%
\label{prop:character-decomposition}
%           II.L07, II.T33, II.D35, II.D48
Every spectral character
$\chi \in \mathrm{Spec}(\hat{\mathbb{Z}}_\tau, H_\tau^{\mathrm{cal}})$
is idempotent-supported.
Specifically:
\begin{enumerate}
    \item \textbf{Existence.}
          There exist unique ring homomorphisms
          $\chi_+ : \hat{\mathbb{Z}}_\tau \to A_\tau^{(B)}$
          and
          $\chi_- : \hat{\mathbb{Z}}_\tau \to A_\tau^{(C)}$
          such that
          $\chi = e_+ \cdot \chi_+ + e_- \cdot \chi_-$.
    \item \textbf{Uniqueness.}
          The decomposition is unique:
          if also
          $\chi = e_+ \cdot \chi_+' + e_- \cdot \chi_-'$,
          then $\chi_+' = \chi_+$ and $\chi_-' = \chi_-$.
    \item \textbf{$B$-channel determination.}
          $\chi_+$ is determined by its values
          on the $B$-coordinate primes
          (the $\gamma$-assigned primes
          from the Prime Polarity Theorem, I.T05).
    \item \textbf{$C$-channel determination.}
          $\chi_-$ is determined by its values
          on the $C$-coordinate primes
          (the $\eta$-assigned primes).
    \item \textbf{Tower coherence.}
          The tower coherence of $\chi$
          follows from the tower coherence
          of $\chi_+$ and $\chi_-$ separately:
          \[
              \chi_n = e_+ \cdot (\chi_+)_n
                     + e_- \cdot (\chi_-)_n
              \quad \text{for every stage } n.
          \]
\end{enumerate}
Consequently:
\[
    \boxed{%
    A_{\mathrm{spec}}(\mathbb{L})
    = \mathrm{Spec}(\hat{\mathbb{Z}}_\tau, H_\tau^{\mathrm{cal}}).}
\]
Every spectral character is idempotent-supported;
the character algebra $A_{\mathrm{spec}}(\mathbb{L})$
is the \emph{full} character algebra.

The proof proceeds in three steps.

\medskip\noindent
\textbf{Step 1: Existence via idempotent projection.}
Let $\chi : \hat{\mathbb{Z}}_\tau \to H_\tau^{\mathrm{cal}}$
be any spectral character.
Define $\chi_+$ and $\chi_-$ by the projections
\eqref{eq:chi-plus-proj}--\eqref{eq:chi-minus-proj}:
\[
    \chi_+(x) := e_+ \cdot \chi(x),
    \qquad
    \chi_-(x) := e_- \cdot \chi(x).
\]
By Lemma~\ref{lem:channel-ring-hom},
both $\chi_+$ and $\chi_-$ are ring homomorphisms.
Since $e_+ + e_- = 1$ in $H_\tau^{\mathrm{cal}}$, we have:
\[
    \chi(x)
    = 1 \cdot \chi(x)
    = (e_+ + e_-) \cdot \chi(x)
    = e_+ \cdot \chi(x) + e_- \cdot \chi(x)
    = \chi_+(x) + \chi_-(x).
\]
This establishes the decomposition
$\chi = e_+ \cdot \chi_+ + e_- \cdot \chi_-$.

\medskip\noindent
\textbf{Step 2: Uniqueness via orthogonality.}
Suppose $\chi = e_+ \cdot \chi_+' + e_- \cdot \chi_-'$
is another decomposition.
Multiplying both sides by $e_+$:
\begin{align*}
    e_+ \cdot \chi(x)
    &= e_+ \cdot \bigl(e_+ \cdot \chi_+'(x)
       + e_- \cdot \chi_-'(x)\bigr) \\
    &= e_+^2 \cdot \chi_+'(x)
       + e_+ \cdot e_- \cdot \chi_-'(x) \\
    &= e_+ \cdot \chi_+'(x) + 0 \\
    &= \chi_+'(x).
\end{align*}
But $e_+ \cdot \chi(x) = \chi_+(x)$
by definition,
so $\chi_+'(x) = \chi_+(x)$ for all $x$.
The same argument with $e_-$
gives $\chi_-' = \chi_-$.

\medskip\noindent
\textbf{Step 3: Channel determination and tower coherence.}
The ring $\hat{\mathbb{Z}}_\tau$
decomposes via the Chinese Remainder Theorem
(the CRT Tower, I.T18)
into a product indexed by primes.
The Prime Polarity Theorem (I.T05)
partitions the primes into $B$-primes ($\gamma$-assigned)
and $C$-primes ($\eta$-assigned).

The projection $\chi_+$ maps into $A_\tau^{(B)}$,
which is the $e_+$-sector of $H_\tau^{\mathrm{cal}}$.
Since $e_+ \cdot e_- = 0$,
the $C$-prime components of $\hat{\mathbb{Z}}_\tau$
are annihilated by $e_+$:
for any $C$-prime $q$ and any $x \in \hat{\mathbb{Z}}_\tau$
whose support is confined to the $q$-component,
$\chi_+(x) = e_+ \cdot \chi(x)$
lies in $e_+ \cdot A_\tau^{(C)} = \{0\}$.
Therefore $\chi_+$ is determined
by its values on the $B$-prime components.
Symmetrically, $\chi_-$ is determined
by the $C$-prime components.

For tower coherence,
the decomposition commutes with reduction modulo $P_n$
because $e_+$ and $e_-$ are global idempotents
(they do not depend on the stage).
The family $\{\chi_n\}$ decomposes stage by stage:
\[
    \chi_n(x)
    = e_+ \cdot (\chi_+)_n(x) + e_- \cdot (\chi_-)_n(x),
\]
and the compatibility conditions
$\chi_n \equiv \chi_{n+1} \pmod{P_n}$
follow from the compatibility of
$(\chi_+)_n$ and $(\chi_-)_n$ separately.