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\label{thm:ontic-closure}
The universe of Category~$\tau$ satisfies:
\begin{enumerate}
    \item \textbf{Exhaustiveness.}
          $\Obj(\tau) = \{\omega\} \cup O_\alpha
          \cup O_\pi \cup O_\gamma \cup O_\eta$.
    \item \textbf{Disjointness.}
          The five sets are pairwise disjoint.
    \item \textbf{Countability.}
          Each $O_g$ is countably infinite; $|\Obj(\tau)| = \aleph_0$.
    \item \textbf{Unique representation.}
          Every $x \in \Obj(\tau)$ is either $\omega$
          or has a unique representation $x = \rho^n(g)$
          for a unique $g \in \{\alpha, \pi, \gamma, \eta\}$
          and a unique $n \geq 0$.
    \item \textbf{Ontic seal.}
          No further objects can be produced.
          The generative act is complete.
\end{enumerate}

\emph{Part~(1)} is $\KAxiom{6}$ (Object Closure).

\emph{Part~(2)} is Proposition~\ref{prop:orbit-disjoint}.

\emph{Part~(3):}
By Proposition~\ref{prop:orbit-countable},
each $O_g$ is in bijection with $\mathbb{N}$.
The beacon singleton has cardinality~$1$.
The total cardinality is
$1 + \aleph_0 + \aleph_0 + \aleph_0 + \aleph_0 = \aleph_0$.

\emph{Part~(4):}
By Part~(1), every $x$ belongs to one of the five sets.
By Part~(2), it belongs to exactly one.
If $x \in O_g$, then $x = \rho^n(g)$ for some~$n$
(by definition of $O_g$).
Uniqueness of $n$ follows from the injectivity of $\varphi_g$
(Proposition~\ref{prop:orbit-countable}).
Uniqueness of $g$ follows from Part~(2).

\emph{Part~(5):}
$\KAxiom{6}$ asserts that every object of $\tau$
is in the decomposition.
Since $\rho$ maps orbit elements to orbit elements
(by $\KAxiom{3}$) and maps $\omega$ to $\omega$ (by $\KAxiom{2}$),
no application of $\rho$ can produce an object
outside the decomposition.
No other operation exists in the signature.
Therefore, the universe is sealed.