%
\label{thm:asymmetric-determination}
In $\tau$-holomorphy ($\jj^2 = +1$),
boundary data on the lemniscate $\Lemniscate$
determines interior values on $\tau^3$
with the following properties:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Directional propagation.}
          Boundary data propagates into the interior
          along the two characteristic families
          of the wave equation,
          not isotropically.
    \item[\textup{(ii)}]
          \textbf{No reverse constraint.}
          Interior values do not constrain
          boundary values:
          the dependency is one-directional
          (boundary $\to$ interior).
    \item[\textup{(iii)}]
          \textbf{Global Hartogs as wave front.}
          The Hartogs extension theorem (I.T31)
          is the statement that wave-front data
          on the boundary determines
          the full interior holomorphic function.
    \item[\textup{(iv)}]
          \textbf{Uniqueness from independence.}
          The two characteristic families
          carry independent data;
          their consistency forces uniqueness
          of the interior extension.
\end{enumerate}

[Proof sketch]
Properties~(i)--(ii) follow from the hyperbolic
character of $\Box f = 0$:
data propagates along characteristics,
and hyperbolic equations have
domain-of-dependence properties
that prevent reverse information flow.
Property~(iii) identifies Hartogs extension
as the constructive analog of the elliptic
maximum principle:
where the Laplacian ``averages'' boundary data,
the wave operator ``propagates'' it.
Property~(iv) follows from the idempotent
decomposition $f = e_+ g + e_- h$:
the two components are independent,
and the boundary data for each sector
determines its interior value uniquely
(II.T27, mutual determination).