Smooth integrable obstructions to hyperboloid support representations

We construct smooth integrable counterexamples to Dyson's conjectured extension of the Jost–Lehmann–Dyson support restriction to arbitrary open sets. For each prescribed finite order, all momentum derivatives have weighted integrable decay, and the Fourier transform is an integrable function supported in a narrower causal cone. The open zero region intersects every complete standard hyperboloid, each complete upper or lower mass-shell branch, and every affine three-dimensional hyperplane, leaving no admissible elementary terms. A connected convex zero region with a real-analytic spacelike boundary also excludes every complete hyperboloid. The construction uses mass differentiation and compactly supported smoothing of a known causal shell distribution. We also prove that an integrable function whose Fourier transform has causal support must vanish if its energy sections have bounded essential support for almost every spatial momentum and its absolute energy moments of every order are finite. Thus a single Schwartz counterexample cannot be confined to the shell tube used here. Acknowledgments. We acknowledge Gewu Intelligence Lab for providing the collaborative research environment in which this project was developed. W.J. acknowledges support from the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB1680102.