%
\label{thm:removable-singularity}
Let $K \subseteq \mathbb{L}$ be primordially thin
(Definition~\ref{def:primorial-thinness}, I.D67),
$f \in \mathrm{HolFun}$ bounded on $\mathbb{L} \setminus K$.
Then $f$ extends uniquely to
$\tilde{f} \in \mathrm{Hol}(\mathbb{L})$
(Definition~\ref{def:hol-L}, I.D49):
\[
    \boxed{%
    K \text{ thin},\;
    f \in \mathrm{HolFun}(\mathbb{L} \setminus K)
    \text{ bounded}
    \;\;\Longrightarrow\;\;
    \exists!\; \tilde{f} \in \mathrm{Hol}(\mathbb{L}):\;
    \tilde{f}\!\big|_{\mathbb{L} \setminus K} = f.}
\]

\textbf{Existence.}
For $t \in K$, define $\tilde{f}(t) \bmod M_d$
by CRT extension (Lemma~\ref{lem:crt-extension}, I.L08)
at each depth $d$.
For $t \notin K$, set $\tilde{f}(t) := f(t)$.
The sequence $(\tilde{f}(t) \bmod M_d)_{d \geq 1}$
is compatible: the CRT reconstruction at $M_{d+1}$
reduces to the one at $M_d$ by transitivity of CRT.
Boundedness of $f$ ensures the reconstructed values
remain in $\hat{\mathbb{Z}}_\tau$.
The extension satisfies tower coherence
(Definition~\ref{def:tower-coherence}, I.D46)
and CRT coherence
(Theorem~\ref{thm:crt-coherence}, I.T18)
by construction, so $\tilde{f}$ is $\tau$-holomorphic.

\textbf{Uniqueness.}
Two extensions $\tilde{f}_1, \tilde{f}_2$
agree on $\mathbb{L} \setminus K$.
Since $K$ is thin,
$\mathbb{L} \setminus K$ is primordially dense,
so $\tilde{f}_1$ and $\tilde{f}_2$ agree at some depth $d_0$.
By the $\tau$-Identity Theorem
(Theorem~\ref{thm:tau-identity}, I.T21),
$\tilde{f}_1 = \tilde{f}_2$.