%
\label{thm:global-hartogs}
%           I.T21, I.T18, I.T12, I.D37
Let $K \subseteq \mathbb{L}$ be primordially thin
(Definition~\ref{def:primorial-thinness}, I.D67),
and let $f \in \mathrm{HolFun}$
be defined on $\mathbb{L} \setminus K$.
Then $f$ extends uniquely to
$\tilde{f} \in \mathrm{Hol}(\mathbb{L})$
(Definition~\ref{def:hol-L}, I.D49):
\[
    \boxed{%
    K \text{ thin},\;
    f \in \mathrm{HolFun}(\mathbb{L} \setminus K)
    \;\;\Longrightarrow\;\;
    \exists!\; \tilde{f} \in \mathrm{Hol}(\mathbb{L}):\;
    \tilde{f}\!\big|_{\mathbb{L} \setminus K} = f.}
\]
No boundedness hypothesis is required.

The proof proceeds in four steps,
each drawing on a different strand
of the machinery developed in Parts~IX--XII.

\medskip
\textbf{Step 1: Spectral reduction.}
The spectral decomposition
(Theorem~\ref{thm:spectral-decomposition}, I.T12)
splits $f$ into spectral components:
\[
    f(t) = f_+(t) \cdot e_+ + f_-(t) \cdot e_-,
    \qquad
    f_\pm(t) := \chi_\pm(f(t)),
\]
for $t \in \mathbb{L} \setminus K$.
It suffices to extend $f_+$ and $f_-$ separately,
recovering $\tilde{f}$ by spectral reassembly.
Each spectral component inherits tower coherence
(the characters are ring homomorphisms
commuting with primorial reduction).
At each stage $M_d$,
$f_\pm(t) \bmod M_d \in \mathbb{Z}/M_d\mathbb{Z}$ ---
a finite set.
Each spectral component is therefore \emph{automatically bounded}.

\medskip
\textbf{Step 2: CRT reconstruction at each stage.}
Fix $f_+$ and a depth $d$.
For each $t \in K$,
the CRT Extension Lemma
(Lemma~\ref{lem:crt-extension}, I.L08)
determines $f_+(t) \bmod M_d$ uniquely:
$K$ is thin, so $K_d$ misses two CRT directions,
and the Chinese Remainder Theorem
assembles the known values along those directions
into $f_+(t) \bmod M_d$.

\medskip
\textbf{Step 3: Tower coherence ensures global consistency.}
The stage-by-stage reconstructions must satisfy:
\[
    \tilde{f}_+(t) \bmod M_d
    \;=\;
    \bigl(\tilde{f}_+(t) \bmod M_{d+1}\bigr) \bmod M_d
    \qquad \text{for all } d \geq 1.
\]
This holds because CRT reconstruction at $M_{d+1} = M_d \cdot p_{d+1}$
decomposes into the reconstruction at $M_d$
plus a new factor.
Tower coherence
(Definition~\ref{def:tower-coherence}, I.D46)
and CRT coherence
(Theorem~\ref{thm:crt-coherence}, I.T18)
force the reconstructed values on $K$
to be compatible.
The sequence $(\tilde{f}_+(t) \bmod M_d)_{d \geq 1}$
defines a valid omega-tail
(Definition~\ref{def:omega-tail}, I.D25).
Similarly for $\tilde{f}_-$.
Set $\tilde{f}(t) := \tilde{f}_+(t) \cdot e_+ + \tilde{f}_-(t) \cdot e_-$.

\medskip
\textbf{Step 4: Uniqueness by the Identity Theorem.}
If $\tilde{f}_1, \tilde{f}_2 \in \mathrm{Hol}(\mathbb{L})$
both extend $f$, they agree on $\mathbb{L} \setminus K$.
Since $K$ is thin, $\mathbb{L} \setminus K$ is primordially dense ---
at each depth, it contains elements in all CRT directions
not blocked by $K$.
Agreement at some depth $d_0$ follows,
and the $\tau$-Identity Theorem
(Theorem~\ref{thm:tau-identity}, I.T21)
gives $\tilde{f}_1 = \tilde{f}_2$.