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\label{prop:ch54-tau3-is-manifold}
The fibered product
$\tau^3 = \tau^1 \times_f T^2$,
equipped with the canonical atlas
of cylinder domains
and identity charts,
is a $\tau$-manifold
of dimension~$4$.

\textbf{(M1)}
The Stone topology on~$\tau^3$
is Hausdorff, totally disconnected,
and compact
(II.D14, Chapter~\ref{ch:stone-space}).
The cylinder domains
$\{C_{k,a} : k \geq 1,\; a \in \mathbb{Z}/P_k\mathbb{Z}\}$
form a countable clopen basis
(since $k$ ranges over~$\mathbb{N}$
and $\mathbb{Z}/P_k\mathbb{Z}$ is finite
for each~$k$).
Hence $M = \tau^3$ is second countable.

\textbf{(M2)}
Each cylinder domain $C_{k,a}$
is clopen in the Stone topology.
The identity inclusion
$\varphi_{k,a} \colon C_{k,a} \hookrightarrow \tau^3$
is a homeomorphism onto its image.
For overlapping cylinders,
the transition function is the identity
(Remark~\ref{rem:ch54-trivial-transitions}),
which satisfies (TA1)--(TA3) trivially.
The atlas extends to a maximal atlas
by adding all charts
compatible with the existing ones.

\textbf{(M3)}
The sheaf property of~$\mathcal{O}_\tau$
was established in Theorem~\ref{thm:sheaf-axioms}
(II.T32, Chapter~\ref{ch:sheaf-coherence}).
Since the charts are identity maps,
transport via charts does not alter
the sheaf structure:
$\mathcal{O}_\tau$ on~$\tau^3$
equals $\mathcal{O}_\tau$ on~$M$.