%
\label{def:omega-readout}
Let $\mathcal{P}_\omega$ denote the set of all paths to $\omega$
--- i.e., sequences $(X_m)_{m \geq 1}$ of $\tau$-objects
whose primorial depth grows without bound.
The \textbf{omega readout} is the map
\[
    \Phi_\omega \;\colon\; \mathcal{P}_\omega
    \;\longrightarrow\;
    \{(\Omega, \Omega)\} \times \mathbb{L}
\]
that assigns to each path approaching $\omega$
its base limit $(\Omega, \Omega)$ and its fiber limit on~$\mathbb{L}$.
Concretely, for a path $(X_m)$
with ABCD readouts $\Phi(X_m) = (A_m, B_m, C_m, D_m)$:
\begin{enumerate}
    \item The \textbf{base component} is
          $\operatorname{pr}_{\mathrm{base}}(\Phi_\omega(X_m))
          = (\Omega, \Omega)$
          for \emph{all} paths (universal collapse).
    \item The \textbf{fiber component}
          $\operatorname{pr}_{\mathrm{fiber}}(\Phi_\omega(X_m))
          \in \mathbb{L}$
          is determined by the asymptotic dominance
          of the B and C coordinates:
          \[
              \operatorname{pr}_{\mathrm{fiber}}(\Phi_\omega(X_m))
              \;=\;
              \begin{cases}
                  e_+\text{-lobe} & \text{if } B_m / C_m \to \infty, \\[2pt]
                  e_-\text{-lobe} & \text{if } C_m / B_m \to \infty, \\[2pt]
                  \omega_{\mathbb{L}} & \text{if } B_m / C_m \to 1
                      \text{ (balanced)},
              \end{cases}
          \]
          where $\omega_{\mathbb{L}}$ is the crossing-point germ
          (I.D18, the node of~$\mathbb{L}$).
\end{enumerate}
The omega readout factors through the canonical projections:
the base projection $\operatorname{pr}_{\mathrm{base}} \circ \Phi_\omega$
is constant (always $(\Omega, \Omega)$),
while the fiber projection
$\operatorname{pr}_{\mathrm{fiber}} \circ \Phi_\omega$
is path-dependent and traces out all of~$\mathbb{L}$.