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\label{lem:polarity-symmetry}
Let $G = G_+ + G_-$
be the Branch Factorization
of an $\omega$-germ transformer \textup{(II.L08)}.
Then the $\jj$-involution interchanges the two sectors:
\begin{enumerate}
    \item[\textup{(i)}]
          $\sigma_\jj(G_+) = G_-$
          and $\sigma_\jj(G_-) = G_+$.
    \item[\textup{(ii)}]
          The spectral data of $G_+$ at a $\gamma$-orbit prime~$p$
          determines the spectral data of $G_-$
          at the corresponding $\eta$-orbit prime,
          and conversely.
    \item[\textup{(iii)}]
          Knowing $G_+$ alone
          determines $G_-$ \textup{(}and hence $G$\textup{)}
          via $G_- = \sigma_\jj(G_+)$
          and $G = G_+ + \sigma_\jj(G_+)$.
\end{enumerate}

\emph{(i) Channel interchange.}
By the channel-swapping property
of $\sigma_\jj$
(Remark~\ref{rem:ch38-sigma-properties}):
\begin{align*}
    \sigma_\jj(G_+)
    &= \sigma_\jj(e_+ \cdot G)
    = \sigma_\jj(e_+) \cdot \sigma_\jj(G)
    = e_- \cdot \sigma_\jj(G).
\end{align*}
We need $\sigma_\jj(G) = G$.
This holds because
$G$ is a $\tau$-holomorphic transformer,
and the $\jj$-involution
is an automorphism of $H_\tau$
that preserves the holomorphic structure.
Concretely:
the Mutual Determination Theorem (II.T27,
Chapter~\ref{ch:mutual-determination})
identifies a holomorphic datum
with a boundary character
$\varphi \colon R_\tau \to H_\tau$.
The boundary ring $R_\tau$ (I.D19, Book~I)
is invariant under $\sigma_\jj$
(it is the profinite completion
of cyclic groups,
which are stable under conjugation).
The holomorphic condition is symmetric
in the B and C channels
(the five-way equivalence of II.T27
treats both channels identically).
Hence $\sigma_\jj$ maps holomorphic transformers
to holomorphic transformers,
and since $G$ is the \emph{total} transformer,
$\sigma_\jj(G) = G$ as a transformation
on the germ algebra.

Therefore:
$\sigma_\jj(G_+) = e_- \cdot \sigma_\jj(G) = e_- \cdot G = G_-$.
Similarly, $\sigma_\jj(G_-) = G_+$.

\emph{(ii) Spectral correspondence.}
By Prime-Split Support (II.L09),
$G_+$ has spectral support in
the $\gamma$-orbit primes
and $G_-$ has spectral support in
the $\eta$-orbit primes.
The $\jj$-involution maps
$\chi_+$ to $\chi_-$ and $\chi_-$ to $\chi_+$
(I.D22--I.D23, Book~I:
$\chi_\pm = e_\pm \cdot \chi$
for any character~$\chi$,
and $\sigma_\jj$ swaps $e_\pm$).
Hence the spectral coefficient of $G_+$
at a $\gamma$-orbit prime~$p$
equals the spectral coefficient of $G_-$
at the partner prime in the $\eta$-orbit.
The correspondence is canonical
(determined by Prime Polarity, I.T05).

\emph{(iii) One channel determines both.}
Since $G_- = \sigma_\jj(G_+)$ by~(i),
and $G = G_+ + G_-$ by Branch Factorization,
we have $G = G_+ + \sigma_\jj(G_+)$.
Knowledge of $G_+$ alone
determines $G$ completely.
Symmetrically, $G = \sigma_\jj(G_-) + G_-$.