Proof of Quantum Geometric Langlands via 2-Fourier–Mukai Transform — E8 Intelligence Research

FINDING: The "fundamental local equivalence" in quantum geometric Langlands is a conjectural categorical substitute for the Satake equivalence, now proven in most cases (Gaitsgory–Lurie conjecture, extended to affine flag varieties by Dhillon et al.), via a 2-Fourier–Mukai transform for non-constant group schemes. MATH: The core object is a categorical equivalence between factorization modules (or D-modules on the affine Grassmannian/flag variety) and representations of the Langlands dual group, twisted by a quantum parameter. Key structures: - **Satake equivalence** (classical): \( Rep({G}) PervG(O)(Gr_G) \) — replaced by a *fundamental local equivalence*: \( QCoh(LocSys_{{G}})ᶠᵃᶜᵗ IndCoh(Gr_G)ᶠᵃᶜᵗ \) (or similar), with a **2-Fourier–Mukai kernel** implementing the transform. - The quantum parameter \( ψ \) (level) enters as a twist; the equivalence is expected to be **se Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com