%
\label{def:spectral-algebra}
The \textbf{spectral algebra} $A_{\mathrm{spec}}(\mathbb{L})$
is the algebra of all idempotent-supported characters
on the algebraic lemniscate $\mathbb{L} = S^1 \vee S^1$,
valued in the calibrated split-complex codomain
$H_\tau^{\mathrm{cal}}$.
Explicitly:
\begin{equation}
\label{eq:ch51-spectral-algebra}
    A_{\mathrm{spec}}(\mathbb{L})
    \;:=\;
    \Bigl\{
        \chi \colon \widehat{\mathbb{Z}}_\tau \to H_\tau^{\mathrm{cal}}
        \;\Big|\;
        \chi = e_+ \cdot \chi_+ + e_- \cdot \chi_-,\;
        \chi \text{ tower-coherent}
    \Bigr\}.
\end{equation}
The structure of $A_{\mathrm{spec}}(\mathbb{L})$ is as follows:
\begin{enumerate}
    \item[\textup{(A)}] \textbf{Elements.}
          Each $\chi \in A_{\mathrm{spec}}(\mathbb{L})$
          is a ring homomorphism
          $\chi \colon \widehat{\mathbb{Z}}_\tau \to H_\tau^{\mathrm{cal}}$
          satisfying two conditions:
          \begin{itemize}
              \item \emph{Idempotent support:}
                    $\chi$ decomposes as
                    $\chi = e_+ \cdot \chi_+ + e_- \cdot \chi_-$,
                    where $\chi_\pm = e_\pm \cdot \chi$
                    are the projections onto the B-channel
                    and C-channel respectively
                    (II.D59,
                    Chapter~\ref{ch:boundary-characters-idempotent}).
              \item \emph{Tower coherence:}
                    the system of restrictions
                    $\chi_k := \chi|_{\mathbb{Z}/P_k\mathbb{Z}}$
                    satisfies
                    $\chi_k \equiv \chi_{k+1} \pmod{P_k}$
                    for all sufficiently large~$k$.
          \end{itemize}

    \item[\textup{(B)}] \textbf{Addition.}
          Pointwise addition in~$H_\tau^{\mathrm{cal}}$:
          $(\chi + \psi)(r) := \chi(r) + \psi(r)$
          for $r \in \widehat{\mathbb{Z}}_\tau$.

    \item[\textup{(C)}] \textbf{Multiplication.}
          Pointwise multiplication in~$H_\tau^{\mathrm{cal}}$:
          $(\chi \cdot \psi)(r) := \chi(r) \cdot \psi(r)$.
          Since $H_\tau^{\mathrm{cal}}$ is a commutative ring
          (with zero divisors along the $e_\pm$-axes),
          so is $A_{\mathrm{spec}}(\mathbb{L})$.

    \item[\textup{(D)}] \textbf{Topology.}
          $A_{\mathrm{spec}}(\mathbb{L})$ carries the
          inverse-limit topology inherited from the tower:
          a net $(\chi^\alpha)$ converges to~$\chi$
          if and only if $\chi^\alpha_k \to \chi_k$
          for every stage~$k$.
          Since each $\chi_k$ takes values
          in a finite-dimensional $H_\tau^{\mathrm{cal}}$-module,
          the convergence at each stage is convergence
          of finitely many split-complex coefficients.

    \item[\textup{(E)}] \textbf{Bipolar decomposition.}
          The algebra decomposes as
          \begin{equation}
          \label{eq:ch51-spectral-bipolar}
              A_{\mathrm{spec}}(\mathbb{L})
              \;=\;
              e_+ \cdot A_{\mathrm{spec}}^+(\mathbb{L})
              \;\oplus\;
              e_- \cdot A_{\mathrm{spec}}^-(\mathbb{L}),
          \end{equation}
          where $A_{\mathrm{spec}}^\pm(\mathbb{L})
          := e_\pm \cdot A_{\mathrm{spec}}(\mathbb{L})$
          are the $e_\pm$-components.
          The two components are independent:
          $e_+ \cdot e_- = 0$,
          so no element of
          $A_{\mathrm{spec}}^+(\mathbb{L})$
          has any effect on
          $A_{\mathrm{spec}}^-(\mathbb{L})$
          and conversely.

    \item[\textup{(F)}] \textbf{Calibration.}
          The four calibration constants
          enter through the $H_\tau^{\mathrm{cal}}$ structure
          (II.D35, Chapter~\ref{ch:split-complex-calibrated}):
          \begin{itemize}
              \item $\pi$ scales the angular periods
                    of the B and C channels.
              \item $e$ scales the radial growth rates
                    along the D-ray.
              \item $\jj$ mediates the polarity flip
                    between $e_+$ and~$e_-$.
              \item $\iota_\tau = 2/(\pi + e)$
                    couples all three
                    as the master spectral invariant.
          \end{itemize}
\end{enumerate}