%
\label{thm:bi-square}
For each pair of primorial depths $k \leq l$,
the pasted bi-square
\[
    \begin{array}{ccccc}
        \mathbb{Z}/M_l\mathbb{Z}
        & \xrightarrow{\;\;T_l\;\;}
        & \mathbb{Z}/M_l\mathbb{Z}[\jj]
        & \xrightarrow{\;\;(\chi_+,\, \chi_-)\;\;}
        & \mathbb{Z}/M_l\mathbb{Z} \times \mathbb{Z}/M_l\mathbb{Z}
        \\[6pt]
        \downarrow\scriptstyle{\pi}
        & &
        \downarrow\scriptstyle{\pi}
        & &
        \downarrow\scriptstyle{(\pi,\pi)}
        \\[6pt]
        \mathbb{Z}/M_k\mathbb{Z}
        & \xrightarrow{\;\;T_k\;\;}
        & \mathbb{Z}/M_k\mathbb{Z}[\jj]
        & \xrightarrow{\;\;(\chi_+,\, \chi_-)\;\;}
        & \mathbb{Z}/M_k\mathbb{Z} \times \mathbb{Z}/M_k\mathbb{Z}
    \end{array}
\]
characterizes $\tau$-holomorphy completely:

\medskip
\noindent
\fbox{\parbox{0.93\textwidth}{%
A family $\{T_d\}_{d \geq 1}$ of functions
$T_d : \mathbb{Z}/M_d\mathbb{Z} \to \mathbb{Z}/M_d\mathbb{Z}[\jj]$
is $\tau$-holomorphic
if and only if both squares of the bi-square commute
for all $k \leq l$.}}

Three equivalences, each already established.
\emph{Left square commutes}
$\Leftrightarrow$ tower coherence (I.D46)
$\Leftrightarrow$ $T \in \mathrm{Nat}(F, F_\jj)$ (I.T40).
\emph{Right square commutes}
$\Leftrightarrow$ characters respect the tower
$\Leftrightarrow$ sector independence (I.P22).
\emph{Both together}
$\Leftrightarrow$ $\mathrm{HolFun}$ (I.D47).
The pasted outer rectangle also commutes
(composition of commuting squares).