%
\label{def:geometric-bisquare}
%   II.T06, II.T07, II.T13, II.D35, II.D60, II.T32, II.T37, II.T40
The \textbf{geometric bi-square}
is the pasted commuting diagram obtained from
the algebraic bi-square (I.T41)
by replacing every algebraic component
with its earned geometric counterpart.
The diagram has two levels:
a \textbf{stage diagram} (for each pair $k \leq \ell$)
and a \textbf{limit diagram} (the limiting row).

\medskip
\noindent
\textbf{Stage diagram} (for stages $k \leq \ell$):
\begin{equation}
\label{eq:ch59-stage-diagram}
    \begin{array}{ccccc}
    \tau^3_\ell
      & \xrightarrow{\;\; f_\ell \;\;}
      & H_\tau^{\mathrm{cal}}
      & \xrightarrow{\;\; (\chi_+,\, \chi_-) \;\;}
      & \widehat{\mathbb{Z}}_\tau \times \widehat{\mathbb{Z}}_\tau
    \\[8pt]
    \bigg\downarrow\vcenter{\rlap{\scriptsize\; $\mathrm{proj}_k$}}
      & & \bigg\downarrow\vcenter{\rlap{\scriptsize\; $\mathrm{proj}_k$}}
      & & \bigg\downarrow\vcenter{\rlap{\scriptsize\; $(\mathrm{proj}_k,\, \mathrm{proj}_k)$}}
    \\[8pt]
    \tau^3_k
      & \xrightarrow{\;\; f_k \;\;}
      & H_\tau^{\mathrm{cal}}
      & \xrightarrow{\;\; (\chi_+,\, \chi_-) \;\;}
      & \widehat{\mathbb{Z}}_\tau \times \widehat{\mathbb{Z}}_\tau
    \end{array}
\end{equation}
where:
\begin{itemize}
    \item $\tau^3_k$ denotes the stage-$k$ truncation of~$\tau^3$:
          the finite set of $\tau$-admissible ABCD quadruples
          modulo $P_k = p_1 \cdots p_k$,
          equipped with the discrete topology.
    \item $H_\tau^{\mathrm{cal}}$ is the calibrated split-complex codomain
          (II.D35, Chapter~\ref{ch:split-complex-calibrated}),
          carrying the constants $\pi$, $e$, $\jj$, $\iota_\tau$.
    \item $f_k \colon \tau^3_k \to H_\tau^{\mathrm{cal}}$
          is the stage-$k$ holomorphic function,
          and $f_\ell$ its stage-$\ell$ counterpart.
    \item $(\chi_+, \chi_-)$ is the bipolar spectral decomposition
          (II.D59, Chapter~\ref{ch:boundary-characters-idempotent}):
          $f = e_+ \cdot f_+ + e_- \cdot f_-$.
    \item $\mathrm{proj}_k$ denotes the \textbf{continuous} projection
          from stage~$\ell$ to stage~$k$
          in the inverse-limit topology
          (continuous by II.T06,
          Chapter~\ref{ch:hol-implies-cont}).
\end{itemize}

\medskip
\noindent
\textbf{Limit diagram} (the Central Theorem row):
\begin{equation}
\label{eq:ch59-limit-diagram}
    \begin{array}{ccccc}
    \Lemniscate = S^1 \vee S^1
      & \xrightarrow{\quad \Phi \quad}
      & \mathcal{O}(\tau^3)
      & \xrightarrow{\quad \Psi \quad}
      & A_{\mathrm{spec}}(\Lemniscate)
    \\[8pt]
    \bigg\downarrow\vcenter{\rlap{\scriptsize\; $\mathrm{proj}_d$}}
      & & \bigg\downarrow\vcenter{\rlap{\scriptsize\; $\mathrm{eval}_d$}}
      & & \bigg\downarrow\vcenter{\rlap{\scriptsize\; $\mathrm{proj}_d$}}
    \\[8pt]
    \Lemniscate_d
      & \xrightarrow{\;\; f_d \;\;}
      & H_\tau^{\mathrm{cal}}
      & \xrightarrow{\;\; (\chi_+,\, \chi_-) \;\;}
      & \mathbb{Z}/M_d\mathbb{Z} \times \mathbb{Z}/M_d\mathbb{Z}
    \end{array}
\end{equation}
where:
\begin{itemize}
    \item $\Lemniscate = S^1 \vee S^1$ is the geometric lemniscate
          (II.T13, Chapter~\ref{ch:torus-degeneration}):
          the torus $T^2$ degenerated to the wedge of two circles
          at the boundary.
    \item $\mathcal{O}(\tau^3)$ is the ring of holomorphic functions
          on~$\tau^3$.
    \item $A_{\mathrm{spec}}(\Lemniscate)$ is the spectral algebra
          (II.D60, Chapter~\ref{ch:central-theorem}).
    \item $\Phi \colon A_{\mathrm{spec}}(\Lemniscate) \to \mathcal{O}(\tau^3)$
          is the boundary-to-interior map
          (Hartogs extension followed by Yoneda identification).
    \item $\Psi \colon \mathcal{O}(\tau^3) \to A_{\mathrm{spec}}(\Lemniscate)$
          is the interior-to-boundary map
          (restriction followed by Mutual Determination).
    \item The composition $\Psi \circ \Phi = \mathrm{id}$
          and $\Phi \circ \Psi = \mathrm{id}$
          \textbf{is} the Central Theorem (II.T40).
    \item $\mathrm{eval}_d$ evaluates a holomorphic function
          at stage~$d$;
          $\mathrm{proj}_d$ projects the inverse limit
          to the $d$-th finite stage.
\end{itemize}