%
\label{thm:mutual-determination}
%   I.D20, I.D21, I.T05, I.T10, I.T18, II.D33, II.D35, II.D36, II.T25
The following five descriptions of a holomorphic datum
on $\tau^3$ are canonically equivalent.
Given any one, the other four are uniquely determined:
\begin{enumerate}
    \item[\textup{(R)}]
          A \textbf{refinement tail}:
          a tower-coherent sequence $(f_k)_{k \geq n}$
          in $H_\tau$ stabilized after stage~$n$.

    \item[\textup{(S)}]
          A \textbf{spectral tail}:
          a stabilized character decomposition
          $f = \sum_{\chi \in S_n} c_\chi \cdot \chi$
          with finite support $S_n \subset \widehat{R}_\tau$.

    \item[\textup{(G)}]
          An \textbf{$\omega$-germ}:
          an equivalence class of tower-coherent sequences
          agreeing on all sufficiently deep stages.

    \item[\textup{(C)}]
          A \textbf{boundary character}:
          a ring homomorphism
          $\varphi \colon R_\tau \to H_\tau$.

    \item[\textup{(H)}]
          A \textbf{Hartogs continuation}:
          a holomorphic extension from boundary to interior
          via iterated $\mathrm{BndLift}_n$.
\end{enumerate}

\noindent
Explicitly, the equivalences are:
\[
    \boxed{%
    \textup{(R)}
    \;\overset{\textup{II.L02}}{\Longleftrightarrow}\;
    \textup{(S)}
    \;\overset{\textup{II.L03}}{\Longleftrightarrow}\;
    \textup{(G)}
    \;\overset{\textup{II.L04}}{\Longleftrightarrow}\;
    \textup{(C)}
    \;\overset{\textup{II.L05}}{\Longleftrightarrow}\;
    \textup{(H)}}
\]
All five descriptions are unified
by the bipolar idempotent decomposition
$1 = e_+ + e_-$:
each description splits into two independent channels,
and each channel carries one-dimensional data
that determines all others uniquely.