\label{prop:duality-mutual-determination-z2}
The automorphic--Galois duality (Definition~\ref{def:automorphic-galois-duality}) is an instance of the Mutual Determination Schema (Definition~\ref{def:mutual-determination-schema}, Ch.~33) with the following identification:
\begin{enumerate}
\item\emph{(B)} \textbf{Boundary datum} $=$ the $m$-axis data $(\operatorname{Fr}_p)_{p}$ (Galois side): the prime-by-prime CRT decomposition of the global character restricted to the boundary $\Lemniscate$.

\item\emph{(S)} \textbf{Spectral datum} $=$ the $n$-axis data $(\lambda_n)_{n}$ (automorphic side): the eigenvalue-by-eigenvalue decomposition of the spectral operator on $A_{\mathrm{spec}}(\Lemniscate)$.

\item\emph{(I)} \textbf{Interior datum} $=$ the full character $\chi_{m,n}$ on $\mathbb{Z}^2$: the complete lattice of spectral characters at enrichment level~$\Elayer{1}$.
\end{enumerate}
The triple $(B, S, I)$ satisfies full bidirectionality:
\begin{equation}
B \;\longleftrightarrow\; I \;\longleftrightarrow\; S.
\label{eq:ch48-mutual-det-bsi}
\end{equation}