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\label{thm:fiber-degeneration-omega}
At the $\omega$-limit of the primorial tower:
\begin{enumerate}
    \item \textbf{Base collapse.}
          The base readout $(D, A)$ collapses to a single point
          $(\Omega, \Omega)$.
    \item \textbf{Fiber locus.}
          The fiber readout $(B, C)$
          traces a one-dimensional locus ---
          the algebraic lemniscate $\mathbb{L}$ (I.D18).
    \item \textbf{Degeneration mechanism.}
          The fiber degeneration $T^2 \to \mathbb{L}$ is forced:
          the two-dimensional fiber parameter space $(B, C)$
          loses one degree of freedom at the boundary
          because the B/C competition
          locks into bipolar lobe structure.
\end{enumerate}

\textbf{(1) Base collapse.}
Along any path $(X_m)$ to $\omega$,
the base coordinates $A_m$ and $D_m$
both grow without bound.
For the primorial path this is immediate:
$A_n = p_n \to \infty$ and $D_n = P_{n-1} \to \infty$.
For a general path, the primorial depth condition
ensures that arbitrarily large primes
must eventually appear in the factorization of $X_m$,
forcing $A_m \to \infty$;
similarly, the cofactor $D_m$ must accommodate
an unbounded collection of smaller primes, forcing $D_m \to \infty$.
Since both coordinates diverge universally,
$\operatorname{pr}_{\mathrm{base}}(\Phi_\omega) = (\Omega, \Omega)$.

\textbf{(2) Fiber locus.}
The primorial path yields $B_n = C_n = 1$ for all~$n$,
but this is only the crossing point $\omega_{\mathbb{L}}$ of $\mathbb{L}$.
Paths with growing $B$ and fixed $C$
(e.g., $X_m = 2^m$) trace the $e_+$-lobe.
Paths with growing $C$ and fixed $B$
(e.g., $X_m = {}^{m}\!2$) trace the $e_-$-lobe.
The totality of fiber limits
is parametrized by the asymptotic dominance ratio $B/C$,
which ranges over all of~$\mathbb{L}$:
$e_+$-lobe for $B \gg C$,
$e_-$-lobe for $C \gg B$,
crossing point for $B \sim C$.
By the Prime Polarity Theorem (I.T05),
every direction in the B/C plane
is realized by some path to~$\omega$,
and the dominance partition
is precisely the bipolar partition
that defines~$\mathbb{L}$ (I.D18).

\textbf{(3) Degeneration mechanism.}
At finite stages, $B$ and $C$ are independent coordinates:
$\dim_\tau = 4$ and the fiber has dimension~$2$.
At the $\omega$-limit, the B and C coordinates
are no longer independent.
The primorial CRT structure (I.T18)
couples them through the coherence conditions
of the inverse system:
for each approach to~$\omega$,
the asymptotic $B/C$ ratio is locked
by the polarity character $\tilde{\chi}$
(the sector assignment of the prime sequence
along the chosen path).
This coupling reduces the two independent
fiber parameters to a single real parameter ---
the position along~$\mathbb{L}$ ---
achieving the degeneration $T^2 \to \mathbb{L}$.
The dimensional reduction is $2 \to 1$,
and the topology of $\mathbb{L}$
(two lobes joined at a node)
reflects the bipolar partition of the coupling.