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\label{lem:crt-extension}
Let $K \subseteq \mathbb{L}$ be primordially thin,
and let $f \in \mathrm{HolFun}$
(Definition~\ref{def:holfun}, I.D47)
be defined on $\mathbb{L} \setminus K$.
Then for each $t \in K$ and each depth $d \geq 1$,
$f(t) \bmod M_d$ is uniquely determined
by the restriction $f\!\big|_{\mathbb{L} \setminus K}$.

Fix $t \in K$ and $d \geq 1$.
Since $K$ is thin,
$K_d$ misses two CRT directions ---
say the $k_1$-th and $k_2$-th.

\textbf{Step 1: Lateral access.}
Modify $t$ at stage $d$
so that its $k_1$-th CRT component equals
the missing residue $r_{k_1}$.
The resulting $t'$ satisfies $t' \bmod M_d \notin K_d$,
so $f(t')$ is known.

\textbf{Step 2: CRT reconstruction.}
Tower coherence
(Definition~\ref{def:tower-coherence}, I.D46) gives
$f(t) \bmod M_d
= \mathrm{CRT}(f(t) \bmod p_1, \ldots, f(t) \bmod p_d)$.
The CRT coherence constraint
(Theorem~\ref{thm:crt-coherence}, I.T18)
forces each $f(t) \bmod p_k$
to be determined by values of $f$
on omega-tails differing from $t$ only in the $k$-th direction.
Two independent missing directions
guarantee that every reconstruction path
routes through $\mathbb{L} \setminus K$.

\textbf{Step 3: Uniqueness.}
CRT reconstruction is unique ---
$f(t) \bmod M_d$ is determined.