%
\label{lem:stagewise-naturality}
Let $\chi$ be an idempotent-supported
boundary character,
and let $f_\chi \colon \tau^3 \to H_\tau$
be its Hartogs extension \textup{(II.T37)}.
Then $f_\chi$ is \textbf{stagewise natural}:
for all $\ell > k \geq 1$,
the diagram
\[
    \begin{array}{ccc}
        \mathbb{Z}/P_\ell\mathbb{Z}
        & \xrightarrow{\;\; f_{\chi,\ell} \;\;}
        & H_\tau \\[6pt]
        \Big\downarrow\vcenter{\rlap{\scriptsize$\rho_{k,\ell}$}}
        & & \Big\| \\[6pt]
        \mathbb{Z}/P_k\mathbb{Z}
        & \xrightarrow{\;\; f_{\chi,k} \;\;}
        & H_\tau
    \end{array}
\]
commutes in the sense that
\[
    \boxed{%
    f_{\chi,k}\bigl(\rho_{k,\ell}(a)\bigr)
    \;=\;
    \mathrm{reduce}_k\bigl(f_{\chi,\ell}(a)\bigr)
    \qquad
    \text{for all } a \in \mathbb{Z}/P_\ell\mathbb{Z},}
\]
where $\mathrm{reduce}_k$ is the stage-$k$
reduction on the codomain $H_\tau$
induced by the primorial tower structure.
In other words:
computing $f_\chi$ at the finer stage~$\ell$
and then reducing the output to stage~$k$
gives the same result
as first reducing the input to stage~$k$
and then computing~$f_\chi$.

The proof uses three ingredients,
each established in previous chapters.

\smallskip
\noindent\textbf{Ingredient 1: Tower coherence of~$\chi$.}
The boundary character $\chi$
is a ring homomorphism
$\chi \colon R_\tau \to H_\tau$,
where $R_\tau = \widehat{\mathbb{Z}}_\tau$
is the profinite boundary ring (I.D19, Book~I).
By definition of the profinite limit,
$\chi$ is determined by a coherent family
$(\chi_k)_{k \geq 1}$
where $\chi_k \colon \mathbb{Z}/P_k\mathbb{Z} \to H_\tau$
satisfies
\[
    \chi_k\bigl(\rho_{k,\ell}(a)\bigr)
    \;=\;
    \mathrm{reduce}_k\bigl(\chi_\ell(a)\bigr)
    \qquad
    \text{for all } a \in \mathbb{Z}/P_\ell\mathbb{Z}.
\]
This is tower coherence (I.D46, Book~I)
for the boundary data.

\smallskip
\noindent\textbf{Ingredient 2: BndLift preserves coherence.}
The $\mathrm{BndLift}_n$ construction
(II.D36, Chapter~\ref{ch:bndlift-construction})
extends boundary data at stage~$n$
to interior data at stage~$n+1$
via the CRT decomposition.
By the Mutual Determination Theorem
(II.T27, Chapter~\ref{ch:mutual-determination}),
the five descriptions of holomorphic data
are equivalent,
and in particular the tower-coherent description~(R)
is preserved by $\mathrm{BndLift}_n$.
Concretely:
if $(g_k)_{k \leq n}$ is tower-coherent,
then the extended sequence
$(g_k)_{k \leq n+1}$
with $g_{n+1} = \mathrm{BndLift}_n(g_n)$
is still tower-coherent.

\smallskip
\noindent\textbf{Ingredient 3: Bipolar channel independence.}
The extension $f_\chi$
decomposes as
$f_\chi = e_+ \cdot f_\chi^+ + e_- \cdot f_\chi^-$
(Proposition~\ref{prop:ch49-bipolar-stage}).
The two channels are independent.
It therefore suffices to verify
stagewise naturality in each channel separately.
In the $e_+$-channel:
$f_{\chi,k}^+ = e_+ \cdot f_{\chi,k}$
depends only on $\chi_+$,
and $\chi_+$ is tower-coherent (Ingredient~1).
The extension in the $e_+$-channel
is the iterated $\mathrm{BndLift}_n^+$ of $\chi_+$,
which preserves coherence (Ingredient~2).
The same argument applies
in the $e_-$-channel with $\chi_-$.

\smallskip
\noindent\textbf{Assembly.}
Combining all three ingredients:
for any $\ell > k \geq 1$
and $a \in \mathbb{Z}/P_\ell\mathbb{Z}$,
\begin{align*}
    \mathrm{reduce}_k\bigl(f_{\chi,\ell}(a)\bigr)
    &= \mathrm{reduce}_k\Bigl(
        e_+ \cdot f_{\chi,\ell}^+(a)
        + e_- \cdot f_{\chi,\ell}^-(a)
    \Bigr) \\
    &= e_+ \cdot \mathrm{reduce}_k\bigl(f_{\chi,\ell}^+(a)\bigr)
    + e_- \cdot \mathrm{reduce}_k\bigl(f_{\chi,\ell}^-(a)\bigr) \\
    &= e_+ \cdot f_{\chi,k}^+\bigl(\rho_{k,\ell}(a)\bigr)
    + e_- \cdot f_{\chi,k}^-\bigl(\rho_{k,\ell}(a)\bigr) \\
    &= f_{\chi,k}\bigl(\rho_{k,\ell}(a)\bigr),
\end{align*}
where the third equality uses
the coherence of each channel
(Ingredients~1 and~2 in each channel independently).