Slice-support obstruction for oscillatory singularities ($a(0,\theta )=0\nRightarrow I_{\Phi }(a)\text{ is smooth near }0$)

In one base and one phase variable, take $\Phi =x\theta$ and $a=x\eta (\theta )/|\theta |$, with an even smooth cutoff function equal to zero near zero and one for large $|\theta |$. This is a symbol of order $-1$. Its oscillatory integral is $-2x\log |x|$ plus a smooth function near zero. Thus it is singular there even though the restricted function $a(0,\cdot )$ has empty support. The joint closed conic support of an oscillatory amplitude retains the limiting base point and its stationary directions. A support-sensitive regularity theorem must use that conic neighborhood information, not merely vanishing of an amplitude at one base point.