Differential Algebraic Methods in Additive Number Theory: A Constructive Framework for Partition Functions and Additive Bases

This paper extends the differential algebraic framework from multiplicative to additive number theory, developing explicit representation theorems for partition functions, additive bases, and related combinatorial objects. We construct the additive number theoretic differential closure Kadd ANT through a carefully staged recursive adjunction process that incorporates generating functions of additive structures, solutions to additive differential equations, and combinatorial correction terms derived from the circle method and modular transformations. Within this closure, we prove that broad classes of additive number theoretic functions admit explicit representations combining particular solutions with spectral expansions. The framework provides certified error bounds through interval arithmetic and establishes rigorous validation protocols. We develop efficient algorithms with precise complexity analysis and demonstrate applications to partition asymptotics, Waring’s problem, and connections with random matrix theory and efficient computation. The work bridges differential algebra, additive number theory, and computational mathematics, providing new constructive perspectives on classical additive problems while maintaining mathematical rigor and practical implementability.