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\label{def:split-complex-codomain}
The \textbf{split-complex codomain} is the split-complex
extension of the boundary ring:
\[
    H_\tau
    \;:=\;
    \bigl\{\, a + b\,j \;:\; a, b \in A_\tau \,\bigr\},
    \qquad j^2 = +1,
\]
where $A_\tau = \hat{\mathbb{Z}}_\tau$ is the boundary ring
(I.D28, Book~I).
Equivalently, via the canonical idempotent decomposition,
\[
    H_\tau
    \;\cong\;
    A_\tau^{(+)} \times A_\tau^{(-)}
\]
where $A_\tau^{(+)} = e_+ \cdot H_\tau$
and $A_\tau^{(-)} = e_- \cdot H_\tau$
are the two sector components,
each isomorphic to $A_\tau$ as a ring.

$H_\tau$ is the scalar codomain for
\textbf{all} holomorphic functions in Book~II.
Every $\tau$-holomorphic function
$f \colon \tau^3 \to H_\tau$
takes values in this split-complex algebra,
and the idempotent decomposition
$f = e_+ f_+ + e_- f_-$
provides the canonical sector components
that the Central Theorem relates to the
spectral algebra of $\mathbb{L}$.