%
\label{prop:projection-formula}
Let $f_k : \mathbb{Z}/P_k\mathbb{Z} \to H_\tau$
be the stage-$k$ component
of a $\tau$-holomorphic function,
and let $p \mid P_k$ be a prime factor.
The spectral coefficient
$\varphi_{p,v}^{(\sigma)}$
in the expansion
of~$f_k$ with respect to the cylinder generator
$E_{k,v}^{(\sigma)}$ is
\[
    \boxed{%
    \varphi_{p,v}^{(\sigma)}
    \;=\;
    \frac{1}{|F_p|}
    \sum_{x \in F_p(v)}
    e_\sigma \cdot f_k(x),}
\]
where
$F_p(v) := \{x \in \mathbb{Z}/P_k\mathbb{Z}
: x \equiv v \pmod{p}\}$
is the fiber over the residue class~$v$,
$|F_p| = P_k / p$ is the fiber cardinality,
and $e_\sigma$ is the channel idempotent
($e_B = e_+$, $e_C = e_-$).

The CRT isomorphism (I.T18, Book~I)
gives
$\mathbb{Z}/P_k\mathbb{Z}
\cong \mathbb{Z}/p\mathbb{Z}
\times \mathbb{Z}/(P_k/p)\mathbb{Z}$.
The fiber $F_p(v)$ is the set of elements
whose projection to the first factor is~$v$.
By CRT independence,
$|F_p(v)| = P_k/p$ for every $v \in \mathbb{Z}/p\mathbb{Z}$.

Projecting $f_k$ to the $\sigma$-channel
via $e_\sigma$ extracts the component
$f_k^{(\sigma)} = e_\sigma \cdot f_k$.
On the fiber $F_p(v)$,
the cylinder generator $E_{k,v}^{(\sigma)}$
is constantly equal to~$e_\sigma$,
and all other generators
$E_{k,w}^{(\sigma)}$ with $w \neq v$
(modulo~$p$) vanish.
Summing $e_\sigma \cdot f_k(x)$
over $x \in F_p(v)$
isolates the contribution of the $(p, v, \sigma)$ component.
Dividing by $|F_p| = P_k/p$ gives the coefficient.