%
\label{def:cylinder-generator}
Let $k \geq 1$ be a stage,
$p$ a prime factor of~$P_k$,
$v \in \mathbb{Z}/p\mathbb{Z}$ a residue class
(the \textbf{peel token}),
and $\sigma \in \{B, C\}$ a channel.
The \textbf{cylinder generator} is the function
\[
    \boxed{%
    E_{k,v}^{(\sigma)}
    \;:\;
    \mathbb{Z}/P_k\mathbb{Z}
    \;\longrightarrow\;
    A_\tau^{(\sigma)},
    \qquad
    E_{k,v}^{(\sigma)}(x)
    \;:=\;
    \begin{cases}
        e_\sigma & \text{if } x \equiv v \pmod{p}, \\[4pt]
        0        & \text{otherwise},
    \end{cases}}
\]
where $e_B := e_+$ and $e_C := e_-$
are the bipolar idempotents (I.D21, Book~I),
and $x \equiv v \pmod{p}$ is computed
via the CRT projection
$\mathbb{Z}/P_k\mathbb{Z} \twoheadrightarrow \mathbb{Z}/p\mathbb{Z}$.

The \textbf{indexing set} at stage~$k$ is
\[
    \Lambda_k
    \;:=\;
    \bigl\{\,(p, v, \sigma)
    \;\big|\;
    p \mid P_k,\;
    v \in \mathbb{Z}/p\mathbb{Z},\;
    \sigma \in \{B, C\}
    \,\bigr\}.
\]