%
\label{thm:crt-coherence}
At each primorial stage $M_k = \underline{p}_1 \cdot \underline{p}_2 \cdots \underline{p}_k$,
the Chinese Remainder Theorem
(Section~\ref{subsec:ch30-crt}, I.D29)
gives the ring isomorphism:
\[
    \varphi_k \colon
    \mathbb{Z}/M_k\mathbb{Z}
    \;\xrightarrow{\;\cong\;}
    \prod_{i=1}^{k} \mathbb{Z}/\underline{p}_i\mathbb{Z}.
\]
Let $T$ be a tower-coherent $\omega$-germ transformer
(Definition~\ref{def:tower-coherence}, I.D46).
Then:
\begin{enumerate}
    \item \textbf{Prime factorization of the action.}
          The action of $T$ on the $k$-th stage
          is completely determined by its action
          on each CRT factor $\mathbb{Z}/\underline{p}_i\mathbb{Z}$
          for $i = 1, \ldots, k$.
          Formally, there exist functions
          $T^{(i)} : \mathbb{Z}/\underline{p}_i\mathbb{Z} \to \mathbb{Z}/\underline{p}_i\mathbb{Z}[j]$
          such that under the CRT isomorphism:
          \[
              T_k(a_1, \ldots, a_k) = \bigl(T^{(1)}(a_1), \ldots, T^{(k)}(a_k)\bigr).
          \]
    \item \textbf{Stability under tower extension.}
          Passing from stage $k$ to stage $k+1 = M_k \cdot \underline{p}_{k+1}$
          adds exactly one new CRT factor
          (for prime $\underline{p}_{k+1}$),
          and tower coherence requires
          the restriction to the existing $k$ factors
          to be unchanged:
          the functions $T^{(1)}, \ldots, T^{(k)}$
          at stage $k+1$ are identical to those at stage $k$.
    \item \textbf{Prime-by-prime determination.}
          By induction on $k$:
          $T$ is completely determined by the family
          $\{T^{(i)}\}_{i \geq 1}$,
          where each $T^{(i)} : \mathbb{Z}/\underline{p}_i\mathbb{Z} \to \mathbb{Z}/\underline{p}_i\mathbb{Z}[j]$
          describes $T$'s action on the $i$-th prime factor.
\end{enumerate}

\textbf{(1) Prime factorization.}
Fix a stage $k$.
By CRT (Section~\ref{subsec:ch30-crt}),
an element of $\mathbb{Z}/M_k\mathbb{Z}$
decomposes uniquely as a tuple
$(a_1, \ldots, a_k)$ with $a_i \in \mathbb{Z}/\underline{p}_i\mathbb{Z}$.
Consider two elements that agree on all factors
except the $i$-th:
$(a_1, \ldots, a_i, \ldots, a_k)$
and $(a_1, \ldots, a_i', \ldots, a_k)$.
The primorial reduction map $\pi_{k \to k-1}$
(which forgets the $k$-th factor)
sends both to the same depth-$(k-1)$ element
when $i = k$, or preserves the difference when $i < k$.
Iterating tower coherence over the chain of reductions
\[
    \pi_{k \to k-1} \circ \pi_{k-1 \to k-2} \circ \cdots,
\]
we see that the action of $T_k$ on each CRT factor
is determined independently.
Define $T^{(i)}(a_i) := \pi_{k \to \{i\}}(T_k(0, \ldots, a_i, \ldots, 0))$
where $\pi_{k \to \{i\}}$ projects to the $i$-th factor.
The CRT orthogonality of basis elements
($e_i \cdot e_j \equiv 0$ for $i \neq j$)
ensures that the total action decomposes as claimed.

\textbf{(2) Stability.}
An element of $\mathbb{Z}/M_{k+1}\mathbb{Z}$
decomposes via CRT as
$(a_1, \ldots, a_k, a_{k+1})$
where $a_i \in \mathbb{Z}/\underline{p}_i\mathbb{Z}$.
The reduction map $\pi_{k+1 \to k}$
simply forgets the last component $a_{k+1}$.
Tower coherence requires:
\[
    \pi_{k+1 \to k}\bigl(T_{k+1}(a_1, \ldots, a_k, a_{k+1})\bigr)
    = T_k\bigl(\pi_{k+1 \to k}(a_1, \ldots, a_k, a_{k+1})\bigr)
    = T_k(a_1, \ldots, a_k).
\]
By statement (1) applied at stage $k+1$:
$T_{k+1}(a_1, \ldots, a_k, a_{k+1})
= (T^{(1)}(a_1), \ldots, T^{(k)}(a_k), T^{(k+1)}(a_{k+1}))$.
Projecting away the $(k+1)$-th component gives
$(T^{(1)}(a_1), \ldots, T^{(k)}(a_k))$,
which must equal $T_k(a_1, \ldots, a_k)$.
Therefore the functions $T^{(1)}, \ldots, T^{(k)}$
appearing at stage $k+1$
are the same as those at stage $k$.

\textbf{(3) Prime-by-prime determination.}
By induction: at stage $1$, $T$ is determined by $T^{(1)}$.
At stage $k+1$, tower coherence fixes $T^{(1)}, \ldots, T^{(k)}$
(by the inductive hypothesis and statement~(2))
and adds exactly one new function $T^{(k+1)}$.
Therefore $T$ is determined by the countable family
$\{T^{(i)}\}_{i \geq 1}$.