%
\label{thm:hartogs-uniqueness}
%   II.D49, I.T31, I.D21
Let $\chi = e_+ \cdot \chi_+ + e_- \cdot \chi_-$
be an idempotent-supported character
on $\widehat{\mathbb{Z}}_\tau$.
Then the extension $f_\chi : \tau^3 \to H_\tau$
of Lemma~\textup{\ref{lem:extension-h-tau}} is \textbf{unique}:
\[
    \boxed{%
    g : \tau^3 \to H_\tau
    \text{ is $\tau$-holomorphic},\;\;
    g\big|_{\mathbb{L}} = \chi
    \;\;\Longrightarrow\;\;
    g = f_\chi.}
\]
No $\tau$-holomorphic function on~$\tau^3$
with boundary value~$\chi$
can differ from~$f_\chi$.

[Proof via Code/Decode]
The Code/Decode bijection
(Theorem~\ref{thm:code-decode-bijection}, II.T35,
Chapter~\ref{ch:code-decode})
states that every $\tau$-holomorphic function
$g : \tau^3 \to H_\tau$
is uniquely determined
by its boundary coefficient stream
$\mathrm{Code}(g) \in \widehat{\mathbb{Z}}_\tau \times \widehat{\mathbb{Z}}_\tau$.

\medskip
\textbf{Step 1.}
By hypothesis, $g\big|_{\mathbb{L}} = \chi$.
The boundary coefficient stream
is the spectral decomposition
of the boundary restriction:
$\mathrm{Code}(g) = (\chi_+, \chi_-)$.

\medskip
\textbf{Step 2.}
By Lemma~\ref{lem:extension-h-tau},
$f_\chi\big|_{\mathbb{L}} = \chi$.
Hence
$\mathrm{Code}(f_\chi) = (\chi_+, \chi_-)$.

\medskip
\textbf{Step 3.}
Since $\mathrm{Code}$ is a bijection (II.T35),
$\mathrm{Code}(g) = \mathrm{Code}(f_\chi)$
implies $g = f_\chi$.