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\label{def:proto-rationality}
A holomorphic function $f \in \mathcal{O}(\tau^3)$
is \textbf{proto-rational} if its spectral coefficients
$\{\varphi_{mn}\}_{(m,n) \in S}$
satisfy the following conditions:
\begin{enumerate}
    \item \textbf{Finite spectral support.}
          The support $S \subset \Lambda_\tau$ is finite.
          (This is already guaranteed by
          Theorem~\ref{thm:finite-spectral-support}, II.T30.)

    \item \textbf{Basis image condition.}
          Each coefficient $\varphi_{mn}$
          lies in the image of the canonical basis
          $\mathcal{B}_\tau$
          (Definition~\ref{def:canonical-basis}, II.D45):
          there exist finitely many
          cylinder generators
          $E_{k,v}^{(B)}$, $E_{l,w}^{(C)}$
          (Definition~\ref{def:cylinder-generator}, II.D46)
          such that $\varphi_{mn}$
          is an $H_\tau^{\mathrm{cal}}$-linear combination
          of their products.

    \item \textbf{Prime determinacy.}
          The coefficients are determined by
          finitely many prime-indexed data:
          there exists a finite set
          $\Pi \subset \mathbb{P}_\tau$ of primes
          such that $\varphi_{mn}$
          is determined by the restriction of $f$
          to the sub-tower
          $\prod_{p \in \Pi} \Z / p \Z$.
\end{enumerate}
We write $\mathcal{O}_{\mathrm{pr}}(\tau^3)$
for the sub-algebra of proto-rational functions
in $\mathcal{O}(\tau^3)$.