%
\label{def:evolution-operator}
For integers $m > n \geq 1$,
the \textbf{evolution operator}
$\mathcal{E}_{n \to m}$
is the composition of successive boundary lifts:
\[
    \boxed{%
    \mathcal{E}_{n \to m}
    \;:=\;
    \mathrm{BndLift}_{m-1}
    \circ
    \mathrm{BndLift}_{m-2}
    \circ \cdots \circ
    \mathrm{BndLift}_n
    \;\colon\;
    \mathrm{Hol}_n(\tau^3)
    \;\longrightarrow\;
    \mathrm{Hol}_m(\tau^3),}
\]
where $\mathrm{Hol}_k(\tau^3)$
is the space of holomorphic data
at stage~$k$
(tower-coherent maps
$f_k \colon \mathbb{Z}/P_k\mathbb{Z} \to H_\tau$).

The operator has the following structure:
\begin{enumerate}
    \item[\textup{(E1)}]
          \textbf{Stage-by-stage construction.}
          Each $\mathrm{BndLift}_k$
          uses the CRT decomposition
          $\mathbb{Z}/P_{k+1}\mathbb{Z}
          \cong \mathbb{Z}/P_k\mathbb{Z}
          \times \mathbb{Z}/p_{k+1}\mathbb{Z}$
          to extend the domain by one prime factor.

    \item[\textup{(E2)}]
          \textbf{Bipolar splitting.}
          At each step,
          $\mathrm{BndLift}_k$
          splits into independent lifts
          in the $e_+$-channel and $e_-$-channel:
          \[
              \mathrm{BndLift}_k
              \;=\;
              e_+ \cdot \mathrm{BndLift}_k^+
              \;+\;
              e_- \cdot \mathrm{BndLift}_k^-.
          \]

    \item[\textup{(E3)}]
          \textbf{Semigroup property.}
          For $\ell > m > n \geq 1$:
          \[
              \mathcal{E}_{n \to \ell}
              \;=\;
              \mathcal{E}_{m \to \ell}
              \circ
              \mathcal{E}_{n \to m}.
          \]

    \item[\textup{(E4)}]
          \textbf{Identity.}
          $\mathcal{E}_{n \to n} = \mathrm{id}$.
\end{enumerate}