%
\label{thm:code-decode-bijection}
The maps
\[
    \mathrm{Code}
    \;:\;
    \mathrm{Hol}_\tau(\tau^3, H_\tau^{\mathrm{cal}})
    \;\longrightarrow\;
    \mathrm{Code}_\tau
    \qquad\text{and}\qquad
    \mathrm{Decode}
    \;:\;
    \mathrm{Code}_\tau
    \;\longrightarrow\;
    \mathrm{Hol}_\tau(\tau^3, H_\tau^{\mathrm{cal}})
\]
are mutually inverse bijections:
\[
    \boxed{%
    \mathrm{Code} \circ \mathrm{Decode}
    \;=\;
    \mathrm{id}_{\mathrm{Code}_\tau},
    \qquad
    \mathrm{Decode} \circ \mathrm{Code}
    \;=\;
    \mathrm{id}_{\mathrm{Hol}_\tau}.}
\]
Equivalently: \textbf{a $\tau$-holomorphic function is its bipolar
boundary coefficient stream}, and every tower-coherent
bipolar stream is a $\tau$-holomorphic function.

We verify both compositions.

\smallskip
\noindent\textbf{Step 1: $\mathrm{Decode} \circ \mathrm{Code} = \mathrm{id}$.}
Let $f \in \mathrm{Hol}_\tau(\tau^3, H_\tau^{\mathrm{cal}})$.
Set $\mathbf{c} := \mathrm{Code}(f)$.
We must show $\mathrm{Decode}(\mathbf{c}) = f$.

The code $\mathbf{c} = (c^+_f, c^-_f)$
records the spectral coefficients of~$f$
in both channels at every stage.
The Decode map reconstructs
stage-$k$ functions from these coefficients
via finite linear combinations
in the canonical basis.
But $f_k$ already \emph{has} this expansion
(by the uniqueness of the stage-$k$ spectral decomposition,
Proposition~\ref{prop:ch35-stage-k-decomposition}).
So the reconstructed stage-$k$ function
equals the original stage-$k$ function,
at every stage.
The inverse limits agree:
$\mathrm{Decode}(\mathbf{c}) = f$.

\smallskip
\noindent\textbf{Step 2: $\mathrm{Code} \circ \mathrm{Decode} = \mathrm{id}$.}
Let $\mathbf{c} = (c^+, c^-) \in \mathrm{Code}_\tau$.
Set $f := \mathrm{Decode}(\mathbf{c})$.
We must show $\mathrm{Code}(f) = \mathbf{c}$.

The function $f$ is built from the code:
at stage~$k$, the coefficients of $f_k$
in the canonical basis
are exactly the entries of $c^\pm_k$
(by construction~(D1) of Definition~\ref{def:decode}).
Applying the Code map extracts
these same coefficients
via the Projection Formula (II.P08).
We verify that Code recovers the original data.

Fix a stage~$k$, a prime $p \mid P_k$,
a residue class $v \in \mathbb{Z}/p\mathbb{Z}$,
and a channel $\sigma \in \{B, C\}$.
The Projection Formula gives:
\[
    \varphi_{p,v}^{(\sigma)}
    \;=\;
    \frac{1}{|F_p|}
    \sum_{x \in F_p(v)}
    e_\sigma \cdot f_k(x).
\]
By step~(D1), $f_k(x)$
is a sum of cylinder generators
with coefficients from~$\mathbf{c}$.
The idempotent projection $e_\sigma$
kills the opposite channel:
$e_+ \cdot E_{k,v}^{(C)}(x) = 0$
and $e_- \cdot E_{k,v}^{(B)}(x) = 0$.
The surviving terms are exactly the $\sigma$-channel
contributions.
Summing over the fiber $F_p(v)$
picks out the coefficient $\varphi_{p,v}^{(\sigma)}$
from~$\mathbf{c}$:
the Projection Formula is the inverse
of the basis expansion,
by the orthogonality of cylinder generators
(Remark~\ref{rem:ch35-idempotent-orthogonality}).
Hence $\mathrm{Code}(f) = \mathbf{c}$.

\smallskip
\noindent\textbf{Step 3: Surjectivity ingredients.}
The argument for Step~2
implicitly uses that every code
actually produces a holomorphic function
(Proposition~\ref{prop:ch41-decode-holomorphic}).
The key inputs are:
\begin{itemize}
    \item \textbf{Sheaf coherence} (II.T32,
          Theorem~\ref{thm:sheaf-axioms},
          Chapter~\ref{ch:sheaf-coherence}):
          the stage-$k$ functions
          assembled from a code
          paste into a global section
          of the holomorphic presheaf~$\mathcal{O}_\tau$.
    \item \textbf{Idempotent decomposition} (II.L07,
          Chapter~\ref{ch:idempotent-decomposition}):
          the bipolar splitting
          is functorial and complete,
          so no information is lost
          in the channel separation.
    \item \textbf{Mutual Determination} (II.T27,
          Theorem~\ref{thm:mutual-determination},
          Chapter~\ref{ch:mutual-determination}):
          the boundary spectral data determines
          the full holomorphic function
          (equivalence of descriptions).
\end{itemize}
These three ingredients combine to ensure
that Decode is surjective onto the holomorphic functions
whose boundary data matches the given code.
Together with injectivity (Step~1),
the bijection is established.