%
\label{def:bndlift}
%   II.D09, II.D33, II.D35
Let $n \geq 1$.
The \textbf{boundary lift at stage~$n$}
is the operator
\[
    \boxed{%
    \mathrm{BndLift}_n
    \;:\;
    \mathrm{Hol}(\mathbb{Z}/P_n\mathbb{Z},\; H_\tau^{\mathrm{cal}})
    \;\longrightarrow\;
    \mathrm{Hol}(\mathbb{Z}/P_{n+1}\mathbb{Z},\; H_\tau^{\mathrm{cal}})}
\]
defined as follows.
Given a holomorphic datum
$f_n : \mathbb{Z}/P_n\mathbb{Z} \to H_\tau^{\mathrm{cal}}$,
the lifted datum
$f_{n+1} := \mathrm{BndLift}_n(f_n)$
is the unique function
$f_{n+1} : \mathbb{Z}/P_{n+1}\mathbb{Z} \to H_\tau^{\mathrm{cal}}$
satisfying:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Compatibility.}
          The restriction of $f_{n+1}$
          to the stage-$n$ component
          equals $f_n$:
          \[
              f_{n+1}(x) \big|_{\bmod P_n}
              \;=\;
              f_n(x \bmod P_n)
          \]
          for all $x \in \mathbb{Z}/P_{n+1}\mathbb{Z}$.

    \item[\textup{(ii)}]
          \textbf{Bipolar splitting.}
          The lift decomposes along the
          bipolar idempotents:
          \[
              f_{n+1}(x)
              \;=\;
              \mathrm{BndLift}_n^{(+)}(f_n^{(+)})(x)\, e_+
              \;+\;
              \mathrm{BndLift}_n^{(-)}(f_n^{(-)})(x)\, e_-,
          \]
          where $f_n^{(\pm)} := e_\pm \cdot f_n$
          are the sector components,
          and $\mathrm{BndLift}_n^{(\pm)}$
          are the \textbf{sector lifts}
          acting independently on each channel.

    \item[\textup{(iii)}]
          \textbf{CRT extension.}
          Each sector lift
          $\mathrm{BndLift}_n^{(\pm)}$
          extends the stage-$n$ sector value
          to stage $n+1$
          using the CRT decomposition:
          \[
              \mathrm{BndLift}_n^{(\pm)}(g)(x)
              \;=\;
              g(x \bmod P_n)
              \;+\;
              \Delta_n^{(\pm)}(x \bmod p_{n+1})
              \cdot P_n,
          \]
          where $\Delta_n^{(\pm)} : \mathbb{Z}/p_{n+1}\mathbb{Z} \to \mathbb{Z}$
          is the \textbf{lift increment}:
          the unique function satisfying
          the tower coherence constraint
          and the diagonal discipline (K5, Book~I).

    \item[\textup{(iv)}]
          \textbf{Coupling.}
          The lift increments
          for the two sectors
          are coupled through $\iota_\tau$:
          \[
              \|\Delta_n^{(+)}\|
              \;=\;
              \iota_\tau \cdot \|\Delta_n^{(-)}\|
              \;=\;
              \frac{2}{\pi + e}
              \cdot \|\Delta_n^{(-)}\|,
          \]
          where $\|\cdot\|$ is the $\ell^2$ norm
          on $\mathbb{Z}/p_{n+1}\mathbb{Z}$.
\end{enumerate}