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\label{lem:character-hartogs}
A boundary character
$\varphi \colon R_\tau \to H_\tau$
determines a unique Hartogs continuation
from the boundary to the interior.
Conversely, every Hartogs continuation
restricts to a unique boundary character.

\emph{Forward ($\mathrm{C} \Rightarrow \mathrm{H}$).}
Given a boundary character~$\varphi$,
construct the Hartogs continuation as follows.
The character $\varphi$ provides boundary datum
at the initial stage:
$\varphi$ determines $f_n$
on $\mathbb{Z}/P_n\mathbb{Z}$
via the spectral decomposition
(Lemma~\ref{lem:germ-character}).
Apply $\mathrm{BndLift}_n$
(II.D36, Chapter~\ref{ch:bndlift-construction}):
the boundary datum at stage~$n$
lifts to interior datum at stage~$n+1$.
The lift uses the CRT decomposition
$\mathbb{Z}/P_{n+1}\mathbb{Z}
\cong \mathbb{Z}/P_n\mathbb{Z}
\times \mathbb{Z}/p_{n+1}\mathbb{Z}$
and the bipolar channel independence
(II.P07, Chapter~\ref{ch:bndlift-construction}).

Iteration: apply $\mathrm{BndLift}_{n+1}$
to produce stage~$n+2$ data,
then $\mathrm{BndLift}_{n+2}$ for stage~$n+3$,
and so on.
The resulting sequence $(f_k)_{k \geq n}$
is tower-coherent by construction
(each $\mathrm{BndLift}_k$
preserves coherence with the previous stage).
The coupling strength at each step
is governed by $\iota_\tau = 2/(\pi + e)$
(II.T25, Chapter~\ref{ch:iota-tau-confirmed}).

Uniqueness follows from the uniqueness
of $\mathrm{BndLift}_n$
at each stage:
the CRT decomposition is canonical,
and the lift in each bipolar channel
is one-dimensional,
leaving no freedom.

\emph{Reverse ($\mathrm{H} \Rightarrow \mathrm{C}$).}
Given a Hartogs continuation $(f_k)_{k \geq n}$,
restrict to the boundary ring:
define $\varphi(r) := \lim_{k \to \infty} f_k(r \bmod P_k)$
for each $r \in R_\tau$.
The limit exists because $(f_k)$
is tower-coherent
and $\tau^3$ is compact (II.T07).
The resulting $\varphi$ is a ring homomorphism
because each $f_k$ respects the CRT ring structure.

\emph{Inverse property.}
Starting from $\varphi$,
constructing the Hartogs continuation,
and restricting back to the boundary
recovers~$\varphi$:
the lift is the unique extension,
and restriction is the inverse operation.
Starting from a Hartogs continuation,
restricting to the boundary,
and lifting back recovers the original continuation
by the uniqueness of $\mathrm{BndLift}_n$.