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\label{thm:minimal-alphabet}
$|\mathrm{Gen}| = 5$ is the unique cardinality
for which the $\tau$-kernel simultaneously achieves:
\begin{enumerate}
    \item[(a)] \textbf{Completeness:}
          All three rewiring levels of the iterator ladder
          have canonical orbit channel assignments
          ($\pi \leftrightarrow +$,
          $\gamma \leftrightarrow \times$,
          $\eta \leftrightarrow \hat{}$\,).
    \item[(b)] \textbf{Rigidity:}
          $\Aut(\tau) = \{\id\}$ ---
          no non-trivial $\rho$-automorphism exists.
    \item[(c)] \textbf{Saturation:}
          Tetration has no channel,
          and is algebraically degraded.
\end{enumerate}

The 5-generator system satisfies all three properties:
completeness by the channel assignments
of Chapter~\ref{ch:iterator-ladder},
rigidity by Theorem~\ref{thm:rigidity},
and saturation by Theorem~\ref{thm:ladder-saturation}
and Theorem~\ref{thm:tetration-degraded}.

Uniqueness follows from the counter-models:
\begin{itemize}
    \item $|\mathrm{Gen}| = 4$ fails completeness
          (Theorem~\ref{thm:four-gen-incomplete}).
    \item $|\mathrm{Gen}| = 6$ fails rigidity
          (Theorem~\ref{thm:six-gen-rigidity-fail}).
\end{itemize}
Since adding generators breaks rigidity
and removing generators breaks completeness,
$5$ is the unique solution
to the simultaneous constraint.