%
\label{prop:sector-inheritance}
Every $\tau$-admissible point $(D, A, B, C) \in \tau^3$
inherits a bipolar sector assignment
from the boundary structure:
\begin{enumerate}
    \item At each finite stage of the primorial ladder,
          the sector assignment is determined by the
          fiber coordinates $(B, C)$ via the
          interior bipolar decomposition
          (Definition~\ref{def:interior-bipolar}).
    \item The sector assignment is compatible with
          the idempotent decomposition of $H_\tau$:
          $\Psi_{\mathrm{int}} = e_+ \cdot s_+ + e_- \cdot s_-$
          with $e_+ \cdot s_- = 0$ and $e_- \cdot s_+ = 0$.
    \item At the $\omega$-limit,
          the sector assignment recovers
          the polarity character
          $\tilde{\chi} \colon \mathbb{P}_\tau \to \{e_+, e_-\}$
          of the prime polarity theorem.
\end{enumerate}

Part~(1) is by construction:
Definition~\ref{def:interior-bipolar} assigns
sector components via the fiber coordinates.
Part~(2) follows from the idempotent properties
$e_+^2 = e_+$, $e_-^2 = e_-$, $e_+ \cdot e_- = 0$:
projecting $\Psi_{\mathrm{int}}$ by $e_+$
gives $e_+ \cdot \Psi_{\mathrm{int}}
= e_+^2 \cdot \Psi(B, A, D)
= e_+ \cdot \Psi(B, A, D) = s_+$,
and similarly $e_- \cdot \Psi_{\mathrm{int}} = s_-$.
Part~(3): at the $\omega$-limit,
the fiber coordinates stabilize
by the tail-stability of the polarity map
(Book~I, Part~VI).
The B-dominant primes have $B \gg C$ asymptotically,
mapping to $e_+$;
the C-dominant primes have $C \gg B$ asymptotically,
mapping to $e_-$.
This recovers the polarity character $\tilde{\chi}$.