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\label{thm:algebraic-lemniscate}
The boundary of $\tau$ carries a canonical algebraic structure
$\mathbb{L}$ consisting of:
\begin{enumerate}
    \item The \textbf{bipolar spectral algebra}
          $H_\tau = \hat{\mathbb{Z}}_\tau[j]$
          (Definition~\ref{def:bipolar-spectral-algebra})
          with two sectors $e_+ H_\tau$ and $e_- H_\tau$
          corresponding to the two polarity channels.
    \item The \textbf{crossing-point germ}
          (Definition~\ref{def:crossing-germ}):
          the unique omega-germ $\omega_{\mathbb{L}}$
          where neither channel is eventually constant ---
          both sectors remain active.
          This germ acts as the \emph{identity element}
          of the bipolar structure.
    \item The \textbf{polarity involution}
          $\sigma \colon H_\tau \to H_\tau$
          defined by $\sigma(j) = -j$,
          which swaps the B-sector and C-sector:
          $\sigma(e_+) = e_-$ and $\sigma(e_-) = e_+$.
\end{enumerate}
We call $\mathbb{L} = (H_\tau, \omega_{\mathbb{L}}, \sigma)$
the \textbf{algebraic lemniscate}.

[Proof sketch]
The bipolar spectral algebra $H_\tau$ is constructed
in Definition~\ref{def:bipolar-spectral-algebra}
from the boundary local ring and the split-complex unit $j$.
The crossing-point germ exists and is unique
by Proposition~\ref{prop:crossing-unique}:
it is the only omega-germ where both channels
refine nontrivially.
The polarity involution $\sigma$ is the ring automorphism
of $H_\tau$ defined by $\sigma(j) = -j$,
fixing $\hat{\mathbb{Z}}_\tau$ pointwise.
It swaps the two idempotent sectors
and corresponds to the exchange of B-dominant
and C-dominant roles.