---
title: "A counterexample to the nearby Lagrangian conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 340; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "differential-geometry"
  - "unformalized"
seed_rank: 2195
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "A counterexample to the nearby Lagrangian conjecture"
    url: "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# A counterexample to the nearby Lagrangian conjecture

## One-sentence takeaway

OpenAI's result family 340 (Differential geometry) claims: Disproves the unrestricted nearby Lagrangian conjecture.

## Why it matters here

Differential geometry is the deep background for geometry processing, curvature flows and mesh work in graphics. 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): For some sufficiently large even N, constructs a closed exact embedded Lagrangian in $T^*(S^9\times S^{N-1})$ that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.
- *A counterexample to the nearby Lagrangian conjecture*: We disprove the unrestricted nearby Lagrangian conjecture. For some sufficiently large even integer N, we construct a closed exact smoothly embedded Lagrangian in $T^*(S^9\times S^{N-1})$ that is diffeomorphic to the base but is not Hamiltonian isotopic to the zero section.

## 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: [A counterexample to the nearby Lagrangian conjecture](https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.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