%
\label{thm:adelic-embedding}
The canonical map
$\iota_{\mathbb{A}} : \widehat{\Z}_\tau \to \mathbb{A}_\tau$
satisfies:
\begin{enumerate}
    \item[\textup{(i)}] \textbf{Injectivity.}
          $\iota_{\mathbb{A}}$ is injective:
          if $(a_p)_p = (b_p)_p$ in $\mathbb{A}_\tau$,
          then $a = b$ in $\widehat{\Z}_\tau$.
    \item[\textup{(ii)}] \textbf{Dense image.}
          The image of $\iota_{\mathbb{A}}$
          is dense in $\mathbb{A}_\tau$
          in the following $\tau$-effective sense:
          for every adelic tuple $(x_p)_p \in \mathbb{A}_\tau$
          and every primorial depth~$k$,
          there exists $a \in \widehat{\Z}_\tau$
          such that $a_p = x_p$
          for all $p \leq p_k$.
    \item[\textup{(iii)}] \textbf{Product formula.}
          For every $a \in \widehat{\Z}_\tau$, $a \neq 0$,
          \begin{equation}\label{eq:ch17-product-formula}
              \prod_{p} |a|_p \;\cdot\; \|a\|_{\mathrm{NF}} \;=\; 1,
          \end{equation}
          where $|a|_p = p^{-v_p(a)}$
          is the $p$-adic absolute value,
          $v_p$ is the $p$-adic valuation
          from the NF decomposition,
          and $\|a\|_{\mathrm{NF}} = \prod_p p^{v_p(a)}$
          is the \emph{NF norm}---the
          $\tau$-internal analogue of the archimedean
          absolute value~$|a|_\infty$.
\end{enumerate}
In particular, $\widehat{\Z}_\tau$ embeds
into $\mathbb{A}_\tau$
as a discrete, dense subring.