%
\label{thm:bc-asymmetry-time}
%   II.D21, II.D22, II.D37, II.D38, II.T27
The B/C asymmetry from Prime Polarity
\textup{(I.T05, Book~I)}
induces a canonical time-like direction on~$\tau^3$:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Existence of time-like direction.}
          The evolution operator
          $\mathcal{E}_{n \to m}$ \textup{(II.D37)}
          defines a forward-directed semigroup action
          on holomorphic data.
          The causal arrow \textup{(II.D38)}
          selects a unique forward direction
          within the null cone of the
          wave-type causal structure \textup{(II.D22)}.

    \item[\textup{(ii)}]
          \textbf{Structural origin.}
          The time-like direction is not chosen
          but \textbf{forced}:
          it arises from the
          exponent--tetration hierarchy in the ABCD chart,
          which is a consequence of
          the peel-off ordering in the program monoid
          \textup{(I.P02, Book~I)}.

    \item[\textup{(iii)}]
          \textbf{Elliptic impossibility.}
          In classical complex analysis \textup{($i^2 = -1$)},
          no time-like direction exists.
          The Laplace equation is elliptic
          \textup{(no null cone)},
          the codomain $\mathbb{C}$ is a field
          \textup{(no nontrivial idempotents)},
          and the angular structure is $SO(2)$
          \textup{(connected, no preferred direction)}.
          No combination of elliptic structures
          can produce a causal arrow.

    \item[\textup{(iv)}]
          \textbf{Compatibility with the wave equation.}
          The time-like direction aligns with the
          characteristic curves of the split-complex
          wave equation
          \textup{(II.D21, Chapter~\ref{ch:wave-causal})}:
          forward propagation along the tower
          corresponds to forward propagation
          along the null lines
          $\xi = x + y = \mathrm{const}$ and
          $\zeta = x - y = \mathrm{const}$.
\end{enumerate}

\emph{(i).}
The evolution operator
$\mathcal{E}_{n \to m}$
is a semigroup (Definition~\ref{def:evolution-operator},
properties E3--E4).
It is forward-directed:
$\mathcal{E}_{n \to m}$ is defined only for $m \geq n$,
and the inverse does not exist
(Remark~\ref{rem:ch32-not-group}).
The causal arrow (Definition~\ref{def:causal-arrow})
selects the forward direction
within the null cone
by the B/C asymmetry.
Within each time-like sector of the null cone
(Definition~\ref{def:causal-structure},
Chapter~\ref{ch:wave-causal}),
the forward half is the one
in which the B-channel (exponent)
precedes the C-channel (tetration).
This is well-defined because the B/C ordering
is canonical (I.T05, Book~I)
and does not depend on the stage number.

\emph{(ii).}
The B/C asymmetry traces back to
Prime Polarity (I.T05, Book~I):
the hyperfactorization
$n = \alpha^{\pi} \cdot \gamma^{\eta} \cdot \omega^{(\cdot)}$
separates the exponent~$\gamma$ from the tetration~$\eta$,
and the peel-off sequence processes them
in a canonical order
determined by the program monoid (I.P02, Book~I).
The exponent is always available
before the tetration
(one cannot tetrate without first exponentiating),
establishing $B \prec C$
as a structural fact.
No external convention selects this ordering.

\emph{(iii).}
In the classical case $i^2 = -1$:

The Laplace equation
$\partial^2 u / \partial x^2 + \partial^2 u / \partial y^2 = 0$
is elliptic.
Its characteristic polynomial
$\xi^2 + \eta^2 = 0$
has no real roots.
No null cone exists,
so no time-like direction can be defined.

Even if we formally seek an asymmetry
in the classical codomain $\mathbb{C}$:
$\mathbb{C}$ is a field,
so $e_\pm = (1 \pm i)/2$
are not idempotents
($e_+^2 = (1 + 2i - 1)/4 = i/2 \neq e_+$).
There is no bipolar decomposition,
hence no independent channels,
hence no structural asymmetry
to select a direction.

The rotation group $SO(2)$
acts on $\mathbb{C}$ by multiplication by $e^{i\theta}$.
$SO(2)$ is connected:
every direction can be continuously
rotated into every other.
No discrete asymmetry (like $B \prec C$)
survives in a connected symmetry group.

\emph{(iv).}
The characteristic curves of the wave equation
(Proposition~\ref{prop:ch21-null-lines},
Chapter~\ref{ch:wave-causal})
are the null lines
$\xi = x + y = \mathrm{const}$
and $\zeta = x - y = \mathrm{const}$.
In idempotent coordinates,
the $e_+$-channel carries $F(\xi)$
(propagating along $\xi$-null lines)
and the $e_-$-channel carries $G(\zeta)$
(propagating along $\zeta$-null lines).
The B/C asymmetry assigns
$e_+$ to the B-channel (exponent)
and $e_-$ to the C-channel (tetration).
Since $B \prec C$,
the $\xi$-propagation (B-channel)
is the ``earlier'' component,
and the $\zeta$-propagation (C-channel)
is the ``later'' component.
Forward propagation along the tower
$n \to n+1 \to n+2 \to \cdots$
corresponds to forward propagation
along both characteristic families,
with the B-characteristic ``leading''
and the C-characteristic ``following.''