Solenoidal Circle

The inverse limit of finite cyclic groups in each angular direction. A compact, totally disconnected, torsion-free profinite group whose Archimedean projection is S^1.

Solenoidal Circle

Summary

The inverse limit of finite cyclic groups in each angular direction. A compact, totally disconnected, torsion-free profinite group whose Archimedean projection is S^1.

Statement

%
\label{def:solenoidal-circle}
For each coordinate axis $X \in \{A, B, C\}$,
the \textbf{solenoidal circle in the $X$-direction} is
the inverse limit
\[
    \boxed{%
    \mathcal{S}^X
    \;:=\;
    \varprojlim_{k}\;
    \mathcal{C}_k^X
    \;=\;
    \Bigl\{\,
    (c_k)_{k \geq 1}
    \;\Big|\;
    c_k \in \mathcal{C}_k^X,\;
    \psi_{k,k+1}^X(c_{k+1}) = c_k
    \text{ for all } k
    \,\Bigr\}.}
\]
$\mathcal{S}^X$ carries:
\begin{enumerate}
    \item[\textup{(i)}]
          a \textbf{profinite group structure}:
          componentwise addition modulo the respective primes;
    \item[\textup{(ii)}]
          an \textbf{ultrametric}:
          $d\bigl((c_k), (c_k')\bigr)
          = 2^{-\min\{k : c_k \neq c_k'\}}$;
    \item[\textup{(iii)}]
          the \textbf{profinite topology}:
          compact, Hausdorff, totally disconnected.
\end{enumerate}

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-02.jsonl line 63
• Manuscript source: 2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch24-circles-solenoidal.tex lines 118-152

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookII.Transcendentals.Circles
• Name: solenoid_circle

Dependencies

• Canonical: II.D25, I.T04

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.