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\label{def:archimedean-bridge}
The \textbf{Archimedean-Non-Archimedean Bridge}
is the pair of measurement systems:
\begin{enumerate}
    \item[\textup{(NA)}]
          \textbf{Non-Archimedean measurement}
          (ultrametric refinement):
          the D-depth $\delta(x,y)$
          measures how many stages
          two $\tau$-admissible points agree.
          Refinement increases depth:
          stage $k+1$ refines stage~$k$
          by a factor of~$p_{k+1}$.
          The metric is $d(x,y) = 2^{-\delta(x,y)}$
          (II.D13).
          Convergence means: increasing depth of agreement.

    \item[\textup{(A)}]
          \textbf{Archimedean measurement}
          (Euclidean resolution):
          the ABCD coordinates,
          normalized by $1/P_k$
          (the approximation sequence,
          Chapter~\ref{ch:orthodox-bridge}),
          measure how precisely the angular
          and radial positions
          are determined.
          Resolution increases precision:
          stage $k+1$ resolves angles
          to $1/P_{k+1}$ instead of $1/P_k$.
          Convergence means: increasing decimal precision.
\end{enumerate}
The constant $\iota_\tau$ converts between the two:
\[
    \boxed{%
    \text{Euclidean resolution at depth } \delta
    \;\approx\;
    \iota_\tau \cdot 2^{-\delta}
    \;=\;
    \frac{2}{\pi + e}\,
    \cdot\, 2^{-\delta}.}
\]
The factor $\iota_\tau$ accounts for the
difference between the base-$2$ ultrametric
and the base-$P_k$ Archimedean normalization.