= Stationary-direction bound for singular support
{title2=$\operatorname{sing\,supp}I_\Phi(a)\subset\pi_X(C_a\cap\{\nabla_\theta\Phi=0\})$}
For a homogeneous <phase function> $\Phi$ and finite-order <symbol class> amplitude $a$, the <singular support> of $I_\Phi(a)$ lies in the spatial projection of the closed conic amplitude support intersected with $\nabla_\theta\Phi=0$. Away from that set, the frequency gradient is uniformly nonzero on compact spatial sets and supported directions. Repeated frequency <integration by parts> lowers the amplitude order until every desired spatial derivative is absolutely integrable. If the ordinary amplitude support is already conic, it gives the same bound.
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