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\label{thm:tetration-degraded}
Tetration fails all three algebraic canonicality conditions:
\begin{enumerate}
    \item \textbf{Non-commutative:}
          ${}^{\underline{3}}\underline{2} = \underline{16}
          \neq \underline{9} = {}^{\underline{2}}\underline{3}$.
    \item \textbf{Non-associative:}
          $({}^{\underline{2}}\underline{2})
          \uparrow\uparrow \underline{2}
          = {}^{\underline{2}}\underline{4} = \underline{256}
          \neq \underline{65536}
          = {}^{\underline{4}}\underline{2}
          = {}^{({}^{\underline{2}}\underline{2})}\underline{2}$.
    \item \textbf{No left identity $\geq \underline{2}$:}
          for any $\underline{e} \geq \underline{2}$,
          ${}^{\underline{2}}\underline{e} = \underline{e}^{\underline{e}}
          \geq \underline{4} > \underline{2}$.
\end{enumerate}
By contrast, addition has identity~$\underline{0}$,
multiplication has identity~$\underline{1}$,
and exponentiation has right identity~$\underline{1}$
($\underline{n}^{\underline{1}} = \underline{n}$).
The algebraic degradation at level~3
is an independent obstruction to extending
the canonical operation hierarchy beyond tetration.

Items (1) and (2) are verified by computation
on small concrete values.
For (3): if $\underline{e} \geq \underline{2}$,
then $\underline{e}^{\underline{e}} \geq \underline{2}^{\underline{2}}
= \underline{4} > \underline{2}$,
so ${}^{\underline{2}}\underline{e}
= \underline{e}^{\underline{e}} \neq \underline{2}$.