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\label{lem:branch-factorization}
Every $\omega$-germ transformer
$G \colon d(\tau^3) \to d(\tau^3)$
factors through the bipolar idempotents.
Concretely:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Decomposition.}
          $G = G_+ + G_-$,
          where $G_+ := e_+ \cdot G$ and $G_- := e_- \cdot G$.
    \item[\textup{(ii)}]
          \textbf{Each factor is an $\omega$-germ transformer.}
          Both $G_+$ and $G_-$ are $\omega$-germ transformers
          in their own right:
          they preserve tower coherence,
          commute with the tower transition maps,
          and produce stabilized $\omega$-germs
          from stabilized inputs.
    \item[\textup{(iii)}]
          \textbf{Orthogonality.}
          The two factors are orthogonal:
          $e_+ \cdot G_- = 0$ and $e_- \cdot G_+ = 0$.
    \item[\textup{(iv)}]
          \textbf{Uniqueness.}
          The decomposition $G = G_+ + G_-$
          is the \textbf{unique} way to write $G$
          as a sum of an $e_+$-valued transformer
          and an $e_-$-valued transformer.
\end{enumerate}

\emph{(i) Decomposition.}
The element $1 \in H_\tau$ decomposes as $1 = e_+ + e_-$,
and the idempotents satisfy
$e_+^2 = e_+$, $e_-^2 = e_-$, $e_+ e_- = 0$
(I.D21, Book~I).
For any $\omega$-germ transformer $G$,
define $G_+ := e_+ \cdot G$ and $G_- := e_- \cdot G$
by pointwise multiplication.
Then:
\[
    G_+ + G_-
    = e_+ \cdot G + e_- \cdot G
    = (e_+ + e_-) \cdot G
    = 1 \cdot G
    = G.
\]

\emph{(ii) Each factor is an $\omega$-germ transformer.}
We verify the three properties for $G_+$
(the argument for $G_-$ is identical
with $e_-$ replacing $e_+$).

\emph{Tower coherence.}
Let $(f_k)_{k \geq n}$ be a tower-coherent input.
Then $G(f_k)_{k \geq n}$ is tower-coherent
because $G$ is an $\omega$-germ transformer.
Projecting:
$(G_+)_k(f) = e_+ \cdot G_k(f)$.
Since $e_+ \in H_\tau$ is a fixed scalar
and the tower transition maps
$\pi_{k,k+1} \colon \mathbb{Z}/P_{k+1}\mathbb{Z}
\to \mathbb{Z}/P_k\mathbb{Z}$
are ring homomorphisms,
the scalar multiplication by $e_+$
commutes with the transition maps:
\[
    \pi_{k,k+1}\bigl(e_+ \cdot G_{k+1}(f)\bigr)
    = e_+ \cdot \pi_{k,k+1}\bigl(G_{k+1}(f)\bigr)
    = e_+ \cdot G_k(f)
    = (G_+)_k(f).
\]
Tower coherence of $G_+$ follows.

\emph{Commutativity with transition maps.}
Immediate from the above calculation:
$e_+$ is central in $H_\tau$
(because $e_+$ is a scalar,
and scalar multiplication commutes
with all ring operations in $H_\tau$).

\emph{Stabilization.}
If $G$ produces a stabilized $\omega$-germ
from a stabilized input
(spectral support fixed after some stage~$n$),
then $G_+ = e_+ \cdot G$ has spectral support
contained in that of~$G$
(multiplication by a fixed element
cannot enlarge the support).
Hence $G_+$ also stabilizes.

\emph{(iii) Orthogonality.}
$e_+ \cdot G_- = e_+ \cdot (e_- \cdot G)
= (e_+ e_-) \cdot G = 0 \cdot G = 0$,
using $e_+ e_- = 0$ (I.D21).
Similarly, $e_- \cdot G_+ = 0$.

\emph{(iv) Uniqueness.}
Suppose $G = H_+ + H_-$
with $e_+ \cdot H_- = 0$ and $e_- \cdot H_+ = 0$.
Then $e_+ \cdot G = e_+ \cdot H_+ + e_+ \cdot H_-
= H_+ + 0 = H_+$,
where the last step uses $e_+ \cdot H_+ = H_+$
(since $H_+$ is $e_+$-valued: $H_+ = e_+ \cdot H_+$).
Hence $H_+ = G_+$, and symmetrically $H_- = G_-$.