%
\label{def:tau-analytic-atlas}
Let $U, V \subseteq \tau^3$
be cylinder domains (II.D10).
A map $f \colon U \to V$ is \textbf{$\tau$-analytic}
if the following three conditions hold:
\begin{enumerate}
    \item[\textup{(TA1)}]
          \textbf{$\tau$-holomorphicity.}
          The map~$f$ is $\tau$-holomorphic
          in the sense of the Mutual Determination Theorem
          (II.T27, Chapter~\ref{ch:mutual-determination}):
          tower-coherent, stagewise natural,
          and sector-independent.
          Equivalently,
          $f$ is a section
          of the holomorphic sheaf
          $\mathcal{O}_\tau$
          (II.D47, II.T32,
          Chapter~\ref{ch:sheaf-coherence}).

    \item[\textup{(TA2)}]
          \textbf{Fibration preservation.}
          The map~$f$ preserves the product structure
          of the fibered product
          $\tau^3 = \tau^1 \times_f T^2$:
          \begin{itemize}
              \item The base components $(D, A)$
                    of $f(x)$
                    depend only on the base components
                    $(D, A)$ of~$x$.
              \item The fiber components $(B, C)$
                    of $f(x)$
                    depend on the full quadruple
                    $(A, B, C, D)$ of~$x$,
                    but map fiber to fiber:
                    for each fixed base point $(D_0, A_0)$,
                    the restricted map
                    $f|_{\{D_0\} \times \{A_0\} \times T^2}$
                    is a $\tau$-holomorphic map
                    of the fiber $T^2$.
          \end{itemize}
          In short: \emph{base maps to base,
          fiber maps to fiber}.

    \item[\textup{(TA3)}]
          \textbf{Finite spectral support.}
          At each stage~$k$,
          the map~$f$ admits a character expansion
          in the canonical basis $\mathcal{B}_\tau$
          (II.D45, Chapter~\ref{ch:canonical-basis})
          with \textbf{finitely many} active spectral
          coefficients
          (II.T31, Chapter~\ref{ch:canonical-basis}).
\end{enumerate}

\medskip\noindent
A collection of pairs
$\{(U_i, \varphi_i)\}_{i \in I}$
is a \textbf{$\tau$-analytic atlas}
on a topological space~$M$ if:
\begin{enumerate}
    \item[\textup{(A1)}]
          Each $U_i \subseteq M$ is open,
          and $\{U_i\}_{i \in I}$ covers~$M$.
    \item[\textup{(A2)}]
          Each $\varphi_i \colon U_i \to \tau^3$
          is a homeomorphism onto its image
          (a cylinder domain or finite union thereof).
    \item[\textup{(A3)}]
          For each pair $i, j \in I$
          with $U_i \cap U_j \neq \varnothing$,
          the \textbf{transition function}
          \[
              \varphi_j \circ \varphi_i^{-1}
              \;\colon\;
              \varphi_i(U_i \cap U_j)
              \;\longrightarrow\;
              \varphi_j(U_i \cap U_j)
          \]
          is $\tau$-analytic in the sense
          of conditions \textup{(TA1)--(TA3)} above.
    \item[\textup{(A4)}]
          The atlas is \textbf{maximal}:
          every chart $(U, \varphi)$
          whose transition functions
          with all charts in the atlas
          are $\tau$-analytic
          is already included.
\end{enumerate}