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\label{thm:six-gen-rigidity-fail}
Consider a hypothetical 6-generator system
$\mathrm{Gen}_6 = \{\alpha, \pi, \gamma, \eta, \zeta, \omega\}$
(adding a sixth generator~$\zeta$).
The transposition $\eta \leftrightarrow \zeta$
defines a non-trivial $\rho_6$-automorphism:
\begin{enumerate}
    \item It commutes with~$\rho_6$
          (both $\eta$ and $\zeta$ are solenoidal,
          so $\rho_6$ increments their depths identically).
    \item It is an involution
          (applying the swap twice returns to the identity).
    \item It is \emph{not} the identity
          ($\langle\eta, 0\rangle \mapsto \langle\zeta, 0\rangle$).
\end{enumerate}
This breaks the rigidity theorem
(Theorem~\ref{thm:rigidity}).

The swap $\sigma_{\eta\zeta}$ acts on $\mathrm{Obj}_6$
by exchanging the seed:
$\sigma_{\eta\zeta}(\langle g, n\rangle) = \langle s(g), n\rangle$
where $s(\eta) = \zeta$, $s(\zeta) = \eta$,
and $s(g) = g$ otherwise.
Since $\rho_6$ fixes $\omega$ and increments depth
for all other seeds, $\sigma_{\eta\zeta}$
commutes with $\rho_6$ by direct verification.
It is an involution and moves $\langle\eta, 0\rangle$,
so it is a non-trivial automorphism.