Sharp Support Thresholds for Smeariness of Absolutely Continuous Measures on Spheres

We investigate support thresholds for fully smeary and directionally smeary absolutely continuous probability measures on the sphere ${\mathbb{S}}^m$. The motivation is inferential: smeariness is caused by degeneracy of the Hessian of the Fréchet function, and such degeneracy can invalidate the classical central limit theorem (CLT) for Fréchet means and the corresponding Wald-type ${\chi }^2$ inference. For rotationally symmetric densities, we show that full and directional smeariness are equivalent. The Hessian and fourth-order terms are governed by two explicit geometry-dependent radii $R_m<S_m$. In dimensions $m=2,3$, rotationally symmetric smeariness cannot occur. For $m\ge 4$, support contained in the geodesic ball of radius $S_m$, centered at the Fréchet mean, rules out smeariness; conversely, for every $\epsilon >0$ with $S_m+\epsilon <\pi$, we construct examples of rotationally symmetric $2$-smeary densities supported in the ball of radius $S_m+\epsilon$. For general densities, closed hemispherical support rules out both full and directional smeariness. Support contained in the closed ball of radius $S_m$ rules out full smeariness, while we construct explicit, directionally $2$-smeary examples supported in balls of radius $\pi \mathrm{/}2+\epsilon$. As a byproduct, the explicit Hessian formulas in this paper also provide a practical diagnostic for detecting proximity to the Hessian-degenerate, non-classical regime.