\label{thm:hinge-theorem}
Let $\operatorname{Enr}\colon \operatorname{Cat}_{\T}(\Elayer{k}) \to \operatorname{Cat}_{\T}(\Elayer{k+1})$ be the self-enrichment functor, and let $\{A, B, C, D\}$ be the four primitive sectors of the $4{+}1$ decomposition.  The content of Books~IV--VII decomposes as follows.
\begin{enumerate}
\item\emph{(Book~IV: Microcosm.)}
The fiber $T^2$ of the fibration $\tau^3 = \tau^1 \times_f T^2$ at enrichment level $\Elayer{1}$, restricted to the $D$-sector, produces the categorical microcosm:
\[
\textup{Book~IV} \;=\; \operatorname{Enr}^1\big|_{D}\bigl(\operatorname{Cat}_{\T}(\Elayer{0})\bigr)\big|_{T^2}.
\]
The $D$-sector characters encode gravity and quantum mechanics on the fiber.

\item\emph{(Book~V: Macrocosm.)}
The base $\tau^1$ at enrichment level $\Elayer{1}$, restricted to sectors $A$, $B$, $C$, produces the categorical macrocosm:
\[
\textup{Book~V} \;=\; \operatorname{Enr}^1\big|_{A \cup B \cup C}\bigl(\operatorname{Cat}_{\T}(\Elayer{0})\bigr)\big|_{\tau^1}.
\]
The three primitive sectors encode the three gauge forces: weak ($A$), electromagnetic ($B$), strong ($C$).

\item\emph{(Book~VI: Life.)}
Enrichment level $\Elayer{2}$, applied to the full sector template, produces the categorical life layer:
\[
\textup{Book~VI} \;=\; \operatorname{Enr}^2\bigl(\operatorname{Cat}_{\T}(\Elayer{0})\bigr).
\]
Proto-codes (III.D61, Ch.~46) provide the carrier; operational closure provides the invariant.

\item\emph{(Book~VII: Metaphysics.)}
Enrichment level $\Elayer{3}$, saturated by Theorem~\ref{thm:saturation-e3} (Ch.~7), produces the categorical metaphysics layer:
\[
\textup{Book~VII} \;=\; \operatorname{Enr}^3\bigl(\operatorname{Cat}_{\T}(\Elayer{0})\bigr).
\]
Self-models provide the carrier; saturation provides the terminal invariant.

\item\emph{(Derivation.)}
The seven-book architecture is derived: the number of books equals the number of irreducible instantiations of the enrichment tower with sector restrictions, which is~$7$.
\end{enumerate}