= Ordinary amplitude support can miss a singular-support limit
For $\Phi(s,t,\theta)=s\theta$, take a smooth symbol that turns on increasingly high frequencies in disjoint spatial bumps centered at $t_j\to0$. Superpolynomially small bump coefficients keep all <symbol class> estimates valid. The amplitude vanishes near every finite-frequency point over $t=0$, yet its <oscillatory integral> has delta singularities at $(0,t_j)$. Closedness of <singular support> forces a singularity at $(0,0)$ as well. A bound using only ordinary, rather than closed conic, amplitude support can consequently fail.
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