# Proof of Geometric Langlands Conjecture Unifies Number Theory and Geometry — E8 Intelligence Research

Article · Andrew Stewart Caldin

- Archive id: doi:10.5281/zenodo.21449565
- Published: 2026-07-20
- Venue: Zenodo (CERN European Organization for Nuclear Research)
- DOI: https://doi.org/10.5281/zenodo.21449565
- Citations: 0
- Source URL: https://doi.org/10.5281/zenodo.21449565
- Provenance: openalex

## Abstract

FINDING: Proof of the geometric Langlands conjecture, a central pillar of the Langlands program, unifying number theory, geometry, and representation theory. \| MATH: The geometric Langlands conjecture establishes an equivalence of derived categories: \\( D(Bun\_G) \\cong QCoh(LocSys\_{^LG}) \\), where \\( Bun\_G \\) is the moduli stack of \\( G \\)-bundles on a Riemann surface and \\( LocSys\_{^LG} \\) is the moduli stack of \\( ^LG \\)-local systems. This is a categorical Fourier–Mukai transform. Key constants: no specific numerical constants, but the dual group \\( ^LG \\) (Langlands dual) encodes root system symmetries (e.g., \\( G = SL\_n \\) dual to \\( PGL\_n \\)). \| CONNECTION: Root systems and Weyl groups of Lie algebras (e.g., \\( E\_8 \\), \\( E\_7 \\), \\( E\_6 \\)) are central. The dual group construction involves the Cartan matrix and Coxeter numbers (e.g., Coxeter number \\( h \\) for \\( E\_8 \\) is 30). The moduli stacks involve affine Grassmannians and Hecke operators, which are deeply tied to affine root Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

_Abstract reproduced under CC-BY (publisher license, via OpenAlex)._

Topics: [Landmark Results in Pure Mathematics](https://space-kitty.com/topics/landmark-math-results)

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