%
\label{def:interior-bipolar}
Let $(D, A, B, C) \in \tau^3$ be a $\tau$-admissible point.
The \textbf{interior bipolar decomposition} assigns
to each such point a pair of sector components:
\[
    (D, A, B, C)
    \;\longmapsto\;
    \bigl(\, s_+(D, A, B, C),\;
    s_-(D, A, B, C) \,\bigr)
\]
where:
\begin{enumerate}
    \item The \textbf{B-component} $s_+$ is the projection
          onto the $e_+$-sector,
          governed by the $\gamma$-orbit coordinate $B$:
          \[
              s_+(D, A, B, C) \;:=\; e_+ \cdot \Psi(B, A, D),
          \]
          where $\Psi(B, A, D)$ is the $\hat{\mathbb{Z}}_\tau$-valued
          encoding of the exponent tower
          at the base point $(D, A)$.
    \item The \textbf{C-component} $s_-$ is the projection
          onto the $e_-$-sector,
          governed by the $\eta$-orbit coordinate $C$:
          \[
              s_-(D, A, B, C) \;:=\; e_- \cdot \Psi(C, A, D).
          \]
    \item The full interior value reconstructs as
          \[
              \Psi_{\mathrm{int}}(D, A, B, C)
              \;=\;
              s_+ + s_-
              \;=\;
              e_+ \cdot \Psi(B, A, D) + e_- \cdot \Psi(C, A, D)
              \;\in\; H_\tau.
          \]
\end{enumerate}
The decomposition is canonical:
$e_+ \cdot e_- = 0$ ensures the two components
are orthogonal, and $e_+ + e_- = 1$ ensures
completeness.