This paper introduces Jiuzhang Constructive Mathematics (JCM), a novel mathematical framework that systematically incorporates finite approximation and computational realizability as foundational principles. The framework addresses the disconnect between classical mathematics with its reliance on actual infinity and computational practice with its finite resource constraints. JCM is built upon three carefully formulated axioms: the Finite Approximation Axiom ensuring effective Cauchy convergence, the Computable Operations Axiom requiring uniform polynomial-time computability with consistent encoding schemes, and the Categorical Realizability Axiom providing semantic interpretation in a rigorously constructed enriched realizability topos. We construct the JCM universe J as a locally Cartesian closed category supporting intuitionistic higher-order logic, with detailed proofs of all categorical properties. A key technical contribution is the resolution of encoding size consistency between approximation sequences and complexity classes through careful design of finite structure representations. The framework provides faithful embeddings of Bishop’s constructive analysis while maintaining explicit computational content. We establish comprehensive complexity theory with precise relationships between JCM complexity classes and their classical counterparts, and discuss limitations regarding non-polynomial-time computable functions and classical non-constructive principles.