\label{def:automorphic-galois-duality}
The \emph{automorphic--Galois duality in~$\T$} is the bidirectional correspondence
\begin{equation}
\text{(Galois datum on the $m$-axis)}
\quad \xleftrightarrow{\;\;\T\text{-Langlands}\;\;} \quad
\text{(automorphic datum on the $n$-axis)}
\label{eq:ch48-ag-duality}
\end{equation}
on the character lattice $\mathbb{Z}^2$ at enrichment level~$\Elayer{1}$.  Concretely:
\begin{enumerate}
\item\emph{(Forward.)} The Galois datum $(\operatorname{Fr}_p)_{p}$ determines the automorphic datum $(\lambda_n)_{n}$ via the local matching conditions~\eqref{eq:ch48-matching-condition}: each $\operatorname{Fr}_p$ decomposes into eigenvalues via the split-complex idempotents, and the collection of eigenvalues constitutes the automorphic spectrum.

\item\emph{(Backward.)} The automorphic datum $(\lambda_n)_{n}$ determines the Galois datum $(\operatorname{Fr}_p)_{p}$ via the Euler product~\eqref{eq:ch48-spectral-determinant}: the spectral determinant $L_{\T}(s)$ encodes the full sequence of Frobenius elements, which can be recovered from the $L$-function by inverting the local factors.
\end{enumerate}
The duality is \emph{exact} at $\Elayer{1}$: no information is lost in either direction.  The matching condition~\eqref{eq:ch48-matching-condition} at each prime is the local witness of the global duality.