%
\label{def:calibrated-H-tau}
The \textbf{calibrated split-complex codomain}
is the ring
\[
    \boxed{%
    H_\tau^{\mathrm{cal}}
    \;:=\;
    \bigl(\mathbb{R}[\jj],\;
    \|\cdot\|_\tau,\;
    (\pi, e, \iota_\tau)\bigr),}
\]
where:
\begin{enumerate}
    \item[\textup{(i)}]
          The underlying ring is the split-complex ring
          $\mathbb{R}[\jj] = \{a + b\jj : a, b \in \mathbb{R}\}$
          with $\jj^2 = +1$
          (I.D20, I.T10, Book~I).

    \item[\textup{(ii)}]
          The \textbf{calibrated norm} is
          \[
              \|z\|_\tau
              \;:=\;
              |z_+|^{\pi/(\pi+e)}
              \cdot
              |z_-|^{e/(\pi+e)},
          \]
          where $z_+ = a + b$ and $z_- = a - b$
          are the sector values.
          The exponents $\pi/(\pi+e)$
          and $e/(\pi+e)$
          are the \textbf{angular weight}
          and \textbf{growth weight},
          respectively.
          They sum to~$1$:
          the norm is a geometric mean
          weighted by the two calibration constants.

    \item[\textup{(iii)}]
          The \textbf{calibration triple}
          $(\pi, e, \iota_\tau)$
          records the angular period,
          the growth rate,
          and their harmonic coupling
          $\iota_\tau = 2/(\pi + e)$.
\end{enumerate}
The sector decomposition becomes
\[
    z
    \;=\;
    z_+\, e_+
    \;+\;
    z_-\, e_-
    \;\in\;
    H_\tau^{\mathrm{cal}},
\]
where $z_+$ is the \textbf{angular sector value}
(calibrated by~$\pi$)
and $z_-$ is the \textbf{growth sector value}
(calibrated by~$e$).