Cubic endpoint stationary-phase scaling

If a phase at an endpoint satisfies $\varphi (u)=\varphi (0)-c{u}^3+O(u^5)$ with $c>0$, the leading region has width $O(t^{-1/3})$. Rescaling $u=(ct{)}^{-1/3}v$ reduces the leading term to a cubic oscillatory integral and produces decay $t^{-1/3}$.