%
\label{thm:simultaneous-rung}
%   II.T06, II.T07, II.T40, II.T41, II.D13, II.D14,
%   II.D68, II.D70, II.D75, II.D76
$\tau^3$ simultaneously exhibits:
\begin{enumerate}
    \item[\textup{(i)}]
          Complete boundary determination
          ($\mathbb{C}^1$ feature):
          boundary data on $\Lemniscate$
          determines the full interior,
          via BndLift and the Central Theorem (II.T40).
    \item[\textup{(ii)}]
          A distinguished boundary
          ($\mathbb{C}^2$ feature):
          the lemniscate $\Lemniscate = S^1 \vee S^1$
          is a proper subset of the topological boundary
          but determines all holomorphic functions.
    \item[\textup{(iii)}]
          Full Hartogs extension
          ($\mathbb{C}^3$ feature):
          functions holomorphic on the boundary
          extend uniquely to the interior (I.T31).
\end{enumerate}
All three features coexist without the pathologies
associated with their individual rungs
in the orthodox ladder
(no removable singularities,
no Cousin problems,
no Levi problem).

[Proof sketch]
Each feature traces to a different aspect
of the $\tau^3$ fibration
$\tau^3 = \tau^1 \times_f T^2$:
\begin{itemize}
    \item (i) follows from the Central Theorem (II.T40):
          $\mathcal{O}(\tau^3) \cong
          A_{\mathrm{spec}}(\Lemniscate)$.
    \item (ii) follows from the fibered product structure:
          the lemniscate is the base-level boundary
          that controls the fiber.
    \item (iii) follows from Global Hartogs (I.T31):
          the CRT-based extension is constructive
          and works in all dimensions simultaneously.
\end{itemize}
The pathologies are absent because
the Archimedean-Elliptic Engine is absent:
without Archimedean density,
there are no removable singularities
(nothing ``between'' the discrete stages),
no Cousin problems
(cohomology is stage-finite),
and no Levi problem
(domains of holomorphy
are determined by the primorial tower,
not by smooth boundary geometry).