\label{thm:finite-factorization-pasting}
Let $X$ be an $\Elayer{1}$ object in $\operatorname{Cat}_{\T}(\Elayer{1})$ with NF address in the primorial tower at depth~$k$.  Then $X$ factors through finitely many primitive sector components:
\begin{equation}
\operatorname{Enr}_{01}(X) \;\cong\; \bigoplus_{S \in \{A, B, C, D\}} X_S,
\label{eq:ch50-finite-factorization}
\end{equation}
where each $X_S = \pi_S \circ \operatorname{Enr}_{01}(X)$ is the sector-$S$ component.  Moreover, the factorization is compatible with both the tower structure and the spectral structure:
\begin{enumerate}
\item\emph{(Tower.)} The restriction $\operatorname{res}_k(X_S) = (\operatorname{res}_k X)_S$ for every sector~$S$ and every primorial depth~$k$: the sector decomposition commutes with tower restriction.
\item\emph{(Spectral.)} The automorphic--Galois duality preserves the factorization: $\operatorname{AG}(\operatorname{Enr}_{01}(X)) = \prod_S \operatorname{AG}_S(X_S)$, where the product on the right runs over the primitive sectors.
\item\emph{(Finiteness.)} The number of non-trivial sector components is bounded: $|\{S : X_S \neq 0\}| \leq 4$, with equality when $X$ has non-trivial projection onto every primitive sector.
\end{enumerate}