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\label{thm:spectral-trichotomy}
Let $n \geq 1$ and $\chi \in \Char_n(\Lemniscate)$.
Then:
\begin{enumerate}
    \item[\textup{(i)}] \textbf{Exact decomposition.}
          $\chi$ decomposes uniquely as
          \begin{equation}\label{eq:ch19-trichotomy}
              \chi \;=\; \chi_B + \chi_C,
              \qquad
              \chi_B = e_+ \cdot \chi \in \Char_B^{(n)},
              \quad
              \chi_C = e_- \cdot \chi \in \Char_C^{(n)}.
          \end{equation}
    \item[\textup{(ii)}] \textbf{Orthogonality.}
          $e_+ \cdot \chi_C = 0$
          and $e_- \cdot \chi_B = 0$.
    \item[\textup{(iii)}] \textbf{Functoriality.}
          If $\pi_{n+1,n} : \Char_{n+1}(\Lemniscate)
          \to \Char_n(\Lemniscate)$
          is the projection, then
          \begin{equation}\label{eq:ch19-functoriality}
              \pi_{n+1,n}(\chi_B)
              = (\pi_{n+1,n}(\chi))_B,
              \qquad
              \pi_{n+1,n}(\chi_C)
              = (\pi_{n+1,n}(\chi))_C.
          \end{equation}
    \item[\textup{(iv)}] \textbf{Trichotomy.}
          Exactly one holds:
          $\chi \in \Char_B^{(n)}$ (pure $B$),
          $\chi \in \Char_C^{(n)}$ (pure $C$),
          or $\chi \in \Char_X^{(n)}$ ($X$-mixing).
          The three classes partition $\Char_n(\Lemniscate)$.
\end{enumerate}