%
\label{thm:associativity}
Let $f = \{f_k\}$, $g = \{g_k\}$,
and $h = \{h_k\}$ be $\omega$-germ transformers.
Then
\[
    \boxed{%
    h \circ (g \circ f)
    \;=\;
    (h \circ g) \circ f.}
\]

The proof proceeds in two steps.

\emph{Step~1: Stagewise reduction.}
At each finite stage~$k$, both sides evaluate to
the same map $h_k \circ g_k \circ f_k$:
\begin{align*}
    \bigl(h \circ (g \circ f)\bigr)_k
    &\;=\;
    h_k \circ (g_k \circ f_k),
    \\[4pt]
    \bigl((h \circ g) \circ f\bigr)_k
    &\;=\;
    (h_k \circ g_k) \circ f_k.
\end{align*}
These are equal by associativity
of set-theoretic function composition
on the finite stage $\mathbb{Z}/P_k\mathbb{Z}$.

\emph{Step~2: Coherent lift via the program monoid.}
Step~1 shows pointwise equality at each finite stage.
But $\omega$-germ transformers are not merely
families of stage maps---they are
coherent lifts of program-monoid elements.
Let $w_f, w_g, w_h \in \mathcal{M}_\tau$
be the program-monoid words
corresponding to $f$, $g$, $h$
(via the factorization I.D49, Book~I).
Then:
\[
    g \circ f
    \;\longleftrightarrow\;
    w_g \cdot w_f,
    \qquad
    h \circ (g \circ f)
    \;\longleftrightarrow\;
    w_h \cdot (w_g \cdot w_f).
\]
By the program monoid's associativity (I.P02, Book~I):
\[
    w_h \cdot (w_g \cdot w_f)
    \;=\;
    (w_h \cdot w_g) \cdot w_f,
\]
which corresponds to $(h \circ g) \circ f$.
The coherent lift is faithful
(program-monoid multiplication
corresponds bijectively
to $\omega$-germ composition,
by the Mutual Determination Theorem II.T27),
so the equality lifts
from the monoid to the $\omega$-germ level.