%
\label{def:code}
A \textbf{code} is a pair $\mathbf{c} = (c^+, c^-)$,
where each component is a
\textbf{bipolar boundary coefficient stream}:
\[
    \boxed{%
    c^\pm
    \;=\;
    \bigl(\,
    c^\pm_1,\;
    c^\pm_2,\;
    c^\pm_3,\;
    \ldots
    \,\bigr),}
\]
subject to the following conditions.

\begin{enumerate}
    \item[\textup{(C1)}]
          \textbf{Stage data.}
          For each $k \geq 1$,
          the entry $c^\pm_k$ is a finite family
          of spectral coefficients:
          \[
              c^\pm_k
              \;=\;
              \bigl\{\,
              \varphi_{p,v}^{(\pm)}
              \;\big|\;
              p \mid P_k,\;
              v \in \mathbb{Z}/p\mathbb{Z}
              \,\bigr\},
          \]
          with each $\varphi_{p,v}^{(+)} \in A_\tau^{(B)}$
          and each $\varphi_{p,v}^{(-)} \in A_\tau^{(C)}$.
          Here $(+)$ refers to the B-channel
          (associated with $e_+$)
          and $(-)$ to the C-channel
          (associated with $e_-$).

    \item[\textup{(C2)}]
          \textbf{Tower coherence.}
          For each $k \geq 1$
          and each prime $p \mid P_k$,
          the coefficient $\varphi_{p,v}^{(\pm)}$
          in the stage-$k$ entry
          equals the coefficient $\varphi_{p,v}^{(\pm)}$
          in the stage-$(k+1)$ entry.
          That is,
          the restriction maps of the primorial tower
          send $c^\pm_{k+1}$ to $c^\pm_k$:
          \[
              \rho_{k+1,k}\bigl(c^\pm_{k+1}\bigr)
              \;=\;
              c^\pm_k.
          \]

    \item[\textup{(C3)}]
          \textbf{Finite spectral support at each stage.}
          At each stage~$k$,
          only finitely many coefficients are nonzero.
          (This is automatic from the finiteness
          of $\Lambda_k$, but we record it
          for emphasis.)
\end{enumerate}

\medskip\noindent
The \textbf{space of codes} is denoted
\[
    \boxed{%
    \mathrm{Code}_\tau
    \;:=\;
    \bigl\{\,
    \mathbf{c} = (c^+, c^-)
    \;\big|\;
    \text{(C1), (C2), (C3) hold}
    \,\bigr\}.}
\]