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\label{thm:unique-infinity}
$\omega$ is the unique non-finite object of Category~$\tau$.
More precisely:
every $x \in \Obj(\tau)$ is either $\omega$
or has the form $\rho^n(g)$
for some generator $g \in \{\alpha, \pi, \gamma, \eta\}$
and some $n \geq 0$.
The latter objects have finite depth.
Therefore $\omega$ is the unique object of infinite depth ---
the unique infinity.

By the Ontic Closure Theorem
(Theorem~\ref{thm:ontic-closure}, I.T01),
\[
    \Obj(\tau)
    = \{\omega\}
    \cup O_\alpha \cup O_\pi \cup O_\gamma \cup O_\eta,
\]
and the five components are pairwise disjoint.
Each orbit element $\rho^n(g)$ with $n \geq 0$
has \textbf{depth} $n$, a natural number.
This is well-defined because the representation is unique
(Ontic Closure, part~4).

It remains to show that $\omega$ has no finite depth.
Suppose for contradiction that $\omega = \rho^n(g)$
for some generator $g$ and some $n \geq 0$.
If $n = 0$, then $\omega = g$,
contradicting Ontic Closure (disjointness of $\{\omega\}$
from each orbit ray, since $g = \rho^0(g) \in O_g$).
If $n \geq 1$, then $\omega = \rho^n(g)$
contradicts $\KAxiom{5}$.
Thus $\omega$ does not belong to any orbit ray
and has no finite depth.

Since every non-$\omega$ object has finite depth,
and $\omega$ does not,
$\omega$ is the unique object of infinite depth.