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\label{lem:pentation-non-inj}
There is no orbit channel available to host a level-4 operation.
Consequently, pentation cannot be equipped with
a canonical injectivity proof within $\tau$.

[Proof sketch]
A canonical injectivity proof for the level-$k$ operation
requires that the operation's ``type information''
be encoded in a dedicated orbit channel.
At levels 0--3, the four channels
$O_\alpha, O_\pi, O_\gamma, O_\eta$
provide this encoding.

At level~4, a fifth channel would be needed.
But by $\KAxiom{6}$ (Object Closure),
$\Obj(\tau) = \{\omega\} \cup O_\alpha \cup O_\pi
\cup O_\gamma \cup O_\eta$.
There is no fifth orbit ray.
The beacon $\omega$ is a singleton and cannot host
a countably infinite type channel.

Therefore, pentation lacks the structural scaffold
for canonical injectivity.
Without canonical injectivity,
the operation cannot serve as a coordinate function
in the normal form decomposition (Part~V),
and the ladder terminates.