%
\label{prop:no-free-cartesian}
In Category~$\tau$,
the self-pairing map
$\Delta \colon \tau\text{-Idx} \to
\tau\text{-Idx} \times \tau\text{-Idx}$,
$\underline{n} \mapsto (\underline{n}, \underline{n})$,
while definable as an arithmetic function on indices,
does not constitute an \emph{earned} morphism
in the categorical sense.
The diagonal discipline
(Definition~\ref{def:diagonal-discipline})
prevents self-referential structure
from generating objects
outside the existing orbit rays.
The self-pairing required by Cantor's argument
is an \textbf{unearned diagonal}:
it demands that an enumeration
be able to interrogate itself at its own index,
which is precisely the kind of
self-application that the $1{+}3$ channel structure
was designed to regulate.

By the diagonal discipline
(Definition~\ref{def:diagonal-discipline}),
the self-product of an operation at level~$k$
cannot be absorbed within the same orbit channel;
it must overflow into a distinct channel at level~$k{+}1$.
The Cantor diagonal requires the composite
$n \mapsto f(n)(n)$,
which is a self-application of the evaluation map:
the same index $n$ serves simultaneously
as the \emph{selector} (which sequence to examine)
and as the \emph{position} (which digit to read).
This double role constitutes a diagonal rewiring
in the sense of~I.D03:
the evaluation map at level~$k$ is being ``fed back''
through its own index domain.

In the $\tau$-framework,
such a rewiring is permitted only when
a dedicated orbit channel absorbs the overflow
(Section~\ref{sec:ch05-first-rewiring}
through~\ref{sec:ch05-third-rewiring}).
The three solenoidal channels
$O_\pi$, $O_\gamma$, $O_\eta$
absorb the three legitimate rewirings
(multiplication, exponentiation, tetration).
A fourth rewiring --- which the Cantor diagonal
would constitute ---
has no channel to absorb it,
because $\KAxiom{6}$ (Object Closure) limits
the universe to exactly four orbit rays.

Hence the self-pairing $\Delta$,
interpreted as a categorical morphism
that enables diagonal self-interrogation,
is not an earned arrow of~$\tau$.
It can be \emph{written down} as arithmetic,
but it cannot be \emph{used} to generate
new objects outside $\Obj(\tau)$.