\label{def:enriched-bi-square}
The \emph{enriched bi-square} at $\Elayer{1}+$ is the $2 \times 3$ pasted diagram
\begin{equation}
\begin{tikzcd}[column sep=3.5em, row sep=2.5em]
\operatorname{Enr}_{01}\bigl(F(\operatorname{Prim}(k{+}1))\bigr)
  \arrow[r, "\operatorname{res}_k"]
  \arrow[d, "\chi_\pm^{(k+1)}"']
& \operatorname{Enr}_{01}\bigl(F(\operatorname{Prim}(k))\bigr)
  \arrow[r, "\bigoplus_S \pi_S"]
  \arrow[d, "\chi_\pm^{(k)}"]
& \displaystyle\bigoplus_{S} F_S(k)
  \arrow[d, "\operatorname{AG}_S"]
\\
\chi_\pm \circ \operatorname{Enr}_{01}\bigl(F(\operatorname{Prim}(k{+}1))\bigr)
  \arrow[r, "\operatorname{res}_k"']
& \chi_\pm \circ \operatorname{Enr}_{01}\bigl(F(\operatorname{Prim}(k))\bigr)
  \arrow[r, "\prod_S \pi_S^{\mathrm{spec}}"']
& \displaystyle\prod_{S} \operatorname{Spec}_S(k)
\end{tikzcd}
\label{eq:ch50-enriched-bi-square}
\end{equation}
where $k$ ranges over the primorial ladder, $S \in \{A, B, C, D\}$ ranges over primitive sectors, and the structural maps are as follows.

\emph{Left square (sector-coupled tower coherence).}
\begin{itemize}
\item The top-left horizontal map $\operatorname{res}_k$ is the tower restriction: the primorial transition from depth $k{+}1$ to depth $k$, applied through the enrichment functor $\operatorname{Enr}_{01}$.
\item The vertical maps $\chi_\pm^{(k)}$ are the bipolar spectral decompositions via the split-complex idempotents $e_\pm = \tfrac{1}{2}(1 \pm \jj)$.
\item The left square commutes because $\operatorname{Enr}_{01}$ preserves tower coherence (Definition~\ref{def:enrichment-functor-e01}, Ch.~44): the spectral decomposition at depth $k{+}1$ restricts to the spectral decomposition at depth~$k$.
\end{itemize}

\emph{Right square (Langlands functoriality).}
\begin{itemize}
\item The top-right horizontal map $\bigoplus_S \pi_S$ is the sector projection: it decomposes the enriched tower datum at depth~$k$ into its primitive sector components $F_S(k)$ for $S \in \{A, B, C, D\}$.
\item The right-hand vertical maps $\operatorname{AG}_S$ are the automorphic--Galois dualities (Definition~\ref{def:automorphic-galois-duality}, Ch.~48) restricted to each sector.
\item The right square commutes by the Functoriality Theorem (Theorem~\ref{thm:functoriality-theorem}, Ch.~49): sector morphisms commute with the automorphic--Galois duality.
\end{itemize}