%
\label{def:idempotent-character}
A spectral character
$\chi : \hat{\mathbb{Z}}_\tau \to H_\tau^{\mathrm{cal}}$
is \textbf{idempotent-supported} if it admits a decomposition
\[
    \boxed{%
    \chi = e_+ \cdot \chi_+ + e_- \cdot \chi_-,}
\]
where:
\begin{enumerate}
    \item $\chi_+ : \hat{\mathbb{Z}}_\tau \to A_\tau^{(B)}$
          is a ring homomorphism
          (the \textbf{$B$-channel component}),
    \item $\chi_- : \hat{\mathbb{Z}}_\tau \to A_\tau^{(C)}$
          is a ring homomorphism
          (the \textbf{$C$-channel component}),
    \item The decomposition recovers $\chi$ pointwise:
          for every $x \in \hat{\mathbb{Z}}_\tau$,
          \[
              \chi(x)
              = \chi_+(x) + \chi_-(x)
              = e_+ \cdot \chi_+(x) + e_- \cdot \chi_-(x).
          \]
\end{enumerate}
The algebra of all idempotent-supported characters is denoted
\[
    A_{\mathrm{spec}}(\mathbb{L})
    := \bigl\{\chi \in
       \mathrm{Spec}(\hat{\mathbb{Z}}_\tau, H_\tau^{\mathrm{cal}})
       : \chi \text{ is idempotent-supported}\bigr\}.
\]