We study covariance functions in the Gauss hypergeometric ($\mathcal{GH}$) class, a flexible family that encompasses the Generalized Wendland ($\mathcal{GW}$) and Matérn ($\mathcal{MT}$) models. We derive sharp validity conditions, providing a complete characterization of the admissible parameter space, and show that the model exhibits structural identifiability issues under both increasing- and fixed-domain asymptotics. To resolve this issue, we introduce a parsimonious compactly supported subclass selected via a maximum integral range criterion. The resulting hypergeometric model can be viewed as a structural refinement of the $\mathcal{GW}$ family and admits compact-support reparameterizations that recover the $\mathcal{MT}$ model as a limit case. We further establish strong consistency and asymptotic normality of the maximum likelihood estimator of the associated microergodic parameter under fixed-domain asymptotics. Simulation experiments and a real-data application to climate data illustrate the finite-sample behavior and practical performance of the proposed model.

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