%
\label{thm:geometric-bisquare}
%   II.D35, II.D60, II.T27, II.T32, II.T37, II.T40
The geometric bi-square
\textup{(}Definition~\textup{\ref{def:geometric-bisquare},}
II.D77\textup{)}
satisfies the following five properties:
\begin{enumerate}
    \item[\textup{(GB1)}] \textbf{Geometric filling.}
          Every algebraic component of the bi-square \textup{(}I.T41\textup{)}
          receives geometric content earned in Book~II:
          \begin{itemize}
              \item the domain $\Lemniscate$
                    receives the geometric body $S^1 \vee S^1$
                    \textup{(}II.T13\textup{)},
              \item the interior $\tau^3$
                    receives the Stone topology
                    \textup{(}II.T07\textup{)},
              \item the codomain $H_\tau$
                    receives calibration
                    \textup{(}II.D35\textup{)},
              \item the spectral side
                    receives $A_{\mathrm{spec}}(\Lemniscate)$
                    \textup{(}II.D60\textup{)}.
          \end{itemize}

    \item[\textup{(GB2)}] \textbf{Continuity.}
          All vertical arrows
          \textup{(}projections $\mathrm{proj}_k$\textup{)}
          are continuous maps
          in the inverse-limit topology
          \textup{(}by II.T06,
          Chapter~\textup{\ref{ch:hol-implies-cont})}.
          All horizontal arrows
          are $H_\tau^{\mathrm{cal}}$-valued ring homomorphisms.

    \item[\textup{(GB3)}] \textbf{Commutativity.}
          Both faces of the pasted diagram commute:
          \begin{itemize}
              \item Left face:
                    $\mathrm{proj}_k \circ f_\ell
                    = f_k \circ \mathrm{proj}_k$
                    for all $k \leq \ell$.
              \item Right face:
                    $(\mathrm{proj}_k, \mathrm{proj}_k) \circ (\chi_+, \chi_-)
                    = (\chi_+, \chi_-) \circ \mathrm{proj}_k$
                    for all $k \leq \ell$.
          \end{itemize}
          These are the sheaf condition
          \textup{(}II.T32\textup{)}
          and the spectral naturality
          \textup{(}inherited from the character decomposition,
          II.D59\textup{)},
          respectively.

    \item[\textup{(GB4)}] \textbf{Limit row = Central Theorem.}
          The limit row of the geometric bi-square
          \textup{(}\eqref{eq:ch59-limit-diagram}\textup{)}
          is the Central Theorem \textup{(}II.T40\textup{)}:
          the composition $\Psi \circ \Phi = \mathrm{id}$
          and $\Phi \circ \Psi = \mathrm{id}$
          establishes
          $\mathcal{O}(\tau^3) \cong A_{\mathrm{spec}}(\Lemniscate)$.

    \item[\textup{(GB5)}] \textbf{Compatibility with I.T41.}
          The geometric bi-square restricts to the algebraic bi-square
          at every finite stage:
          forgetting the topology, geometry, and calibration
          recovers the diagram of I.T41 exactly.
          The algebraic bi-square is the skeleton;
          the geometric bi-square is the same skeleton
          clothed in the geometric content of Book~II.
\end{enumerate}

The proof is an assembly of earned results.
We verify each property.

\medskip
\noindent
\textup{(GB1)}:
The domain $\Lemniscate$ receives the body $S^1 \vee S^1$
by the Torus Degeneration Theorem
(II.T13, Chapter~\ref{ch:torus-degeneration}):
the fiber $T^2$ degenerates to $S^1 \vee S^1$
via the pinch map (II.D18).
The interior $\tau^3$ receives the Stone topology
by Theorems~II.T07--II.T09
(Chapter~\ref{ch:stone-space}):
compact, Hausdorff, totally disconnected.
The codomain receives calibration
by Definition~II.D35
(Chapter~\ref{ch:split-complex-calibrated}):
the four earned transcendentals
$\pi$, $e$, $\jj$, $\iota_\tau$
equip $H_\tau^{\mathrm{cal}}$
with numerical content.
The spectral algebra $A_{\mathrm{spec}}(\Lemniscate)$
is constructed in Definition~II.D60
(Chapter~\ref{ch:central-theorem}).

\medskip
\noindent
\textup{(GB2)}:
The projections $\mathrm{proj}_k$
are continuous by II.T06
(Chapter~\ref{ch:hol-implies-cont}):
holomorphic implies continuous
in the inverse-limit topology on~$\tau^3$.
The horizontal arrows are ring homomorphisms
by construction:
$f_k$ is a stage function valued in $H_\tau^{\mathrm{cal}}$,
and $(\chi_+, \chi_-)$ is the idempotent decomposition
(II.D59, Chapter~\ref{ch:boundary-characters-idempotent}),
which is a ring homomorphism
because $e_\pm$ are idempotents.

\medskip
\noindent
\textup{(GB3)}:
The left face commutes
because holomorphic functions on~$\tau^3$
form a sheaf
(II.T32, Chapter~\ref{ch:sheaf-coherence}):
the sheaf axiom on the clopen cylinder basis
is precisely the statement
that restriction to a smaller stage
commutes with function evaluation.
The right face commutes
because the bipolar decomposition
$\chi = e_+ \cdot \chi_+ + e_- \cdot \chi_-$
is natural with respect to ring homomorphisms:
reduction maps $\mathrm{proj}_k$ are ring homomorphisms,
and idempotents are preserved by ring homomorphisms.

\medskip
\noindent
\textup{(GB4)}:
The limit row~\eqref{eq:ch59-limit-diagram}
is the Central Theorem (II.T40,
Chapter~\ref{ch:central-theorem}).
The map $\Phi$ is the boundary-to-interior Hartogs extension
(constructed in the proof of II.T40
using II.T37 and II.T39);
the map $\Psi$ is the interior-to-boundary restriction
(constructed using II.T27).
The mutual inverse property
$\Psi \circ \Phi = \mathrm{id}$
and $\Phi \circ \Psi = \mathrm{id}$
was proved in Chapter~\ref{ch:central-theorem}.

\medskip
\noindent
\textup{(GB5)}:
At each finite stage~$k$,
the geometric bi-square restricts to the diagram
\[
    \mathbb{Z}/M_k\mathbb{Z}
    \;\xrightarrow{\;\; f_k \;\;}\;
    H_\tau
    \;\xrightarrow{\;\; (\chi_+,\, \chi_-) \;\;}\;
    \widehat{\mathbb{Z}}_\tau \times \widehat{\mathbb{Z}}_\tau.
\]
The topology is discrete at finite stages
(every function on a finite set is continuous),
the calibration constants are still formal
at any single stage,
and the spectral decomposition
restricts to the finite spectral decomposition
of I.T41.
Hence the geometric bi-square
restricts to the algebraic bi-square.