A new criterion for positivity of Sturm–Liouville generalized translations

We study positivity of generalized translations for one-dimensional Sturm–Liouville operators on the half-line. We introduce a Riccati-positive condition on the logarithmic derivative of the weight and prove that it implies the Chébli sign condition, hence positivity of the associated hyperbolic translation problem. Under the standard Sturm–Liouville support and regularity assumptions, this yields a commutative hypergroup convolution. The condition also admits a curvature interpretation. In finite dimension, the Riccati profile coincides with the negative of the Bakry–Emery Ricci curvature. Consequently, the new positivity criterion is naturally associated with an upper-curvature regime, whereas the classical Chébli–Trimèche monotonicity condition corresponds to the case N=∞ with nonnegative Bakry–Emery curvature. We prove sharpness, showing that the Bessel–Kingman and hyperbolic Bessel–Kingman weights are the extremal equality cases. We also establish stability and comparison results, and develop a gauge-reduction method that produces positive gauge-reduced translations for weights which are not Riccati-positive in their original form.