%
\label{thm:residue-theorem}
Let $f$ be $\tau$-meromorphic
with singular locus $S = \{x_1, \ldots, x_N\}$.
Then the spectral trace equals
the sum of residues:
\[
    \boxed{%
    \mathrm{Tr}_{\mathrm{spec}}(f)
    \;=\;
    \sum_{i=1}^N \mathrm{Res}_{x_i}(f).}
\]
Both sides lie in~$H_\tau$;
in components:
\[
    e_+ \cdot \mathrm{Tr}_{\mathrm{spec}}(f_+)
    \;+\;
    e_- \cdot \mathrm{Tr}_{\mathrm{spec}}(f_-)
    \;=\;
    \sum_{i=1}^N
    \bigl(
    a_{-1}^{(+)}(x_i)\, e_+
    \;+\;
    a_{-1}^{(-)}(x_i)\, e_-
    \bigr).
\]

The proof splits along the bipolar decomposition.

\emph{Step~1: Channel decomposition.}
By idempotent orthogonality
($e_+ \cdot e_- = 0$),
it suffices to prove the theorem
separately for the B-channel ($e_+$)
and C-channel ($e_-$).
We give the B-channel argument;
the C-channel is symmetric.

\emph{Step~2: Finite-stage identity.}
At stage~$k$, the B-channel component
$f_+^{(k)} : \mathbb{Z}/P_k\mathbb{Z} \to \mathbb{R}$
decomposes via the discrete Fourier transform:
\[
    f_+^{(k)}(a)
    \;=\;
    \sum_{n=0}^{P_k - 1}
    \hat{f}_+^{(k)}(n)\,
    \chi_n^{(\gamma)}(a).
\]
The stage average is
\[
    \frac{1}{P_k}
    \sum_{a=0}^{P_k - 1}
    f_+^{(k)}(a)
    \;=\;
    \hat{f}_+^{(k)}(0),
\]
the zeroth Fourier coefficient.
This is an identity of finite sums.

\emph{Step~3: Subtraction of holomorphic part.}
Write $f = f^{\mathrm{hol}} + \mathrm{pp}(f)$,
where $f^{\mathrm{hol}}$ is the holomorphic (regular) part
and $\mathrm{pp}(f)$ is the principal part.
The holomorphic part has $\hat{f}^{\mathrm{hol}}(0) = 0$
by the tower normalization
(the constant term of a holomorphic $\omega$-germ
is determined by the identity germ).
Hence:
\[
    \hat{f}_+^{(k)}(0)
    \;=\;
    \widehat{\mathrm{pp}(f)}_+^{(k)}(0).
\]

\emph{Step~4: Principal part contributes residues.}
The principal part at each singular point $x_i$
has the form
$\mathrm{pp}_{x_i}(f_+)
= \sum_{n < 0} a_n^{(+)}(x_i)\, \phi_{n,k}^{(+)}$.
Its zeroth DFT coefficient
extracts the ``total weight'' of the principal part.
By the orthogonality of characters on
$\mathbb{Z}/P_k\mathbb{Z}$
and the CRT factorization (I.T18):
\[
    \widehat{\mathrm{pp}(f)}_+^{(k)}(0)
    \;\xrightarrow{k \to \infty}\;
    \sum_{i=1}^N a_{-1}^{(+)}(x_i).
\]
Higher-order principal part terms ($n < -1$)
contribute zero to the zeroth coefficient
in the limit,
because they oscillate
and average to zero
over the growing cyclic group.

\emph{Step~5: Assembly.}
Combining Steps~2--4 for both channels:
$\mathrm{Tr}_{\mathrm{spec}}(f)
= \sum_{i=1}^N \mathrm{Res}_{x_i}(f)$.