%
\label{def:decode}
The \textbf{Decode} map takes a code
$\mathbf{c} = (c^+, c^-)
\in \mathrm{Code}_\tau$
and produces a $\tau$-holomorphic function
$f \in \mathrm{Hol}_\tau(\tau^3, H_\tau^{\mathrm{cal}})$
via the following three-step construction:
\[
    \boxed{%
    \mathrm{Decode}
    \;:\;
    \mathrm{Code}_\tau
    \;\longrightarrow\;
    \mathrm{Hol}_\tau(\tau^3, H_\tau^{\mathrm{cal}}),
    \qquad
    \mathbf{c}
    \;\longmapsto\;
    f = \lim_{\longleftarrow} f_k.}
\]

\begin{enumerate}
    \item[\textup{(D1)}]
          \textbf{Stage-$k$ reconstruction.}
          For each $k \geq 1$,
          use the stage-$k$ coefficient data
          $c^\pm_k$ to build the stage-$k$ function:
          \[
              f_k(x)
              \;:=\;
              \sum_{(p,v) :\, p \mid P_k}
              \Bigl(
              \varphi_{p,v}^{(+)} \cdot E_{k,v}^{(B)}(x)
              \;+\;
              \varphi_{p,v}^{(-)} \cdot E_{k,v}^{(C)}(x)
              \Bigr)
          \]
          for all $x \in \mathbb{Z}/P_k\mathbb{Z}$.
          This is a finite sum
          (condition~(C3) of Definition~\ref{def:code}),
          so $f_k$ is a well-defined function
          on the stage-$k$ cyclic group.

    \item[\textup{(D2)}]
          \textbf{Tower compatibility.}
          The tower coherence condition~(C2) guarantees
          that the family $(f_k)_{k \geq 1}$
          is compatible under the tower projection maps:
          $\pi_{k+1,k} \circ f_{k+1} = f_k$.
          This is verified directly:
          the shared primes $p \mid P_k$
          contribute identical coefficients at stages~$k$ and $k+1$
          (by~(C2)),
          while the new prime $p_{k+1}$
          contributes terms that are projected away by $\pi_{k+1,k}$.

    \item[\textup{(D3)}]
          \textbf{Inverse limit.}
          Define $f := \varprojlim_k f_k$
          as the unique element of the inverse limit
          whose stage-$k$ projection is~$f_k$
          for every~$k$.
          Tower compatibility~(D2) ensures existence.
          Uniqueness is automatic from the inverse limit construction.
\end{enumerate}