%
\label{prop:lemniscate-coordinate-limit}
The algebraic lemniscate $\mathbb{L}$ (I.D18)
is isomorphic to the fiber readout of~$\omega$:
\[
    \mathbb{L}
    \;\cong\;
    \operatorname{pr}_{\mathrm{fiber}}(\Phi_\omega).
\]
More precisely:
the bipolar spectral algebra $H_\tau$
with crossing-point germ $\omega_{\mathbb{L}}$
and polarity involution~$\sigma$
(I.D18) is canonically isomorphic,
as an algebra with involution,
to the set of all fiber limits of paths to~$\omega$,
equipped with the sector decomposition
inherited from the $B/C$ dominance partition
and the natural involution that swaps $B$-dominance
with $C$-dominance.

The isomorphism sends each point of~$\mathbb{L}$ to a fiber limit
and vice versa.
We verify the three components.

\textbf{(i) Bipolar sectors.}
The two idempotent sectors $e_+ H_\tau$ and $e_- H_\tau$
of the bipolar spectral algebra (I.D27)
correspond to $B$-dominated and $C$-dominated fiber limits,
respectively.
This correspondence is natural:
the idempotent $e_+ = (1+j)/2$ projects onto the $B$-channel
(the $\gamma$-orbit sector),
and $e_- = (1-j)/2$ projects onto the $C$-channel
(the $\eta$-orbit sector).
The split-complex structure $j^2 = +1$ (I.T10)
is forced by the same bipolar partition
that governs the $B/C$ dominance
in the fiber readout.

\textbf{(ii) Crossing point.}
The crossing-point germ $\omega_{\mathbb{L}}$
corresponds to the fiber limit of balanced paths
(where $B/C \to 1$).
In both descriptions, this is the unique point
where neither sector dominates:
algebraically, it is the identity element
of the bipolar structure;
coordinately, it is the node where the two lobes meet.
The primorial path itself
--- with $B_n = C_n = 1$ for all~$n$ ---
converges to this crossing point.

\textbf{(iii) Polarity involution.}
The involution $\sigma \colon j \mapsto -j$ on $H_\tau$
corresponds to the involution on fiber limits
that swaps B-dominance and C-dominance:
$\sigma(e_+) = e_-$ and $\sigma(e_-) = e_+$.
In the fiber readout, this is the map
that sends a path with asymptotic ratio $B/C = r$
to the path with asymptotic ratio $C/B = 1/r$,
exchanging the two lobes.

Since both structures carry the same bipolar algebra,
the same crossing point,
and the same involution,
the canonical identification
$\mathbb{L} \cong \operatorname{pr}_{\mathrm{fiber}}(\Phi_\omega)$
is an isomorphism of algebras with involution (I.P08).