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\label{thm:applied-saturation}
The enrichment ladder
\[
    \Elayer{0} \;\xrightarrow{\;\mathcal{F}_{E}\;}
    \Elayer{1} \;\xrightarrow{\;\mathcal{F}_{E}\;}
    \Elayer{2} \;\xrightarrow{\;\mathcal{F}_{E}\;}
    \Elayer{3} \;\xrightarrow{\;\mathcal{F}_{E}\;}
    \Elayer{3}
\]
is \textbf{complete}: each of the first three arrows
produces irreducible new structure,
and the fourth arrow is the identity.
Concretely:
\begin{enumerate}
    \item \textbf{$\Elayer{0} \to \Elayer{1}$}:
          split-complex enrichment of $\Hom$~spaces;
          physics emerges.
    \item \textbf{$\Elayer{1} \to \Elayer{2}$}:
          discrete rational carriers via the spectral bridges;
          computation emerges.
    \item \textbf{$\Elayer{2} \to \Elayer{3}$}:
          self-referential codes acquire self-modelling capacity;
          proof theory and metaphysics emerge.
    \item \textbf{$\Elayer{3} \to \Elayer{3}$}:
          self-modelling applied to self-modelling
          yields self-modelling.
          The paradox diagnostics
          (Theorem~\ref{thm:four-paradox-diagnostic})
          exhaust the boundary crossings.
          No new structural mode appears.
\end{enumerate}