 Stationary-direction bound for singular support

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.