%
\label{thm:extensions-omega-germs}
There is a canonical bijection
\[
    \boxed{%
    \left\{\;
    \parbox{0.4\linewidth}{\centering
        Hartogs extensions $f_\chi$\\
        of boundary characters $\chi$\\
        {\small(II.T37)}
    }
    \;\right\}
    \;\;\xleftrightarrow{\;\;\sim\;\;}\;\;
    \left\{\;
    \parbox{0.4\linewidth}{\centering
        $\omega$-germ transformers $G$\\
        regular at the boundary\\
        {\small(I.D45 + regularity)}
    }
    \;\right\}}
\]
given by:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Forward.}
          $f_\chi \;\longmapsto\; G_{f_\chi}
          := \varprojlim_k f_{\chi,k}$.
          Stagewise naturality \textup{(II.L13)}
          ensures tower coherence;
          bipolar splitting ensures sector independence;
          the limit exists and gives a $\mathrm{HolFun}$
          \textup{(I.D47)}.

    \item[\textup{(ii)}]
          \textbf{Converse.}
          $G \;\longmapsto\; f_{\chi_G}$,
          where $\chi_G = \varprojlim_k (G_k\big|_{R_\tau})$
          is the boundary restriction.
          Regularity ensures $\chi_G$ is idempotent-supported;
          \textup{II.T37} produces the unique extension.

    \item[\textup{(iii)}]
          \textbf{Inverse property.}
          The two maps are mutually inverse:
          $G_{f_{\chi_G}} = G$
          \textup{(Proposition~\ref{prop:ch49-converse})}
          and $f_{\chi_{G_{f_\chi}}} = f_\chi$
          \textup{(by uniqueness of the Hartogs extension,
          II.T37)}.
\end{enumerate}

\noindent
Moreover, the bijection respects the bipolar decomposition:
\[
    G_{f_\chi}
    \;=\;
    e_+ \cdot G_{f_\chi}^+
    \;+\;
    e_- \cdot G_{f_\chi}^-,
\]
where $G_{f_\chi}^\pm$
depends only on $\chi_\pm$.

The forward direction is
Proposition~\ref{prop:ch49-g-is-holfun}:
$G_{f_\chi}$ is a $\tau$-holomorphic function
(sector-independent and tower-coherent),
hence an $\omega$-germ transformer.
It is regular at the boundary
because its boundary restriction
recovers~$\chi$
(by construction:
$G_{f_\chi}\big|_{R_\tau}
= \varprojlim_k f_{\chi,k}\big|_{R_\tau}
= \chi$,
since $f_\chi$ is the extension of~$\chi$).

The converse direction is
Proposition~\ref{prop:ch49-converse}:
regularity gives $\chi_G$,
II.T37 gives $f_{\chi_G}$,
and II.T27 gives $G_{f_{\chi_G}} = G$.

The inverse property in direction (iii):
starting from $f_\chi$,
we construct $G_{f_\chi}$,
restrict to the boundary to get $\chi_{G_{f_\chi}}$,
and extend again.
The boundary restriction of $G_{f_\chi}$
is~$\chi$ (shown above),
so $\chi_{G_{f_\chi}} = \chi$,
and the extension of~$\chi$ is $f_\chi$ (II.T37 uniqueness).
Hence $f_{\chi_{G_{f_\chi}}} = f_\chi$.

The bipolar decomposition
follows from
Proposition~\ref{prop:ch49-bipolar-stage}
and the fact that the inverse limit
commutes with the idempotent projection
(projections preserve limits).