Illuminating the Breakthrough: A Dual Verification of the Geometric Langlands Conjecture Proof

 The geometric Langlands conjecture (GLC) establishes a deep correspondence between
 automorphic D-modules on the moduli stack of G-bundles over a smooth projective curve
 X and quasi-coherent sheaves on the stack of G-local systems. In 2024, a complete proof
 of the GLC for reductive groups was presented in a series of papers by D. Arinkin, D.
 Beraldo, L. Chen, J. Faegerman, D. Gaitsgory, K. Lin, S. Raskin, and N. Rozenblyum.
 This paper presents an independent, dual verification of their argument. We achieve this
 through a three-fold process: (1) we recapitulate the essential steps of the original proof,
 focusing on the construction of the Langlands functor; (2) we furnish a detailed analysis and
 alternative derivations of critical lemmas, including the spectral action, the construction of
 the vacuum Poincar´e sheaf, and the role of the Whittaker model; and (3) we assemble these
 components into a coherent reconstruction of the equivalence of categories. Our verification
 confirms each foundational component via distinct methods, reinforcing the proof’s validity
 and highlighting implications for representation theory, conformal field theory, and p-adic
 extensions. We conclude by advocating for continued collective effort to explore the broader
 landscape unlocked by this milestone