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 \(χ^2\) 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 \(m\ge4\), support contained in the geodesic ball of radius \(S_m\), centered at the Fréchet mean, rules out smeariness; conversely, for every \(\varepsilon>0\) with \(S_m+\varepsilon<π\), we construct examples of rotationally symmetric \(2\)-smeary densities supported in the ball of radius \(S_m+\varepsilon\). 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 \(π/2+\varepsilon\). 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.