Abstract: Stability is a fundamental issue in solid mechanics theory. As an emerging non-local theory, peridynamics (PD), particularly ordinary state-based peridynamics (OSBPD), still lacks a comprehensive framework for stability analysis. Rigorous theoretical support is currently missing for the stability of elastic-brittle bodies in complex scenarios involving variable horizons, surface effects, damage and fracture, heterogeneity, and anisotropy. To address this challenge, this paper aims to establish a general stability analysis framework for the dual-horizon OSBPD (DH-OSBPD) theory. Based on the strain energy density function of the linearized DH-OSBPD material model and utilizing the Cauchy-Schwarz inequality, the fundamental material stability theorems are generalized to cover arbitrary integration domains, global stability of elastic bodies, heterogeneity, presence of cracks and damage, and anisotropy. On this basis, the necessary and sufficient conditions for the vanishing second variation of the positive semi-definite strain energy function are derived, thereby unifying the mathematical formulation of all zero-energy modes. The analysis indicates that for an ideal elastic body satisfying the theorem conditions, the zero-energy modes consist solely of rigid body motions. Conversely, for a fractured elastic-brittle body, these modes naturally encompass the independent relative rigid body motions among fragments. Eigenvalue analysis of representative numerical examples validates the theorem's predictions regarding positive semi-definiteness and the number of zero-energy modes. The primary contribution of this work lies in extending the scope of fundamental stability theorems—previously limited to internal material points of ideal elastic bodies—to global stability of complex systems involving surface effects, elastic-brittle fracture, anisotropy, and heterogeneity. This lays a solid theoretical foundation for the robustness of the DH-OSBPD model in numerical simulations across a wide range of applications.