%
\label{thm:liouville-dodge}
The classical Liouville theorem does not apply to~$\tau^3$.
Specifically:
\begin{enumerate}
    \item[\textup{(i)}]
          The split-complex structure $\jj^2 = +1$
          \textup{(I.T10, Book~I)}
          induces a \textbf{wave-type} PDE operator
          $\Box = \partial^2/\partial x^2
          - \partial^2/\partial y^2$
          on the holomorphic functions of~$\tau^3$,
          not the elliptic Laplacian.
    \item[\textup{(ii)}]
          The maximum principle fails
          for wave-type operators
          \textup{(Proposition~\ref{prop:ch52-maximum-principle-failure})}.
    \item[\textup{(iii)}]
          The bipolar idempotent decomposition
          $f = e_+ f_+ + e_- f_-$
          \textup{(II.L07)}
          decomposes every holomorphic function
          into two independent channel components,
          each admitting non-constant bounded solutions.
    \item[\textup{(iv)}]
          Therefore, the compactness of~$\tau^3$
          \textup{(II.T06)}
          is compatible with a rich function algebra
          $\mathcal{O}(\tau^3) \cong
          A_{\mathrm{spec}}(\Lemniscate)$
          \textup{(II.T40)}.
\end{enumerate}

The proof assembles the ingredients
already established in this chapter.

\smallskip
\noindent\textbf{PDE type.}
The split-complex unit~$\jj$ satisfies
$\jj^2 = +1$ (I.T10).
The associated Cauchy--Riemann-type operator
$\bar{\partial}_\jj
= \tfrac{1}{2}(\partial_x + \jj\,\partial_y)$
has principal symbol $\sigma(\xi, \eta)
= \xi^2 - \eta^2$
(indefinite quadratic form, signature $(1,1)$).
This is a \textbf{hyperbolic} PDE,
not an elliptic one.
Liouville's theorem requires elliptic PDE type
(hypothesis~(E) in Remark~\ref{rem:ch52-classical-liouville});
this hypothesis is not satisfied.

\smallskip
\noindent\textbf{Maximum principle failure.}
For hyperbolic operators,
the maximum principle does not hold
(Proposition~\ref{prop:ch52-maximum-principle-failure}).
Without the maximum principle,
Liouville's argument cannot force
holomorphic functions to be constant.

\smallskip
\noindent\textbf{Channel decomposition.}
The idempotent decomposition lemma (II.L07)
gives $f = e_+ f_+ + e_- f_-$,
where $e_\pm = (1 \pm \jj)/2$.
The B-channel component~$f_+$
and the C-channel component~$f_-$
are independent
(Proposition~\ref{prop:bipolar-channel-independence}, II.P07):
each lives in its own sector
and is unconstrained by the other.
Non-constant bounded solutions in each channel
(the standing-wave phenomenon of
Remark~\ref{rem:ch52-standing-waves})
combine to produce a rich function algebra.

\smallskip
\noindent\textbf{Compatibility.}
The three ingredients---hyperbolic PDE type,
maximum principle failure,
and independent channel decomposition---jointly
ensure that the compactness of~$\tau^3$
poses no obstruction
to the existence of non-constant holomorphic functions.
The Central Theorem (II.T40) is consistent.