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\label{def:ccc-linear-dichotomy}
The \textbf{CCC--linear dichotomy}
is the categorical location
of the self-hosting barrier:
\begin{enumerate}
    \item\textbf{CCC side.}:
          The diagonal $\Delta_A : A \to A \times A$
          exists for all~$A$.
          Contraction is free:
          any resource may be duplicated at will.
          Lawvere's fixed-point theorem applies:
          every point-surjection $A \to B^A$
          forces fixed points on all endomorphisms of $B$.
          Self-reference produces incompleteness.
          Full self-hosting is blocked:
          a system that is cartesian closed
          and rich enough to encode its own morphisms
          cannot be both consistent and complete
          about its own provability.
    \item\textbf{$*$-Autonomous side.}:
          No general $\Delta_A : A \to A \otimes A$ exists.
          Contraction is absent:
          resources are consumed, not copied.
          Lawvere's fixed-point theorem does not apply:
          the diagonal morphism that the proof requires
          is unavailable.
          Self-reference must be \emph{controlled} ---
          introduced in specific, bounded cases
          without the universal incompleteness
          that Lawvere guarantees.
\end{enumerate}
This dichotomy is not a loophole to be exploited
but a structural divide to be respected.
The absence of Lawvere's obstruction
does \emph{not} mean self-hosting is easy ---
it means the standard proof of impossibility
does not go through.
The categorical barrier is absent;
other barriers may still apply.