Two boundary traces for reverse-dispersion Airy flow (${\tilde{g}}_2=Q_0(-ir)-r{\tilde{g}}_1-r^2{\tilde{g}}_0,r^3=p$)

For $q_t-q_{xxx}=0$ on $x>0$, evaluate the global relation at $k=-ir/2$, where $\mathrm{Re}p>0$ and the principal cube root $r=p^{1/3}$ has positive real part. The time exponent becomes $-p$, giving the displayed Laplace transform identity. Thus the second-derivative trace is determined once the value and first-derivative traces are prescribed. The homogeneous transformed equation $u^{\prime \prime \prime }-pu=0$ has exactly two spatially decaying roots, so two independent admissible conditions are required. One condition leaves a free decaying mode; three independent conditions generally violate the global constraint.