%
\label{lem:prime-split-support}
Let $G = G_+ + G_-$
be the Branch Factorization of an $\omega$-germ transformer
\textup{(II.L08)}.
Then:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{B-channel support.}
          The spectral support of $G_+$
          \textup{(with respect to the canonical basis
          $\mathcal{B}_\tau$, II.D45)}
          is contained in the B-channel primes:
          $\operatorname{supp}(G_+) \subseteq \Lambda_\tau^{(B)}$,
          where $\Lambda_\tau^{(B)}$
          is the set of indexing triples
          $(p, v, B)$ with $p$ a $\gamma$-orbit prime.
    \item[\textup{(ii)}]
          \textbf{C-channel support.}
          The spectral support of $G_-$
          is contained in the C-channel primes:
          $\operatorname{supp}(G_-) \subseteq \Lambda_\tau^{(C)}$,
          where $\Lambda_\tau^{(C)}$
          is the set of indexing triples
          $(p, v, C)$ with $p$ an $\eta$-orbit prime.
    \item[\textup{(iii)}]
          \textbf{Forced partition.}
          The assignment of primes to channels
          is \textbf{canonical}:
          it is determined by the ABCD chart's
          B/C coordinate structure
          and by Prime Polarity \textup{(I.T05, Book~I)}.
          No choice is involved.
\end{enumerate}

\emph{(i) B-channel support.}
The component $G_+ = e_+ \cdot G$
is valued in the B-channel:
$G_+(x) \in A_\tau^{(B)} = e_+ \cdot H_\tau$
for all $x \in d(\tau^3)$.
We must show that its spectral support
(the set of active cylinder generators
in the canonical basis expansion)
involves only B-channel indices.

Consider the stage-$k$ component
$(G_+)_k \colon \mathbb{Z}/P_k\mathbb{Z} \to A_\tau^{(B)}$.
By the Projection Formula
(II.P08, Chapter~\ref{ch:canonical-basis}),
the spectral coefficient of $(G_+)_k$
at index $(p, v, \sigma)$ is
\[
    \varphi_{p,v}^{(\sigma)}
    = \frac{1}{|F_p|}
    \sum_{x \in F_p(v)}
    e_\sigma \cdot (G_+)_k(x).
\]
For $\sigma = C$ (i.e., $e_\sigma = e_-$):
$e_- \cdot (G_+)_k(x) = e_- \cdot e_+ \cdot G_k(x) = 0$,
since $e_- e_+ = 0$ (I.D21, Book~I).
Hence $\varphi_{p,v}^{(C)} = 0$
for all $(p, v)$.
The spectral support of $G_+$ contains no C-channel indices.

For $\sigma = B$ (i.e., $e_\sigma = e_+$):
$e_+ \cdot (G_+)_k(x) = e_+ \cdot e_+ \cdot G_k(x)
= e_+ \cdot G_k(x) = (G_+)_k(x)$,
since $e_+^2 = e_+$.
The coefficient $\varphi_{p,v}^{(B)}$
is the average of $(G_+)_k$
over the fiber $F_p(v)$,
which is nonzero only when $G_+$
has nontrivial data
at the corresponding residue class.

The key observation:
at each stage $k$,
the new prime $p_{k+1}$
enters the CRT decomposition
$\mathbb{Z}/P_{k+1}\mathbb{Z}
\cong \mathbb{Z}/P_k\mathbb{Z}
\times \mathbb{Z}/p_{k+1}\mathbb{Z}$.
Prime Polarity (I.T05, Book~I)
assigns $p_{k+1}$ to either the $\gamma$-orbit
or the $\eta$-orbit.
The spectral character $\chi_+$ (I.D22, Book~I)
acts on the $\gamma$-orbit primes,
and $\chi_-$ (I.D23, Book~I)
acts on the $\eta$-orbit primes.
Since $G_+$ is valued in the $e_+$-channel,
and $\chi_+$ is the character
that detects $e_+$-valued data
(by the spectral decomposition
of Chapter~\ref{ch:canonical-basis}),
the support of $G_+$
is contained in the set of primes
on which $\chi_+$ acts nontrivially---the
$\gamma$-orbit primes.
Hence
$\operatorname{supp}(G_+) \subseteq \Lambda_\tau^{(B)}$.

\emph{(ii) C-channel support.}
The argument for $G_-$ is symmetric:
$G_-$ is valued in $A_\tau^{(C)} = e_- \cdot H_\tau$,
the B-channel projection vanishes
($e_+ \cdot G_- = 0$),
and the character $\chi_-$ (I.D23, Book~I)
detects $e_-$-valued data
supported on the $\eta$-orbit primes.
Hence
$\operatorname{supp}(G_-) \subseteq \Lambda_\tau^{(C)}$.

\emph{(iii) Forced partition.}
Prime Polarity (I.T05, Book~I)
establishes a canonical partition
of the set of primes
into $\gamma$-orbit and $\eta$-orbit.
This partition is not chosen
but forced by the peel-off structure
of the program monoid (I.P02, Book~I):
the exponent~$\gamma$ and the tetration~$\eta$
extract different arithmetic information
from each prime,
and the two extractions are functorially distinct.
The assignment of a prime to a channel
is determined by which generator
($\gamma$ or $\eta$)
governs its peel-off behavior.
No additional convention or choice is needed.