---
title: "Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 014; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "number-theory"
  - "unformalized"
seed_rank: 1874
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Global Arthur Enhancements of Cuspidal Excursion Parameters"
    url: "https://github.com/openai/math/blob/main/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Rationality of the Canonical Unramified Arthur Filtration"
    url: "https://github.com/openai/math/blob/main/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level"
    url: "https://github.com/openai/math/blob/main/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Temperedness at ramified places for globally generic exceptional groups"
    url: "https://github.com/openai/math/blob/main/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "The Restricted Geometric Langlands Equivalence in Positive Characteristic"
    url: "https://github.com/openai/math/blob/main/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Constructible tame Hecke eigensheaves in positive characteristic"
    url: "https://github.com/openai/math/blob/main/preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026/constructible-tame-hecke-eigensheaves-positive-characteristic.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Tame Hecke Eigensheaves with Several Marked Points"
    url: "https://github.com/openai/math/blob/main/preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026/Tame-Hecke-Eigensheaves-with-Several-Marked-Points.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Frobenius Structures on Tame Hecke Eigensheaves"
    url: "https://github.com/openai/math/blob/main/preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

## One-sentence takeaway

OpenAI's result family 014 (Number theory) claims: Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over $\overline{\mathbb F}_q$ under the four stated Lie-theoretic characteristic hypotheses.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and $p\nmid |W_G|$. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.
- *Global Arthur Enhancements of Cuspidal Excursion Parameters*: Under the finite-level Ramanujan–Arthur decomposition stated in Theorem 1.1, we construct a global Arthur enhancement for every occurring cuspidal excursion parameter at a specified full finite level for a split connected semisimple group over a global function field. A single algebraic SL2 and a commuting Weil centralizer map recover the given parameter by diagonal specialization on the entire Weil group, including inertia. The result does not assert ellipticity, Arthur-packet classification, or a multiplicity formula.
- *Rationality of the Canonical Unramified Arthur Filtration*: Under the characteristic hypotheses of restricted geometric Langlands theory, we prove that the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over ℚ for split connected semisimple groups. The result includes the noncuspidal part and every closed invariant nilpotent support, establishing the rationality conjecture of Gaitsgory–Lafforgue–Raskin in this setting.
- *Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level*: We prove rational and $\overline{\mathbb Q}_\ell$ decompositions of cuspidal automorphic functions for split semisimple groups over global function fields into subspaces indexed by nilpotent orbits of the dual group. The decompositions hold at every full finite level, with arbitrary divisor multiplicities. Each summand is governed by the same orbit at every unramified place and every complex embedding.
- *Temperedness at ramified places for globally generic exceptional groups*: We prove the generalized Ramanujan conjecture for globally generic cuspidal automorphic representations of split connected adjoint exceptional groups over global function fields. Every local component is tempered, without restrictions on characteristic or ramification depth. The proof extends the unramified Ramanujan theorem of a companion paper to all ramified places.
- *The Restricted Geometric Langlands Equivalence in Positive Characteristic*: We prove the $\overline{\mathbb Q}_\ell$-linear restricted geometric Langlands equivalence for smooth projective connected curves and connected reductive groups in characteristic p > 0, with ℓ ≠ p, in two regimes. Over $\overline{\mathbb F}_q$, we assume the four characteristic conditions of the restricted theory stated below. Over arbitrary algebraically closed fields, we assume these conditions and additionally that p is very good and does not divide the Weyl-group order.
- Plus 3 more companion manuscripts in this family; see Links.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Global Arthur Enhancements of Cuspidal Excursion Parameters](https://github.com/openai/math/blob/main/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf)
- Manuscript: [Rationality of the Canonical Unramified Arthur Filtration](https://github.com/openai/math/blob/main/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf)
- Manuscript: [Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level](https://github.com/openai/math/blob/main/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf)
- Manuscript: [Temperedness at ramified places for globally generic exceptional groups](https://github.com/openai/math/blob/main/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf)
- Manuscript: [The Restricted Geometric Langlands Equivalence in Positive Characteristic](https://github.com/openai/math/blob/main/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf)
- Manuscript: [Constructible tame Hecke eigensheaves in positive characteristic](https://github.com/openai/math/blob/main/preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026/constructible-tame-hecke-eigensheaves-positive-characteristic.pdf)
- Manuscript: [Tame Hecke Eigensheaves with Several Marked Points](https://github.com/openai/math/blob/main/preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026/Tame-Hecke-Eigensheaves-with-Several-Marked-Points.pdf)
- Manuscript: [Frobenius Structures on Tame Hecke Eigensheaves](https://github.com/openai/math/blob/main/preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math