# A Proof of the Langlands Correspondence via Information Geometry

Article · Wengang Yu

- Archive id: doi:10.5281/zenodo.20459676
- Published: 2026-05-30
- Venue: Zenodo (CERN European Organization for Nuclear Research)
- DOI: https://doi.org/10.5281/zenodo.20459676
- Citations: 0
- Source URL: https://doi.org/10.5281/zenodo.20459676
- Provenance: openalex

## Abstract

We establish a categorical equivalence between the information space category $\\Info$---whose objects are locally ringed spaces equipped with an entanglement distance structure derived from quantum mutual information---and the perfectoid space category $\\Perfd$ in the sense of Scholze. The equivalence is realized by a contravariant spectral functor $\\D: \\Info \\to \\Perfd$ that assigns to each information space its spectral space. We prove that $\\D$ is fully faithful and essentially surjective (Theorem~A). Within this framework, we prove the geometric Langlands duality for all reductive groups over $p$-adic fields as the fibration of algebraic versus geometric coordinates on the $G$-information space $X\_p^G$ (Theorem~B). Extending this duality to the global setting via the restricted product of local valued information spaces, we establish the global Langlands correspondence for all reductive groups over all number fields---the central conjecture of the Langlands program---as an equivalence between the category of global Galois representations and the category of global automorphic forms (Theorem~C). The proof of essential surjectivity reduces to a purely categorical lemma: the restricted product of local equivalences is an equivalence. We further prove the Langlands functoriality conjecture via quantum state extension, characterize arithmetic $L$-functions as regularized characteristic functions of the entanglement spectrum of associated quantum states, and establish the analytic continuation and functional equation of global $L$-functions within the information space framework. This provides a unified information-theoretic foundation for the entire Langlands program: the dualities between Galois representations and automorphic forms, between $L$-functions and entanglement spectra, and between functorial lifting and quantum state extension all emerge as manifestations of a single underlying principle---the algebraic-geometric binarity of quantum information structures.

_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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