%
\label{lem:refinement-spectral}
Let $(f_k)_{k \geq n}$ be a tower-coherent sequence
in~$H_\tau$ stabilized after stage~$n$.
Then there exists a unique finite spectral decomposition
\[
    f_k
    \;=\;
    \sum_{\chi \in S_n} c_\chi \cdot \chi_k,
    \qquad
    k \geq n,
\]
where $S_n \subset \widehat{R}_\tau$ is a finite
support set
and $\chi_k$ is the restriction
of the character~$\chi$ to stage~$k$.
Conversely, every spectral tail
with finite support
defines a tower-coherent refinement tail.

\emph{Forward direction ($\mathrm{R} \Rightarrow \mathrm{S}$).}
At each stage~$k$, the map
$f_k \colon \mathbb{Z}/P_k\mathbb{Z} \to H_\tau$
is a function on a finite cyclic group.
By the Chinese Remainder Theorem (I.T18, Book~I),
$\mathbb{Z}/P_k\mathbb{Z} \cong
\prod_{j=1}^k \mathbb{Z}/p_j\mathbb{Z}$,
and the characters of $\mathbb{Z}/P_k\mathbb{Z}$
are products of characters
of the individual factors.
The discrete Fourier transform
on $\mathbb{Z}/P_k\mathbb{Z}$
decomposes $f_k$ into a finite sum
of characters with coefficients in $H_\tau$.

The bipolar decomposition applies:
each coefficient $c_\chi \in H_\tau$
splits as $c_\chi = e_+ c_\chi^+ + e_- c_\chi^-$.
Tower coherence
($f_k \equiv f_{k+1} \pmod{P_k}$)
forces the spectral coefficients
at stage~$k+1$ to reduce
to those at stage~$k$
modulo the new prime~$p_{k+1}$.
Since the tail is stabilized after stage~$n$,
no new spectral components appear
for $k > n$:
the support $S_n$ is fixed,
and the coefficients refine consistently.

\emph{Reverse direction ($\mathrm{S} \Rightarrow \mathrm{R}$).}
Given a spectral tail with finite support $S_n$
and coefficients $(c_\chi)_{\chi \in S_n}$ in~$H_\tau$,
define $f_k := \sum_{\chi \in S_n} c_\chi \cdot \chi_k$.
Tower coherence follows from the coherence
of the characters:
$\chi_{k+1} \equiv \chi_k \pmod{P_k}$
by construction of the dual group.
Stabilization after stage~$n$ is immediate
since $S_n$ is fixed.

\emph{Uniqueness.}
The DFT on a finite abelian group
is invertible,
so the spectral decomposition
at each stage is unique.
Coherence forces the decompositions at different stages
to be compatible,
and stabilization ensures
the system has a unique limit.