%
\label{prop:bipolar-channel-independence}
The two sector lifts
$\mathrm{BndLift}_n^{(+)}$ and $\mathrm{BndLift}_n^{(-)}$
are \textbf{informationally independent}:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Algebraic independence.}
          The lift increment $\Delta_n^{(+)}$
          is determined solely by the B-sector data
          $f_n^{(+)} = e_+ \cdot f_n$.
          The lift increment $\Delta_n^{(-)}$
          is determined solely by the C-sector data
          $f_n^{(-)} = e_- \cdot f_n$.
          Neither increment depends on the other sector's data.

    \item[\textup{(ii)}]
          \textbf{Norm independence.}
          The calibrated norm of the lift factorizes:
          \[
              \|\mathrm{BndLift}_n(f_n)\|_\tau
              \;=\;
              \bigl\|
              \mathrm{BndLift}_n^{(+)}(f_n^{(+)})
              \bigr\|^{\pi/(\pi+e)}
              \;\cdot\;
              \bigl\|
              \mathrm{BndLift}_n^{(-)}(f_n^{(-)})
              \bigr\|^{e/(\pi+e)}.
          \]
          The B-channel contributes a $\pi$-weighted factor;
          the C-channel contributes an $e$-weighted factor.

    \item[\textup{(iii)}]
          \textbf{Information capacity.}
          At stage $n+1$,
          each sector carries $\log_2 p_{n+1}$ new bits
          of information.
          The total information increment
          is $2 \log_2 p_{n+1}$ bits
          (one copy per sector).
          The two copies are independently determined:
          knowing the B-sector increment
          gives zero information about the C-sector increment,
          and conversely.

    \item[\textup{(iv)}]
          \textbf{Coupling structure.}
          The only coupling between the two sectors
          is the norm constraint
          $\|\Delta_n^{(+)}\| = \iota_\tau \cdot \|\Delta_n^{(-)}\|$
          \textup{(}Definition~\textup{\ref{def:bndlift}(iv))}.
          This constrains the \textbf{magnitudes}
          of the increments
          but not their \textbf{values}:
          the direction of $\Delta_n^{(+)}$
          is independent of $\Delta_n^{(-)}$.
\end{enumerate}

\textbf{(i)}
The CRT decomposition
is applied independently to each ABCD component.
The B-coordinate maps to the $e_+$-sector
and the C-coordinate maps to the $e_-$-sector
(by the spectral decomposition of~$\jj$,
II.T24, Chapter~\ref{ch:j-replaces-i}).
Since the CRT isomorphism
$\mathbb{Z}/P_{n+1}\mathbb{Z}
\cong \mathbb{Z}/P_n\mathbb{Z} \times \mathbb{Z}/p_{n+1}\mathbb{Z}$
acts componentwise on the ABCD channels,
the B-sector lift depends only on B-data,
and the C-sector lift depends only on C-data.

\textbf{(ii)}
The calibrated norm is
$\|z\|_\tau = |z_+|^{\pi/(\pi+e)} \cdot |z_-|^{e/(\pi+e)}$
(Definition~\ref{def:calibrated-H-tau}, II.D35).
Since the lift factorizes sectorwise,
$(f_{n+1})_+ = \mathrm{BndLift}_n^{(+)}(f_n^{(+)})$
and
$(f_{n+1})_- = \mathrm{BndLift}_n^{(-)}(f_n^{(-)})$.
Substituting into the norm formula
gives the stated factorization.

\textbf{(iii)}
The $p_{n+1}$-component of each sector
is a function
$\Delta_n^{(\pm)} : \mathbb{Z}/p_{n+1}\mathbb{Z} \to \mathbb{Z}$.
Once the diagonal discipline and holomorphicity
constraints are imposed,
$\Delta_n^{(\pm)}$ is uniquely determined
(Theorem~\ref{thm:bndlift-existence}(ii)).
In the information-theoretic sense,
the $p_{n+1}$ residue classes
contribute $\log_2 p_{n+1}$ bits per sector.
The two sectors are CRT-orthogonal
(by the product structure
$\mathbb{R}[\jj] \cong \mathbb{R} \times \mathbb{R}$),
so the mutual information is zero.

\textbf{(iv)}
The coupling
$\|\Delta_n^{(+)}\| = \iota_\tau \cdot \|\Delta_n^{(-)} \|$
constrains the ratio of the $\ell^2$ norms.
But the $\ell^2$ norm is a scalar invariant:
it constrains the magnitude
without constraining the direction
(the specific values at each residue class).
The direction of $\Delta_n^{(+)}$
is determined by the B-sector holomorphicity;
the direction of $\Delta_n^{(-)}$
is determined by the C-sector holomorphicity.
These two holomorphicity conditions
are independent.