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AI systems begin producing original mathematics

AI systems begin producing original mathematics

New Capabilities

Frontier models move from solving olympiad problems to publishing verified proofs

2 days ago: Nature covers the proof; verifier calls it the first AI result 'interesting in itself'

Overview

For 80 years, mathematicians believed square grids were the best way to pack points at distance exactly 1 apart on a plane. On May 20, an internal OpenAI reasoning model produced a counterexample, using algebraic number theory to beat the grid by a small polynomial factor. Nine outside mathematicians verified the proof.

Nature and Scientific American covered the result within days. Fields Medalist Tim Gowers said it deserves publication in the Annals of Mathematics. Kevin Weil, whose October 2025 post erroneously claimed GPT-5 had solved Erdős problems, had left OpenAI a month before this proof appeared.

Why it matters

If frontier AI can produce original mathematics that survives expert scrutiny, the bottleneck on every formal science starts to move.

Key Indicators

80 years
Conjecture stood unsolved
Paul Erdős posed the planar unit distance problem in 1946.
9
Outside mathematicians verified the proof
Including Tim Gowers, Noga Alon, and Thomas Bloom.
n^1.014
New lower bound on unit-distance pairs
Beats the square-grid construction by a small polynomial factor.
19 pages
Companion verification paper
Independent write-up translating the AI proof into human-readable form.

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People Involved

Organizations Involved

Timeline

1946 May 2026

8 events Latest: 2 days ago
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  1. Nature covers the proof; verifier calls it the first AI result 'interesting in itself'

    Latest Media

    Nature publishes coverage of the unit-distance disproof. Verifier Daniel Litt tells Nature it is 'the first result produced autonomously by an AI that I find interesting in itself.'

  2. OpenAI announces verified unit-distance disproof

    Breakthrough

    An internal OpenAI reasoning model produces an infinite family of point configurations that beats the square grid. Nine outside mathematicians, including Bloom and Gowers, verify the proof.

  3. Kevin Weil departs OpenAI

    Personnel

    OpenAI dissolves its OpenAI for Science unit and Weil, who authored the discredited October 2025 Erdős claim, leaves the company. His departure comes 33 days before the verified unit-distance proof appears.

  4. Weil deletes the post

    Retraction

    After pushback from Bloom, Yann LeCun, and Demis Hassabis, the OpenAI executive removes the X post.

  5. OpenAI's discredited Erdős claim

    Setback

    Kevin Weil claims GPT-5 solved ten unsolved Erdős problems. Thomas Bloom shows the answers already existed in published work.

  6. Gemini Deep Think gets IMO gold

    Milestone

    Google's Gemini system officially reaches gold-medal scoring at the 2025 International Mathematical Olympiad.

  7. DeepMind's AlphaProof gets IMO silver

    Milestone

    AlphaProof and AlphaGeometry 2 jointly solve 4 of 6 International Mathematical Olympiad problems. First AI medal at the IMO.

  8. Erdős poses the unit distance problem

    Research

    Paul Erdős asks: given n points in a plane, how many pairs can be exactly distance 1 apart?

Historical Context

2 moments from history that rhyme with this story — and how they unfolded.

June 1976

Four color theorem proved by computer (1976)

Kenneth Appel and Wolfgang Haken at the University of Illinois reduced the four-color map problem to roughly 1,936 cases and checked each by computer. The result needed 1,200 hours of mainframe time. Many mathematicians refused to accept the proof because no human could verify it by hand.

Then

Illinois began stamping outgoing mail 'Four Colors Suffice.' Philosophy journals ran years of arguments about what counts as a proof.

Now

Computer-assisted proofs gained slow acceptance. Georges Gonthier produced a fully machine-verified version in 2005 using the Coq proof assistant, ending the dispute.

Why this matters now

The Appel-Haken proof showed mathematicians would accept machine-aided work if it could be independently verified. OpenAI's unit-distance proof is following the same playbook with a 19-page human-readable verification paper.

July 2024

AlphaProof scores IMO silver (2024)

DeepMind's AlphaProof and AlphaGeometry 2 solved four of six problems at the International Mathematical Olympiad, scoring 28 of 42 points. It was the first AI medal at the IMO and required formalizing solutions in the Lean proof assistant.

Then

AlphaProof became the reference point for serious AI math. DeepMind followed with a gold-medal Gemini Deep Think result in July 2025.

Now

The IMO results convinced funders and researchers that AI could handle structured proof. They also set a higher bar: olympiad problems have known answers, while research problems do not.

Why this matters now

OpenAI's unit-distance proof is the step AlphaProof did not take: producing a new theorem rather than reproducing a known one. It moves the field from solving math to discovering it.

Sources

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