%
\label{def:boundary-to-interior-functor}
The \emph{boundary-to-interior functor}
is the functor
\begin{equation}\label{eq:ch09-phi-functor}
    \Phi \;:\; \Char(\Lemniscate) \;\longrightarrow\; \mathcal{O}(\tau^3)
\end{equation}
defined as follows.
\begin{enumerate}
    \item \textbf{On objects}:
          For each character $\chi_{(m,n)} \in \Char(\Lemniscate)$,
          the functor produces
          the holomorphic function
          $\Phi(\chi_{(m,n)}) \in \mathcal{O}(\tau^3)$
          whose stage-$k$ representative is
          \begin{equation}\label{eq:ch09-phi-stages}
              \Phi(\chi_{(m,n)})_k
              \;=\;
              \chi_{(m,n)} \big|_{\Char_k(\Lemniscate)},
          \end{equation}
          the restriction of the character
          to the $k$-th primorial level.
          Tower coherence is automatic:
          $\chi_{(m,n)}\big|_k$
          reduces to $\chi_{(m,n)}\big|_\ell$
          under the canonical projection
          $\Z / M_k\Z \to \Z / M_\ell\Z$
          for $\ell \leq k$.
    \item \textbf{On morphisms}:
          A lattice translation
          $\chi_{(a,b)} : \chi_{(m,n)} \to \chi_{(m+a,\, n+b)}$
          maps to the holomorphic map
          $\Phi(\chi_{(a,b)})$
          that multiplies
          each stage representative
          by the corresponding
          stage-$k$ value of~$\chi_{(a,b)}$.
\end{enumerate}