\label{def:local-langlands-instance}
For each prime $p$ in the primorial tower, the \emph{local Langlands instance at $p$} is the matched pair
\begin{equation}
\Lambda_p \;=\; \bigl(\operatorname{Fr}_p,\; \lambda_p\bigr),
\label{eq:ch48-local-langlands-pair}
\end{equation}
where $\operatorname{Fr}_p = \chi_\pm^{(p)}(1) \in H_{\T}$ is the Frobenius element (the local Galois datum) and $\lambda_p$ is the eigenvalue of the spectral operator on the $p$-th fiber $V_p$ (the local automorphic datum).

The pair is \emph{matched} if and only if the Frobenius determines the eigenvalue and vice versa:
\begin{equation}
\operatorname{Fr}_p \;=\; e_+ \cdot \lambda_p + e_- \cdot \bar{\lambda}_p,
\label{eq:ch48-matching-condition}
\end{equation}
where $e_\pm = \tfrac{1}{2}(1 \pm \jj)$ are the split-complex idempotents and $\bar{\lambda}_p$ denotes the conjugate eigenvalue under $\jj \mapsto -\jj$.  The matching condition asserts that the Galois datum and the spectral datum are two readings of the same split-complex element.