%
\label{thm:split-complex-forced}
The bipolar prime partition forces
split-complex scalars ($j^2 = +1$)
over elliptic scalars ($i^2 = -1$).

[Proof sketch: three arguments]
\textbf{(1) Bipolar encoding.}
Elliptic $\mathbb{C}$ is structurally \emph{unipolar}:
the unit circle $S^1 \subset \mathbb{C}$
has a single connected component.
It cannot encode two independent sectors.
Split-complex $H$ with $j^2 = +1$
is structurally \emph{bipolar}:
the canonical idempotents
$e_+ = (1+j)/2$ and $e_- = (1-j)/2$
decompose $H$ into two orthogonal sectors
$H = e_+ H \oplus e_- H$.
This decomposition mirrors
the B/C channel partition of the primes
(Theorem~\ref{thm:prime-polarity}).

\textbf{(2) Wave vs.\ diffusion.}
Elliptic holomorphy yields the Laplace equation
$\Delta f = 0$ (elliptic PDE):
information diffuses isotropically,
with no preferred direction.
Split-complex holomorphy yields the wave equation
(hyperbolic PDE):
information propagates along characteristics ---
directionally, reflecting the causal/propagative
structure that the omega-germs carry
via their refinement direction.

\textbf{(3) Diagonal-discipline compatibility.}
Split-complex numbers have zero divisors:
$e_+ \cdot e_- = 0$.
In classical mathematics, this makes them
pathological for analysis.
In $\tau$, the diagonal-free discipline
(Part~I) prevents the formation of arbitrary
projections that would collapse
the split-complex structure.
The idempotents $e_\pm$ \emph{exist}
(they are earned from the prime polarity partition),
but the pathological collapse
(projecting onto one sector while annihilating the other)
cannot be performed
because the required diagonal is not earned.
The zero divisors are \emph{managed}, not avoided.