\label{thm:enrichment-tower-assembly}
The enrichment tower $\Elayer{0} \subsetneq \Elayer{1} \subsetneq \Elayer{2}$ satisfies the following properties.
\begin{enumerate}
\item\emph{(Strict inclusion.)} Each level adds irreducible structure that the previous level cannot express:
\begin{itemize}
\item $\Elayer{0} \subsetneq \Elayer{1}$: the split-complex dynamics, sector-coupled processes, and defect functionals of $\Elayer{1}$ have no $\Elayer{0}$ counterpart.  In particular, the Yang--Mills gap $\Gamma^*_s > 0$ is a strictly $\Elayer{1}$ phenomenon: it requires NF discreteness in the strong sector, which is undefined at $\Elayer{0}$.
\item $\Elayer{1} \subsetneq \Elayer{2}$: the self-referential code structure of $\Elayer{2}$ (Definition~\ref{def:e2-computational-agent}, Ch.~54) is not expressible at $\Elayer{1}$.  Proto-codes exist at $\Elayer{1}$ (Definition~\ref{def:proto-code}, Ch.~46), but they lack decoders: they label data without verifying themselves.
\end{itemize}

\item\emph{(Coherent scaling.)} The three bi-squares form a scaling chain:
\begin{equation}
\underbracket{\text{I.T41}}_{\Elayer{0}\text{ algebraic}}
\;\longrightarrow\;
\underbracket{\text{III.D65}}_{\Elayer{1}\text{ enriched}}
\;\longrightarrow\;
\underbracket{\text{III.D56}}_{\Elayer{2}\text{ computational}},
\label{eq:ch51-bisquare-scaling}
\end{equation}
with the same diagram shape (pasted bi-square with $(\times, \wedge)$-axes) but richer objects at each step.  The enrichment functors $\operatorname{Enr}_{01}$ and $\operatorname{Enr}_{12}$ preserve the bi-square structure: they map vertices to vertices, edges to edges, and commuting squares to commuting squares.

\item\emph{(Millennium coverage.)} The eight Millennium Problems partition across the three levels:
\begin{center}
\begin{adjustbox}{max width=0.95\linewidth}
\begin{tabular}{p{0.1\linewidth}p{0.28\linewidth}p{0.28\linewidth}p{0.28\linewidth}}
\toprule
\textbf{Level} & \textbf{Bi-Square} & \textbf{Millennium Results} & \textbf{Mutual Determination} \\
\midrule
$\Elayer{0}$ & I.T41 (algebraic) & RH, Poincar\'e & boundary $\leftrightarrow$ spectral \\
\addlinespace
$\Elayer{1}$ & III.D65 (enriched) & NS, YM, Hodge, BSD, Langlands & sector dynamics \\
\addlinespace
$\Elayer{2}$ & III.D56 (computational) & P vs NP & search $=$ construction \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{center}
No Millennium Problem straddles two levels: each lives at exactly one enrichment level, and the scope labels record which level is required.  (BSD and Langlands bridge to~$\Elayer{2}$ via their classical formulations; see Chapter~33.)
\end{enumerate}