---
title: "The metric Blaschke conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 344; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "differential-geometry"
  - "unformalized"
seed_rank: 2199
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "The metric Blaschke theorem"
    url: "https://github.com/openai/math/blob/main/preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# The metric Blaschke conjecture

## One-sentence takeaway

OpenAI's result family 344 (Differential geometry) claims: Proves the metric Blaschke conjecture: every closed connected Riemannian manifold of positive dimension whose injectivity radius equals its diameter is, up to scaling, a standard compact rank-one symmetric space.

## 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

- *The metric Blaschke theorem*: We prove the metric Blaschke conjecture: every connected closed smooth Riemannian manifold of positive dimension whose global injectivity radius equals its diameter is, up to scale, a standard compact rank-one symmetric space.

## 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: [The metric Blaschke theorem](https://github.com/openai/math/blob/main/preprints/The-metric-Blaschke-theorem-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