{"id":"doi:10.5281/zenodo.23131590","kind":"article","title":"Proof of Geometric Langlands: Categorical Equivalence Established — E8 Intelligence Research","authors":["Andrew Stewart Caldin"],"author_count":1,"year":2026,"published":"2026-10-04","venue":"Zenodo (CERN European Organization for Nuclear Research)","cited_by":0,"doi":"10.5281/zenodo.23131590","arxiv_id":null,"nct_id":null,"source_url":"https://doi.org/10.5281/zenodo.23131590","archive_url":"https://space-kitty.com/papers/doi:10.5281/zenodo.23131590","updated":null,"pmid":null,"pmcid":null,"openalex_id":"W7219788566","pdf_url":null,"abstract":"FINDING: Recent proof of a central component of the Geometric Langlands Conjecture (by Gaitsgory et al., 2024) establishes a categorical equivalence between D-modules on the moduli stack of G-bundles and ind-coherent sheaves on the Langlands dual stack. | MATH: The core statement is an equivalence of derived categories: \\( \\mathrm{D-mod}(\\mathrm{Bun}_G) \\simeq \\mathrm{IndCoh}(\\mathrm{LocSys}_{{}^L G}) \\). This is not a numerical equation but a functorial isomorphism of ∞-categories. Key structures: the Langlands dual group \\( {}^L G \\), the moduli of local systems \\( \\mathrm{LocSys} \\), and the categorical trace / Hecke eigensheaf condition. No new constants or ratios appear; the depth is structural, not scalar. | CONNECTION: The Langlands dual group \\( {}^L G \\) is built from the **root system** of \\( G \\) — specifically, the dual root system (exchange long/short roots). This directly ties to **crystallographic root systems** (A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2). The Weyl gro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com","abstract_license":"CC-BY (publisher license, via OpenAlex)","license":"cc-by","is_open_access":true,"retracted":false,"peer_reviewed":null,"categories":["Advanced Algebra and Geometry"],"details":null,"provenance":"openalex","attribution":"Abstract of \"Proof of Geometric Langlands: Categorical Equivalence Established — E8 Intelligence Research\" by Andrew Stewart Caldin, https://doi.org/10.5281/zenodo.23131590, used under its Creative Commons Attribution license (CC-BY); markup removed, otherwise unmodified.","topics":[{"slug":"landmark-math-results","name":"Landmark Results in Pure Mathematics","domain_slug":"math-cs"}],"cited_in_dossiers":[]}