A Differential-Algebraic Closure Framework for Combinatorial Representation Theory: A Unified Constructive Approach

This paper establishes a comprehensive differential-algebraic framework for constructing explicit parameterizations in combinatorial representation theory. We define the combinatorial representation closure KCR and the quantum affine closure Kq,aff, which are differentially closed field extensions constructed through recursive adjunction processes, encompassing characters, crystal bases,quantum groups, and geometric structures.Within these closures, we prove that solutions to representation-theoretic problems—including character formulas, crystal parameterizations, and quantum group realizations—admit uniform representations respecting the underlying algebraic and combinatorial structures. The framework rigorously handles infinite-dimensional algebras, connections to geometric Langlands, modular representation theory in characteristic p, and provides foundations for efficient symbolic-numerical hybrid computation.We provide detailed constructive proofs using ordinal recursion and model-theoretic methods, derive explicit formulas with careful convergence analysis in appropriate topologies, and establish well definedness criteria. The constructions maintain mathematical rigor while synthesizing algebraic, geometric, and computational perspectives.This work demonstrates that explicit formal solutions exist within suitably constructed differential algebraic closures, offering a novel algebraic perspective on representability while maintaining consistency with classical theory. Subsequent papers in this series will develop the computational framework, algorithms, and verification methods.