%
\label{thm:sector-preservation}
The boundary-to-interior functor
$\Phi : \Char(\Lemniscate) \to \mathcal{O}(\tau^3)$
preserves the bipolar decomposition
in the following sense:
\begin{enumerate}
    \item[\textup{(i)}]
          $\chi_+$-characters map to B-sector holomorphic functions:
          \[
              m = 0 \;\implies\;
              \Phi(\chi_{(0,n)}) \;=\; e_+ \cdot \Phi(\chi_{(0,n)})_+.
          \]
    \item[\textup{(ii)}]
          $\chi_-$-characters map to C-sector holomorphic functions:
          \[
              n = 0 \;\implies\;
              \Phi(\chi_{(m,0)}) \;=\; e_- \cdot \Phi(\chi_{(m,0)})_-.
          \]
    \item[\textup{(iii)}]
          Mixed characters map to the $\omega$-coupling sector:
          \[
              m \neq 0,\; n \neq 0 \;\implies\;
              \Phi(\chi_{(m,n)}) \;=\;
              e_+ \cdot \Phi(\chi_{(m,n)})_+
              \;+\;
              e_- \cdot \Phi(\chi_{(m,n)})_-,
          \]
          with both components non-trivial.
\end{enumerate}
In particular, $\Phi$ maps
the $m$-axis of~$\Z^2$
into the C-sector of~$\mathcal{O}(\tau^3)$,
the $n$-axis into the B-sector,
and the interior of the lattice
into the $\omega$-coupling sector.

The proof proceeds in three steps.

\emph{Step 1: Stage-level verification.}
At primorial depth~$k$,
the functor $\Phi_k$ sends a character
$\chi_{(m,n)} \in (\Z / M_k\Z)^2$
to the element
$\Phi_k(\chi_{(m,n)}) \in (\Z / M_k\Z)[\,j\,]/(j^2 - 1)$.
By the definition of the split-complex extension
(equation~\eqref{eq:ch09-phi-stages}),
this element decomposes as
\[
    \Phi_k(\chi_{(m,n)})
    \;=\;
    e_+ \cdot \bar{n}_k
    \;+\;
    e_- \cdot \bar{m}_k,
\]
where $\bar{m}_k$ and $\bar{n}_k$
denote the reductions of $m$ and $n$
modulo~$M_k$.
When $m = 0$, the $e_-$-component vanishes;
when $n = 0$, the $e_+$-component vanishes.
This establishes claims~(i) and~(ii) at each finite level.

\emph{Step 2: Tower coherence.}
The primorial projections
$\pi_{\ell \to k} : \Z / M_\ell\Z \to \Z / M_k\Z$
commute with the split-complex extension:
\[
    \pi_{\ell \to k}(e_+ \cdot \bar{n}_\ell + e_- \cdot \bar{m}_\ell)
    \;=\;
    e_+ \cdot \bar{n}_k + e_- \cdot \bar{m}_k.
\]
This is exactly the left square
of the bi-square (I.T41).
The sector assignment---whether
a character maps to the B-sector, the C-sector,
or the $\omega$-coupling sector---is
therefore stable under reduction
and passes to the inverse limit.

\emph{Step 3: The mixed case.}
When $m \neq 0$ and $n \neq 0$,
both $\bar{m}_k$ and $\bar{n}_k$
are non-zero for all sufficiently large~$k$
(since $M_k \to \infty$).
Therefore both idempotent components
of $\Phi_k(\chi_{(m,n)})$ are non-trivial
for all~$k$ beyond a finite threshold.
In the inverse limit,
both components survive,
placing $\Phi(\chi_{(m,n)})$
in the $\omega$-coupling sector.
This establishes claim~(iii).