This paper establishes a comprehensive constructive framework that unifies differential topology and algebraic topology within an extended differential algebraic closure, denoted KDAT. We develop explicit constructive representations for fundamental geometric and topological objects, including harmonic forms, characteristic classes, homotopy groups, and homology groups, all accompanied by certified error bounds. The framework is built upon a recursive adjunction process that incorporates both smooth manifold structures and combinatorial topological data, ensuring compatibility between local differential geometric information and global topological invariants.The framework is extended to infinite-dimensional contexts (loop spaces, mapping spaces), singular spaces (orbifolds, manifolds with boundary, and stratified spaces), and applications in quantum field theory (topological field theories, anomaly computations, instanton moduli spaces). We integrate geometric deep learning and topological data analysis with certified computations, and provide formal verification methods using proof assistants. Detailed algorithms with complexity analysi and stability guarantees are presented, along with high-performance implementations using GPU acceleration and distributed computing. Extensive numerical experiments validate the theoretical results, demonstrating the practical effectiveness of the proposed approach on classical examples including spheres, complex projective spaces, tori, and Calabi-Yau manifolds. The work reconciles classical impossibility results by showing that explicit constructive representations exist within appropriately extended algebraic closures, providing a new paradigm for certified computational topology.
shifa liu (Wed,) studied this question.