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\label{thm:slippage-breaks-omega}
Any foundational stack that permits identity slippage
(Definition~\ref{def:identity-slippage}, I.D90)
at the substrate level
cannot internalize a unique, absolute infinity $\omega$.
Such a system can only generate
``many infinities'' ---
model-relative, non-canonical limit objects ---
via limit/quotient constructions.

[Proof sketch]
The argument has three parts.

\textbf{Part 1: The three roles of $\omega$.}
In $\tau$, $\omega$ serves three roles simultaneously:
\begin{enumerate}
    \item[(a)] \textbf{Absorber.}
          $\omega$ is the unique absorbing element
          of the primorial tower:
          $\omega \cdot n = \omega$ for all finite $n$.
          It is the ``top'' of the tower.
    \item[(b)] \textbf{Unique closure point.}
          The primorial tower $\{p_k\#\}_{k \geq 1}$
          has $\omega$ as its unique limit.
          No other element serves as the closure point
          of the tower.
    \item[(c)] \textbf{Unique absolute reference.}
          $\omega$ is globally identifiable:
          it is invariant under all admissible symmetries
          of the tower,
          not susceptible to equivalence-class ambiguity,
          and not relativized to a model or interpretation.
\end{enumerate}
These three roles are mutually reinforcing:
(a) makes $\omega$ algebraically unique,
(b) makes it topologically unique
(as a limit in the tower),
and (c) makes it ontically unique
(as a reference point
that the system can canonically identify).

\textbf{Part 2: Uniqueness requires identity coherence.}
Being ``the unique absolute reference'' (role~(c))
means that $\omega$ must be globally identifiable:
any admissible construction
that produces an element playing the absorber role
must produce the \emph{same} element.
This is a strong condition.
It requires that the system can determine,
for any candidate $x$:
does $x = \omega$, or does $x \neq \omega$?
The determination must be canonical ---
not dependent on a choice of representative,
not relative to a model,
not subject to equivalence-class ambiguity.

Identity slippage
(Definition~\ref{def:identity-slippage}, I.D90)
is precisely the negation of this condition.
A system with identity slippage
harbors shadow identities
(Definition~\ref{def:shadow-identity}, I.D91):
objects that the system cannot canonically distinguish
from one another.
If $\omega$ could have shadow identities ---
if there existed objects $\omega'$, $\omega''$, $\ldots$
that play the absorber role
but cannot be canonically identified with $\omega$ ---
then $\omega$ would not be unique.
It would be an equivalence class
of absorber-candidates,
mediated by an identification
that the system cannot decide.

\textbf{Part 3: Identity slippage produces many infinities.}
In a foundation with identity slippage,
the construction of infinite objects
proceeds through limits and quotients.
Each construction path may produce
a different ``infinity'':
$\aleph_0$ as the cardinality of $\mathbb{N}$,
$\aleph_1$ as the next infinite cardinal
(whose identity depends on the continuum hypothesis
and thus on the model),
$\omega_1$ as the first uncountable ordinal
(well-defined but not canonically constructible
from the natural numbers alone).
These are not shadow identities of a single $\omega$.
They are genuinely different objects ---
different infinities ---
generated by different construction paths
in a system where no single, canonical closure point
absorbs all others.

The proliferation is not accidental.
It is a structural consequence of identity slippage
at the substrate level.
If the system cannot canonically determine
``this is the same as that,''
then limits constructed by different paths
need not converge to the same point.
Different paths produce different infinities.
The continuum hypothesis is undecidable in ZFC
precisely because ZFC's identity slippage
(at the level of set-membership and extensionality)
prevents the system from canonically resolving
the relationship between $\aleph_0$ and $2^{\aleph_0}$.

In $\tau$, identity slippage is zero
(Corollary~\ref{cor:no-identity-decoherence}, I.C03).
Every construction path converges
to a unique normal form (I.L02).
The primorial tower has one closure point,
not many.
$\omega$ is unique because the system
that produces it has no mechanism
for generating alternatives.