%
\label{thm:bndlift-existence}
%   II.D33, II.D35, II.D36
For every $n \geq 1$
and every holomorphic datum
$f_n \in \mathrm{Hol}(\mathbb{Z}/P_n\mathbb{Z},\, H_\tau^{\mathrm{cal}})$:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Existence.}
          The boundary lift
          $f_{n+1} = \mathrm{BndLift}_n(f_n)$
          exists.

    \item[\textup{(ii)}]
          \textbf{Uniqueness.}
          $f_{n+1}$ is the \textbf{unique}
          holomorphic extension of $f_n$
          to $\mathbb{Z}/P_{n+1}\mathbb{Z}$
          that satisfies the tower coherence
          and diagonal discipline constraints.

    \item[\textup{(iii)}]
          \textbf{Sector factorization.}
          The lift factors as
          \[
              \mathrm{BndLift}_n
              \;=\;
              \mathrm{BndLift}_n^{(+)}
              \;\times\;
              \mathrm{BndLift}_n^{(-)},
          \]
          where each sector lift acts
          independently on its channel.
          The factors commute:
          the B-sector lift does not depend
          on the C-sector input,
          and conversely.

    \item[\textup{(iv)}]
          \textbf{Norm bound.}
          The lifted function satisfies
          \[
              \|f_{n+1}\|_\tau
              \;\leq\;
              \|f_n\|_\tau
              \;\cdot\;
              \bigl(1 + C_\tau / p_{n+1}\bigr),
          \]
          where $C_\tau = 2\iota_\tau = 4/(\pi+e)$
          is a universal constant.
          In particular, $\|f_{n+1}\|_\tau \to \|f_n\|_\tau$
          as $p_{n+1} \to \infty$:
          the lift is \textbf{asymptotically isometric}.
\end{enumerate}

\textbf{(i) Existence.}
The CRT decomposition
$\mathbb{Z}/P_{n+1}\mathbb{Z}
\cong
\mathbb{Z}/P_n\mathbb{Z} \times \mathbb{Z}/p_{n+1}\mathbb{Z}$
(I.T18, Book~I)
reduces the extension problem
to finding the $p_{n+1}$-component
of~$f_{n+1}$.
The sector decomposition
$f_{n+1} = f_{n+1}^{(+)} e_+ + f_{n+1}^{(-)} e_-$
splits this into two independent problems,
one per sector.

For each sector,
the extension from $\mathbb{Z}/P_n\mathbb{Z}$
to $\mathbb{Z}/P_n\mathbb{Z} \times \mathbb{Z}/p_{n+1}\mathbb{Z}$
amounts to choosing a function
$\Delta_n^{(\pm)} : \mathbb{Z}/p_{n+1}\mathbb{Z} \to \mathbb{Z}$.
The tower coherence constraint requires
that the CRT reduction from stage $n+1$ to stage~$n$
recovers~$f_n^{(\pm)}$.
This is automatically satisfied
by the CRT structure:
$f_{n+1}^{(\pm)}(x) \bmod P_n = f_n^{(\pm)}(x \bmod P_n)$
holds by construction.

The diagonal discipline (K5)
constrains the lift increment~$\Delta_n^{(\pm)}$
further.
K5 forbids the use of the diagonal argument
on the tower:
concretely,
$\Delta_n^{(\pm)}$ must be \emph{compatible}
with the NF reduction
$\mathbb{Z}/p_{n+1}\mathbb{Z} \to \{0, 1, \ldots, p_{n+1}-1\}$,
in the sense that
$\Delta_n^{(\pm)}(r) \neq r$
for all $r \in \{1, \ldots, p_{n+1} - 1\}$
(the off-diagonal condition).
The number of such functions is $(p_{n+1}-1)^{p_{n+1}-1}$
(with $\Delta_n^{(\pm)}(0)$ free),
but the holomorphicity condition
(tower coherence at \emph{all} subsequent stages)
pins $\Delta_n^{(\pm)}$ down uniquely.

\textbf{(ii) Uniqueness.}
Suppose $f_{n+1}$ and $f'_{n+1}$
are two holomorphic extensions of~$f_n$.
Their difference $\delta = f_{n+1} - f'_{n+1}$
vanishes on the stage-$n$ component
(by compatibility)
and satisfies the diagonal discipline.
Write $\delta = \delta^{(+)} e_+ + \delta^{(-)} e_-$.
Each sector difference
$\delta^{(\pm)} : \mathbb{Z}/p_{n+1}\mathbb{Z} \to \mathbb{Z}$
satisfies:
\begin{itemize}
    \item $\delta^{(\pm)}(0) = 0$
          (compatibility at $x \equiv 0 \bmod p_{n+1}$),
    \item $\delta^{(\pm)}$ is holomorphic
          (tower-coherent at all subsequent stages).
\end{itemize}
A holomorphic function on $\mathbb{Z}/p_{n+1}\mathbb{Z}$
that vanishes at~$0$
is determined by its values at $1, \ldots, p_{n+1}-1$.
The diagonal discipline imposes
$p_{n+1} - 1$ independent constraints,
one for each non-zero residue class.
This leaves exactly one solution: $\delta = 0$.
Hence $f_{n+1} = f'_{n+1}$.

\textbf{(iii) Sector factorization.}
Since $e_+ e_- = 0$,
the B-sector and C-sector decouple completely:
$\mathrm{BndLift}_n(f_n)
= \mathrm{BndLift}_n^{(+)}(e_+ f_n) \cdot e_+
+ \mathrm{BndLift}_n^{(-)}(e_- f_n) \cdot e_-$.
The B-sector lift depends only on~$f_n^{(+)}$;
the C-sector lift depends only on~$f_n^{(-)}$.
The two lifts compute independently
and assemble via the idempotent sum.

\textbf{(iv) Norm bound.}
In each sector,
the lift increment satisfies
$|\Delta_n^{(\pm)}(r)| \leq p_{n+1} - 1$
for $r \in \mathbb{Z}/p_{n+1}\mathbb{Z}$.
The calibrated norm of the increment
is bounded by
$\|\Delta_n^{(\pm)} \cdot P_n\|
\leq P_n \cdot (p_{n+1} - 1)
= P_{n+1} - P_n$.
Hence
$\|f_{n+1}\|_\tau
\leq \|f_n\|_\tau + \|f_{n+1} - f_n\|_\tau
\leq \|f_n\|_\tau + C_\tau \cdot P_n / P_{n+1}
\leq \|f_n\|_\tau \cdot (1 + C_\tau / p_{n+1})$,
where $C_\tau = 2\iota_\tau$
accounts for the two-sector geometric mean
in the calibrated norm.
Since $\sum_{n} 1/p_{n+1}$ diverges logarithmically,
the product $\prod_n (1 + C_\tau / p_{n+1})$
diverges---but this is the correct behavior:
the norm of the fully lifted function
(the profinite limit)
is finite only for functions
whose lift increments decay sufficiently fast.
The bound says that each individual lift step
changes the norm by at most $C_\tau / p_{n+1}$,
which tends to zero.