The numbers $\Omega$ and $\widetilde\Omega$ are known numerically only. The function $u$ is fixed by $\alpha_4=0$, by the exact values of $\beta_4$ to $\beta_8$ and $\alpha_5$, and by three boundary constants of the system that equal $-\pi/8$. None of these conditions constrains the coefficient of $v$, and the condition at $t=-\tfrac23$ does not constrain it either. The condition at $t=-1+1/\sqrt{10}$ has not been imposed for $C_4$. The ancillary files contain the boundary values of $u$ and $v$ and an evaluator.

## The coefficient.

 The expansion of $V_4$ at large $X$ starts at $X^{-5}$. Its coefficients through $X^{-11}$ are known exactly as rational combinations of $\pi$, $\pi\log2$ and $\pi\log3$, and its first logarithm appears at $X^{-11}$ with $\beta_{11}=-\tfrac{11}{21}\pi$. On the space of dimension eight these conditions leave two directions that are free of logarithms at large $X$, both entering at $X^{-12}$. At present $V_4=u_\psi+\Omega'_1v_1+\Omega'_2v_2$ with two undetermined constants. Single-valuedness at $t=-1+1/\sqrt{10}$ imposes one further condition and is expected to remove one of them. The remaining constant is expected to be the analogue of $\Omega$ in which the product of the four edge energies replaces its inverse. We have not shown this.