%
\label{def:canonical-decomposition}
Let $f \in \mathcal{O}_\tau(\tau^3)$
be a $\tau$-holomorphic function
on~$\tau^3$
(in the sense of the Mutual Determination Theorem,
Theorem~\ref{thm:mutual-determination}, II.T27).
The \textbf{canonical decomposition} of~$f$
is the pair
\[
    \boxed{%
    f \;=\; f_+ \;+\; f_-,
    \qquad
    f_+ := e_+ \cdot f,
    \quad
    f_- := e_- \cdot f,}
\]
where the multiplication is pointwise
in the calibrated split-complex codomain
$H_\tau^{\mathrm{cal}}$
(Definition~\ref{def:calibrated-H-tau}, II.D35,
Chapter~\ref{ch:split-complex-calibrated}).

We call $f_+$ the \textbf{B-channel component}
and $f_-$ the \textbf{C-channel component} of~$f$.
The decomposition $f \mapsto (f_+, f_-)$
is the \textbf{canonical decomposition map}:
\[
    \Delta_\tau
    \colon
    \mathcal{O}_\tau(\tau^3)
    \;\longrightarrow\;
    \mathcal{O}_\tau^{(+)}(\tau^3)
    \times
    \mathcal{O}_\tau^{(-)}(\tau^3),
    \qquad
    \Delta_\tau(f) := (f_+, f_-),
\]
where $\mathcal{O}_\tau^{(\pm)}(\tau^3)$
denotes the space of $\tau$-holomorphic functions
taking values in the channel $H_\tau^{(\pm)} = e_\pm \cdot H_\tau$.