%
\label{def:geo-topo-unification}
The \textbf{geometric-topological unification}
of the circle in direction~$X$
is the identification
\[
    \boxed{%
    S^1_{\mathrm{geo}}
    \;=\;
    S^1_{\mathrm{top}}
    \;=\;
    \mathrm{den}^X(\mathcal{S}^X),}
\]
where:
\begin{itemize}
    \item $S^1_{\mathrm{geo}}$
          is the set of limit points
          at constant ultrametric ``radius''
          from the center of the $X$-direction cylinders
          (constant D-depth, varying $X$-angle);
    \item $S^1_{\mathrm{top}}$
          is the quotient $\mathbb{R}/\mathbb{Z}$
          (the unique compact connected $1$-manifold);
    \item $\mathrm{den}^X(\mathcal{S}^X)$
          is the image of the solenoidal circle
          under the angular denotation map.
\end{itemize}
All three are the same object:
the Archimedean limit of the angular tower
in direction~$X$.