%
\label{lem:gluing}
Let $U \subseteq \tau^3$ be a cylinder domain,
and let $\{U_i\}_{i \in I}$
be a cover of~$U$ by cylinder sub-domains.
Suppose that for each $i \in I$
we are given $f_i \in \mathcal{O}_\tau(U_i)$,
and that these local sections are \textbf{compatible}:
\[
    f_i\big|_{U_i \cap U_j}
    \;=\;
    f_j\big|_{U_i \cap U_j}
    \qquad
    \text{for all } i, j \in I.
\]
Then there exists a unique $f \in \mathcal{O}_\tau(U)$
with $f\big|_{U_i} = f_i$ for all $i \in I$.

The proof proceeds in two steps:
construction of~$f$ as a set-theoretic function,
then verification that~$f$ is $\tau$-holomorphic.

\smallskip
\noindent\textbf{Step 1: Construction.}
Define $f : U \to H_\tau$ by setting,
for each $x \in U$:
\[
    f(x) := f_i(x)
    \qquad\text{for any } i \in I \text{ with } x \in U_i.
\]
This is well-defined:
if $x \in U_i \cap U_j$,
then $f_i(x) = f_j(x)$
by the compatibility condition.

\smallskip
\noindent\textbf{Step 2: $\tau$-holomorphicity.}
We verify that $f$ is $\tau$-holomorphic on~$U$,
i.e., that the stagewise components
$(f_k)_{k \geq 1}$ satisfy tower coherence
and sector independence.

\emph{Tower coherence.}
The cylinder cover $\{U_i\}$
can be refined to a cover by stage-$N$ cylinders
for some sufficiently large~$N$
(because every cylinder domain
is a union of deeper-stage cylinders).
At each stage $k \leq N$,
the component $f_k$
is determined by the $f_i$
on the stage-$k$ projection of each~$U_i$.
The individual $f_i$ are $\tau$-holomorphic,
so each satisfies tower coherence.
On overlaps, the coherence conditions agree
(because the $f_i$ agree on overlaps),
and the coherence data pastes consistently.

The key observation is that cylinder overlaps
in the ultrametric topology
are themselves cylinders:
\[
    C_{k,a} \cap C_{k,b}
    \;=\;
    \begin{cases}
        C_{k,a} & \text{if } a = b, \\
        \varnothing & \text{if } a \neq b.
    \end{cases}
\]
Two cylinders at the same stage
are either identical or disjoint---there
are no ``partial overlaps.''
Overlaps between cylinders at different stages
reduce to containment:
$C_{k,a} \cap C_{\ell, b}$
is either empty or equals the deeper cylinder.
This clean overlap structure
is what makes gluing straightforward:
the compatibility condition
reduces to agreement on nested sub-cylinders,
which is exactly tower coherence.

\emph{Sector independence.}
Each $f_i$ satisfies the bipolar sector independence
of the Mutual Determination Theorem (II.T27):
the B-channel and C-channel components
of~$f_i$ are determined independently.
Since $f$ agrees with~$f_i$ on each~$U_i$,
the sector independence of the local pieces
carries over to~$f$ on all of~$U$.

\emph{Uniqueness.}
Uniqueness follows from locality
(Proposition~\ref{prop:ch36-locality}):
if $f$ and $g$ both satisfy
$f\big|_{U_i} = g\big|_{U_i} = f_i$,
then $(f - g)\big|_{U_i} = 0$ for all~$i$,
so $f - g = 0$ on~$U$.