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\label{lem:spectral-germ}
A stabilized spectral decomposition
with finite support $S_n \subset \widehat{R}_\tau$
determines a unique $\omega$-germ.
Conversely, every $\omega$-germ
arises from a unique stabilized spectral tail.

\emph{Forward ($\mathrm{S} \Rightarrow \mathrm{G}$).}
A spectral tail defines a tower-coherent sequence
$(f_k)_{k \geq n}$ by Lemma~\ref{lem:refinement-spectral}.
Two tower-coherent sequences that differ
only at finitely many stages
define the same $\omega$-germ
(they agree eventually).
Since the spectral tail is stabilized,
the induced sequence is eventually constant
in its spectral structure:
the coefficients $(c_\chi)$ do not change
for $k \geq n$.
This determines a unique equivalence class
at the profinite limit---an $\omega$-germ.

The bipolar decomposition ensures uniqueness:
the $e_+$-component and $e_-$-component
of the spectral data
each produce a one-dimensional datum
at the limit,
and the $\omega$-germ is the pair
of these two limiting data.

\emph{Reverse ($\mathrm{G} \Rightarrow \mathrm{S}$).}
An $\omega$-germ is an equivalence class
of tower-coherent sequences
that agree on all sufficiently deep stages.
Pick any representative $(f_k)_{k \geq m}$.
By the compactness of the profinite space
$\tau^3$ (II.T07, Chapter~\ref{ch:stone-space}),
the spectral support of $(f_k)$
stabilizes at some stage~$n \geq m$:
for $k \geq n$, no new characters enter the support.
The resulting spectral tail
is independent of the representative chosen,
since any two representatives agree
for $k$ sufficiently large.