%
\label{lem:extension-h-tau}
%   II.D35, I.D21, I.T31
Let $\chi = e_+ \cdot \chi_+ + e_- \cdot \chi_-$
be an idempotent-supported character
on the boundary ring~$\widehat{\mathbb{Z}}_\tau$.
Then the function
\[
    \boxed{%
    f_\chi
    \;:=\;
    e_+ \cdot f_+ \;+\; e_- \cdot f_-
    \;:\; \tau^3 \;\longrightarrow\; H_\tau}
\]
satisfies:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Well-defined.}
          $f_\chi$ is a well-defined
          $\tau$-holomorphic function
          valued in $H_\tau^{\mathrm{cal}}$
          (the calibrated split-complex codomain, II.D35).

    \item[\textup{(ii)}]
          \textbf{Boundary recovery.}
          The restriction of $f_\chi$
          to the boundary $\mathbb{L}$
          recovers~$\chi$:
          \[
              f_\chi\big|_{\mathbb{L}}
              \;=\; \chi.
          \]

    \item[\textup{(iii)}]
          \textbf{Tower coherence.}
          The system $\{(f_\chi)_k\}_{k \geq 1}$
          is tower-coherent:
          $(f_\chi)_{k+1} \bmod P_k = (f_\chi)_k$
          for all $k \geq 1$.

    \item[\textup{(iv)}]
          \textbf{$\tau$-regularity.}
          The function $f_\chi$ is $\tau$-regular
          (Definition~\ref{def:tau-regularity}, II.D49)
          at every interior point of~$\tau^3$.
\end{enumerate}

\textbf{(i) Well-defined.}
Each sector lift $f_\pm$
is the inverse limit
of the iterated $\mathrm{BndLift}$ operator
(Proposition~\ref{prop:ch30-convergence},
Chapter~\ref{ch:bndlift-construction}).
The $\mathrm{BndLift}$ existence theorem
(Theorem~\ref{thm:bndlift-existence}, II.T26)
guarantees that each individual lift step exists.
Tower coherence holds by construction,
so the inverse limit is a well-defined function
$f_\pm : \tau^3 \to \mathbb{R}$.

The recombination
$f_\chi = e_+ \cdot f_+ + e_- \cdot f_-$
takes values in
$H_\tau \cong \mathbb{R} \cdot e_+ + \mathbb{R} \cdot e_-$
(the idempotent decomposition
of the calibrated split-complex codomain).
Since $e_+ + e_- = 1$ and $e_+ \cdot e_- = 0$,
the sum is well-defined
and lies in~$H_\tau^{\mathrm{cal}}$.

\textbf{$\tau$-holomorphy.}
The Hol\,$\Leftrightarrow$\,Idemp equivalence
(Theorem~\ref{thm:hol-iff-idempotent}, II.T33)
states that a function $\tau^3 \to H_\tau$
is $\tau$-holomorphic
if and only if it is idempotent-supported.
By construction,
$f_\chi = e_+ \cdot f_+ + e_- \cdot f_-$
is manifestly idempotent-supported:
the $e_+$-component is $f_+$,
the $e_-$-component is $f_-$,
and each component is tower-coherent.
Hence $f_\chi$ is $\tau$-holomorphic.

\textbf{(ii) Boundary recovery.}
At the boundary $\mathbb{L}$,
the sector lifts reduce
to the boundary character components:
$f_+\big|_{\mathbb{L}} = \chi_+$
and $f_-\big|_{\mathbb{L}} = \chi_-$.
This holds because the $\mathrm{BndLift}$ operator
is constructed precisely to be compatible
with the boundary data ---
condition~(i) of Definition~\ref{def:bndlift} (II.D36)
requires
$f_{n+1}\big|_{\bmod P_n} = f_n$
at every stage.
At the profinite limit,
$f_\pm\big|_{\mathbb{L}} = \varprojlim_n f_{\pm,n}\big|_{\mathrm{bnd}}
= \chi_\pm$.
Hence
$f_\chi\big|_{\mathbb{L}}
= e_+ \cdot \chi_+ + e_- \cdot \chi_-
= \chi$.

\textbf{(iii) Tower coherence.}
At stage~$k$,
$(f_\chi)_k
= e_+ \cdot (f_+)_k + e_- \cdot (f_-)_k$.
Tower coherence of $f_+$ and $f_-$ individually
(from the $\mathrm{BndLift}$ construction)
implies tower coherence of their
idempotent-weighted sum:
\begin{align*}
    (f_\chi)_{k+1} \bmod P_k
    &\;=\;
    e_+ \cdot \bigl((f_+)_{k+1} \bmod P_k\bigr)
    \;+\;
    e_- \cdot \bigl((f_-)_{k+1} \bmod P_k\bigr) \\
    &\;=\;
    e_+ \cdot (f_+)_k
    \;+\;
    e_- \cdot (f_-)_k \\
    &\;=\;
    (f_\chi)_k.
\end{align*}
The key step uses
$e_\pm \bmod P_k = e_\pm$
(the idempotents are independent of the stage).

\textbf{(iv) $\tau$-regularity.}
At every interior point $x \in \tau^3 \setminus \mathbb{L}$,
the function $f_\chi$ has a stabilized $\omega$-germ:
the $\mathrm{BndLift}$ system converges
at finite depth,
meaning $x$ lies in some
stage-$k$ cylinder domain (II.D10)
on which $f_\chi$
is already determined by the stage-$k$ data.
This is precisely the positive regularity condition
(Definition~\ref{def:tau-regularity}, II.D49):
$f_\chi$ has a canonical interior extension
at~$x$,
arising from stabilized lift data.