For a one-parameter exponential family on a compact Riemannian manifold, the spatial Fisher information I and the Shannon entropy S are both determined by the log-partition function. The balance condition \(I = T\, S,\) with T a dimensionless parameter, selects a concentration at which gradient sensitivity equals a prescribed multiple of logarithmic uncertainty; uniqueness follows from strict convexity of the log-partition function, equivalently from positivity of the Fisher metric on the statistical manifold. For the von Mises–Fisher family on unit-radius spheres \(\mathbb {S}^{d},\) existence and uniqueness hold exactly in the positive entropy-volume regime \({{\,\textrm{Vol}\,}}\left( \mathbb {S}^{d} \right) > 1;\) in the standard unit-radius convention this is \(1 \le d \le 17,\) while \(\mathbb {S}^{18}\) is the first sphere for which the positive balance point disappears. On \(\mathbb {S}^{1}\) with \(T = 1,\) \(\kappa ^{*} = 1.9048\) and \(I^{*} = S^{*} = 1.2981.\) The Bakry–Emery log-Sobolev inequality constrains the balance constants from below in terms of Ricci curvature and volume. On the flat torus \(\mathbb {T}^{d},\) the isotropic product von Mises family yields a balance parameter that is exactly dimension-independent, illustrating the role of product geometry in the balance construction.
