%
\label{thm:circle-profinite}
For each coordinate axis $X \in \{A, B, C\}$:
\begin{enumerate}
    \item[\textup{(i)}]
          The angular denotation map
          $\mathrm{den}^X : \mathcal{S}^X \to S^1$
          is a continuous surjection
          of topological groups.
    \item[\textup{(ii)}]
          The image
          $\mathrm{den}^X(\mathcal{S}^X) = S^1$:
          every point on the classical circle
          is the limit of some coherent
          angular sequence.
    \item[\textup{(iii)}]
          $S^1$ is recovered as the
          \textbf{Archimedean angular limit}:
          \[
              \boxed{%
              S^1
              \;\cong\;
              \varprojlim_k\;
              \frac{1}{Q_k^X}\,\mathbb{Z}
              \Big/
              \mathbb{Z}
              \;=\;
              \Bigl\{\,
              (\theta_k)_{k \geq 1}
              \;\Big|\;
              \theta_k \in \tfrac{1}{Q_k^X}\mathbb{Z}/\mathbb{Z},\;
              \theta_k \equiv \theta_{k+1}
              \!\!\!\pmod{1/Q_k^X}
              \,\Bigr\}.}
          \]
          Every element of this angular inverse limit
          determines a unique point on~$S^1$:
          the common value of the nested arcs.
    \item[\textup{(iv)}]
          No uncountable continuum is needed:
          $S^1$ arises from countable approximations
          (consistent with I.T35, Book~I).
\end{enumerate}

\textbf{(i).}
Continuity of $\mathrm{den}^X$ follows from
the ultrametric estimate:
if $(c_k) = (c_k')$ for $k \leq K$,
then $|\theta_K^X - \theta_K'^X| \leq 1/Q_K^X$,
so the images are within $1/Q_K^X$ on~$S^1$.
Group homomorphism: componentwise addition maps
to addition of angles.

\smallskip
\noindent\textbf{(ii).}
Surjectivity: for any $\theta \in S^1 = [0,1)$,
define $c_k$ by the CRT reconstruction
of $\lfloor \theta \cdot Q_k^X \rfloor$
modulo the primes $p_{j(1)}, \ldots, p_{j(k)}$.
The resulting coherent sequence maps to~$\theta$.
(This uses the density of the grids
$\frac{1}{Q_k^X}\mathbb{Z}/\mathbb{Z}$
in~$S^1$.)

\smallskip
\noindent\textbf{(iii).}
The angular inverse limit is defined
by the same nested-arc construction
as Theorem~\ref{thm:real-inverse-limit}
for lines,
but on the quotient $\mathbb{R}/\mathbb{Z}$
instead of~$\mathbb{R}$.
Each coherent sequence $(\theta_k)$
determines a nested sequence of arcs
of length $1/Q_k^X$,
whose intersection is a single point on~$S^1$.

\smallskip
\noindent\textbf{(iv).}
At each stage, the grid
$\frac{1}{Q_k^X}\mathbb{Z}/\mathbb{Z}$
is finite (cardinality~$Q_k^X$).
The inverse limit is constructed
from countable data
with coherence constraints.
The same reasoning as in
Chapter~\ref{ch:lines-countable},
Section~\ref{sec:ch23-no-uncountable}
applies: Cantor's diagonal
is inapplicable to this structured construction
(I.T35, Book~I).