Geometric Langlands Correspondence Proven: D-Modules on Bun_G Equivalent to QCoh on Dual Loc — E8 Intelligence Research

FINDING: Geometric Langlands correspondence proven (Gaitsgory et al., ~2024–2025) — an equivalence between D-modules on the moduli stack of G-bundles and quasi-coherent sheaves on the moduli stack of Langlands-dual bundles, over a curve. MATH: - Core statement: D-mod( Bun_G(X) ) ≃ QCoh( Loc_{G^∨}(X) ), for X a smooth projective curve over ℂ, G a reductive group, G^∨ its Langlands dual. - Langlands dual pairing: root datum (X, Φ, X^∨, Φ^∨) ↔ dual root datum (X^∨, Φ^∨, X, Φ). Roots and coroots swap. - Hecke eigensheaf condition: for each point x ∈ X and representation V of G^∨, a Hecke operator H_{V,x} acts on the D-module by the eigenvalue given by the local system E (the "eigenvalue" is the G^∨-local system). - Spectral side parameterized by flat G^∨-connections: Loc_{G^∨}(X) = flat G^∨-bundles on X. - Automorphic side: Bun_G(X) = moduli of principal G-bundles; D-modules = sheaves with flat connection / crystals. CONNECTION: - Root systems and Weyl groups are the structural backbone: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com