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\label{lem:germ-character}
An $\omega$-germ determines a unique
boundary character
$\varphi \colon R_\tau \to H_\tau$.
Conversely, every boundary character
determines a unique $\omega$-germ.

\emph{Forward ($\mathrm{G} \Rightarrow \mathrm{C}$).}
An $\omega$-germ has a unique stabilized spectral decomposition
$\sum_{\chi \in S_n} c_\chi \cdot \chi$
by Lemma~\ref{lem:spectral-germ}.
Define the boundary character
$\varphi \colon R_\tau \to H_\tau$ by
\[
    \varphi(r)
    \;:=\;
    \sum_{\chi \in S_n} c_\chi \cdot \chi(r),
    \qquad r \in R_\tau.
\]
This is a ring homomorphism because
each $\chi$ is a character
(multiplicative homomorphism)
and $H_\tau$ is a ring under the split-complex
multiplication.
The bipolar decomposition splits $\varphi$
into $\varphi^+ = e_+ \circ \varphi$
and $\varphi^- = e_- \circ \varphi$,
each of which is a character of $R_\tau$
valued in a single channel.

\emph{Reverse ($\mathrm{C} \Rightarrow \mathrm{G}$).}
Given a boundary character
$\varphi \colon R_\tau \to H_\tau$,
decompose via bipolar idempotents:
$\varphi = e_+ \varphi^+ + e_- \varphi^-$.
Each $\varphi^\pm$ is a character of $R_\tau$
valued in one channel.
Since $R_\tau$ is the profinite completion
of $\varinjlim \mathbb{Z}/P_k\mathbb{Z}$
(I.D19, Book~I),
each character is determined by its values
on the finite quotients $\mathbb{Z}/P_k\mathbb{Z}$.
These values form a tower-coherent system,
which by the CRT (I.T18, Book~I)
has a unique spectral decomposition
with eventually stable support.
Lemma~\ref{lem:spectral-germ}
converts this spectral tail to an $\omega$-germ.

\emph{Uniqueness.}
In each direction, the bipolar decomposition
reduces the problem to one-dimensional data
(one real coefficient per channel per character).
The constructions are inverse to each other
by the uniqueness of the DFT
and the uniqueness of profinite limits.