Jiuzhang Constructive Mathematics: A Finitary Framework for Computable Approximation in Mathematical Practice with Physical Applications

This work introduces Jiuzhang Constructive Mathematics (JCM), a novel constructive framework designed to formalize mathematical reasoning under explicit finitary and computable constraints. Unlike classical systems that rely on actual infinities or non-constructive existence principles, JCM mandates that every mathematical object be presented as a fast-converging sequence of finite approximants, and every operation be realized by a resource-bounded computational procedure. The system is grounded in three foundational axioms: the Finitary Approximation Axiom, ensuring Cauchy convergence with exponential error decay; the Computable Operation Axiom, requiring polynomial-time computability of approximations; and the Categorical Realizability Axiom, anchoring semantics in a realizability topos over modest sets enriched with metric approximation data. We construct the JCM-universe J as a locally cartesian closed category supporting intuitionistic logic, and prove that it conservatively extends Bishop’s constructive analysis while admitting controlled approximations of classically nonconstructive structures–including elementary embeddings and large cardinal properties–via finite signature truncations. Applications are developed in constructive number theory, topological data analysis, quantum information, quantum gravity, and black hole physics, demonstrating how JCM provides a rigorous foundation for approximation-driven mathematical modeling in computationally constrained environments and physically realizable systems.