arXiv:2611.14346v1 Announce Type: new Abstract: The Kullback-Leibler (KL) divergence and exponential family have laid the foundation for information geometry and have been widely generalized. Using Eguchi's theory, the Fisher information metric can be derived from the KL divergence. Based on this metric, Amari and Nagaoka established an information-geometric proof of the classical Cram\'er-Rao (CR) inequality for finite-support models. Among the various generalizations of the KL divergence, this paper focuses on the $(h,\tau)$-divergence and establishes the associated CR inequality, referred to as the $(h,\tau)$-CR inequality, using information geometry. We also provide a proof of this inequality based on the Cauchy-Schwarz inequality, thereby extending its applicability to models with more general support sets. Under suitable regularity conditions, equality holds in the classical CR inequality if and only if the underlying family is exponential. However, for the $(h,\tau)$-CR inequality, this property does not generally extend to the $(h,\tau)$-exponential family. We therefore establish an attainable CR inequality for this family. This new inequality extends the attainment result of the classical CR inequality by the sufficient statistic of the exponential family to the $(h,\tau)$-exponential family.