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\label{def:tau-manifold}
A \textbf{$\tau$-manifold} is a pair
$(M, \mathcal{A}_\tau)$ where:
\begin{enumerate}
    \item[\textup{(M1)}]
          $M$ is a topological space
          with the $\tau$-topology:
          Hausdorff, totally disconnected,
          compact, and second countable,
          with a clopen basis
          (as established for $\tau^3$
          in Chapter~\ref{ch:stone-space}, II.D14,
          and Chapter~\ref{ch:topology-invariant}, II.T07).

    \item[\textup{(M2)}]
          $\mathcal{A}_\tau$ is a maximal
          $\tau$-analytic atlas on~$M$
          (Definition~\ref{def:tau-analytic-atlas}, II.D64):
          a collection of charts
          $\{(U_i, \varphi_i)\}_{i \in I}$
          covering~$M$,
          with each $\varphi_i \colon U_i \to \tau^3$
          a homeomorphism onto a cylinder domain,
          and all transition functions
          $\varphi_j \circ \varphi_i^{-1}$
          $\tau$-analytic.

    \item[\textup{(M3)}]
          The atlas $\mathcal{A}_\tau$
          is compatible with the sheaf structure:
          the holomorphic presheaf $\mathcal{O}_\tau$
          (II.D47, II.T32,
          Chapter~\ref{ch:sheaf-coherence})
          on~$M$, defined by transport via charts,
          is a sheaf.
\end{enumerate}

\medskip\noindent
The \textbf{dimension} of a $\tau$-manifold
is the dimension of the model space~$\tau^3$,
which is~$4$
(II.D15, Chapter~\ref{ch:dimension-four}):
one radial (D), one base-angular (A),
and two fiber (B, C) coordinates.