= Solution
Every generalized frequency in the <inviscid Couette continuous spectrum> is real: $\omega_\xi=\alpha\xi$ for real $\alpha,\xi$. Thus every time factor has unit modulus and \b[there is no exponentially growing normal mode]:
$$
\boxed{\operatorname{Im}\omega_\xi=0.}
$$
The vorticity equation also gives $q(z,t)=q_0(z)e^{-i\alpha zt}$, preserving its $L^2$ norm. For fixed nonzero $\alpha$, the bounded <Dirichlet Green function> inversion therefore gives a uniform bound on the reconstructed velocity in terms of the initial vorticity norm. Superpositions can still rearrange their velocity energy and display <transient growth from non-normal modes>; neutral <hydrodynamic stability> does not require every velocity component to decrease monotonically. Treating only smooth discrete modes would miss this continuous neutral family.
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