A Complete Constructive Proof of the Geometric Langlands Correspondence via Differential Algebraic Closures

This paper presents a complete constructive proof of the geometric Langlands correspondence using the frame work of differential algebraic closures developed in our previous work. We establish an explicit equivalence between the category of holonomic D-modules on the moduli stack of G-bundles and the category of quasi-coherent sheaves with nilpotent singular support on the moduli stack of Langlands dual local systems. Our approach synthesizes the theory of D-module differential closures and coherent sheaf differential closures to provide explicit constructions of all objects and functors involved in the correspondence. The proof proceeds through detailed constructions of Hecke eigensheaves, explicit computation of monodromy data, and certified verification of the equivalence relations using interval arithmetic. Key innovations include: constructive implementation of the geometric Satake equivalence with explicit error bounds, rigorous parameterization of automorphic D-modules within differential closures, and computational verification of the Hecke eigenvalue conditions with certified complexity bounds. All constructions are accompanied by rigorous error analysis and termination proofs.