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\label{prop:counting-structural}
The countability of $\Obj(\tau)$
(Theorem~\ref{thm:ontic-closure})
is not a limitation imposed from outside
but a structural consequence
of the generative counting principle
(Definition~\ref{def:generative-counting}):
a universe built by counting is a counted universe.
The question ``is $\tau$ countable?''
presupposes an external vantage point
that the ontically sealed universe ($\KAxiom{6}$)
does not provide.

By the Ontic Closure Theorem (I.T01),
$\Obj(\tau) = \{\omega\} \cup O_\alpha
\cup O_\pi \cup O_\gamma \cup O_\eta$.
Each orbit $O_g$ is countably infinite
(Proposition~\ref{prop:orbit-countable}),
and a finite union of countable sets is countable.
Thus $\Obj(\tau)$ is countable.

But this is more than an external observation.
The bijection $\varphi_g(n) = \rho^n(g)$
(Definition~\ref{def:generative-counting})
is not applied \emph{after} the universe is built
to ``measure'' its size.
It \emph{is} the construction.
Each application of $\rho$
simultaneously creates an object
and assigns it an index.
The universe is therefore not merely
``countable in the sense that a bijection to $\mathbb{N}$ exists''
--- it is \emph{counted} in the sense that
the generative process itself is the counting.

The second claim follows from Object Closure ($\KAxiom{6}$):
the universe is ontically sealed.
There is no ``view from outside'' $\Obj(\tau)$
from which to classify its cardinality
relative to other infinite sets.
The only infinite cardinal
that can be formulated within $\tau$
is $\aleph_0$, and it is the cardinality
of the entire universe.