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\label{def:structural-sign-classification}
The \textbf{structural sign classification} is the assignment,
at each of twelve levels of mathematical structure,
of the qualitative outcome forced by the scalar unit equation
$u^2 = \pm 1$.
Explicitly, let $\mathcal{S} = \{1, 2, \ldots, 12\}$
be the set of structural levels
as enumerated in the table of
Section~\ref{sec:ch61-twelve-levels}.
For each level $\ell \in \mathcal{S}$,
define:
\begin{enumerate}
    \item[\textup{(a)}]
          $\mathcal{E}_\ell$ = the structural outcome
          when $u^2 = -1$ (elliptic/orthodox).
    \item[\textup{(b)}]
          $\mathcal{H}_\ell$ = the structural outcome
          when $u^2 = +1$ (hyperbolic/$\tau$).
\end{enumerate}
The pair $(\mathcal{E}_\ell, \mathcal{H}_\ell)$
at each level is called the
\textbf{level-$\ell$ sign trade-off}.
The twelve trade-offs collectively
constitute the structural sign classification.

The classification satisfies:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Completeness.}
          Every structural difference
          between orthodox complex analysis
          and $\tau$-holomorphy
          documented in Books~I--II
          appears as an instance of some
          level-$\ell$ trade-off.
    \item[\textup{(ii)}]
          \textbf{Traceability.}
          Each trade-off $(\mathcal{E}_\ell, \mathcal{H}_\ell)$
          traces, through at most three intermediate steps,
          to the sign equation $u^2 = \pm 1$.
    \item[\textup{(iii)}]
          \textbf{Monotonicity.}
          The levels are ordered so that
          level $\ell + 1$ depends on
          the outcome at level $\ell$
          (or earlier levels):
          scalar algebra determines the PDE type,
          the PDE type determines propagation character,
          and so on.
\end{enumerate}