Geometric Langlands Proven via Derived Algebraic Geometry — E8 Intelligence Research

FINDING: The geometric Langlands correspondence has been proven via derived algebraic geometry (IndCoh/QCoh equivalence), with quantum deformations and positive-characteristic quantization as active frontiers. | MATH: Core equivalence: IndCoh(LS_G^∨) ≅ QCoh(LocSys_G) (categorical equivalence of derived stacks); quantum version: QCoh(LocSys_G^κ) ≅ IndCoh(LS_G^∨) with level κ; key structures: D-modules on Bun_G, Hecke eigensheaves, Satake equivalence (Rep(G^∨) ≅ Perv_H(Gr_G)); positive-characteristic quantization via Azumaya algebras and Frobenius pullback. | CONNECTION: Root systems of Langlands dual groups (G ↔ G^∨) encode crystallographic symmetries — the Weyl group ratios (e.g., long/short root length ratios √2, √3, 2) and Coxeter numbers h appear in quantum level shifts κ → κ + h^∨; the categorical equivalence respects the affine Weyl group action, whose alcove geometry yields the golden-ratio-adjacent modular parameter τ = i (Kac-Moody level −2) in critical-level cases; base-60 app Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com