Primary evidence
- An OpenAI model has disproved a central conjecture in discrete geometryS2 · Official institutional · Official announcement · 2026-05-20
- Planar Point Sets with Many Unit DistancesS1 · Primary research · Paper · 2026-05-20
OpenAI reports an autonomous disproof of Erdős's unit-distance growth conjecture
The model found a construction using tools from algebraic number theory; OpenAI released the proof and a companion paper by external mathematicians who checked and contextualized the argument.
What happened?
OpenAI reported that an internal general reasoning model disproved the longstanding conjectured n^(1+o(1)) upper-growth behavior for planar unit distances by constructing infinitely many configurations with at least n^(1+delta) unit distances for fixed delta>0.
Why it matters
Unlike benchmark progress, the event concerns a longstanding open problem in active mathematics and an AI-generated argument judged substantive by independent experts.
Significance rationale: A substantive AI-generated disproof of a longstanding Erdős conjecture is a historical AI4Math milestone because the central contribution is new mathematics rather than benchmark performance or a general model release; the H1 designation does not itself assert independent verification.
Historical context
Earlier developments
Later developments
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Related milestones
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Evidence & status at a glance
- Significance
- H1 · Historical milestone
- Verification
- Independently corroborated
- Evidence level
- E3 · Inspectable artifact
- AI role
- Autonomous primary role
- Formal assurance
- Artifact available
Contribution & verification
- Mathematical novelty
- New result
- Interface
- Informal, Symbolic
- AI contribution
- OpenAI states that the core construction and proof were generated autonomously by an internal general-purpose reasoning model.
- Human contribution
- External mathematicians checked the proof and wrote companion analysis; OpenAI researchers ran the evaluation and publication process.
- Verification
- Independently corroborated · E3 · Inspectable artifact · Artifact available
- Last reviewed
- 2026-09-11
Technical details
- Systems
- openai-internal-reasoning-model
- Organizations
- openai
- Problems
- erdos-unit-distance-growth-conjecture
- Methods
- algebraic-number-theory, autonomous-research, proof-generation
Sources & artifacts
- An OpenAI model has disproved a central conjecture in discrete geometryS2 · Official institutional · Official announcement · accessed 2026-09-10
- Planar Point Sets with Many Unit DistancesS1 · Primary research · Paper · accessed 2026-09-10
- Remarks on the disproof of the unit distance conjectureS3 · Independent scholarly · Paper · accessed 2026-09-10
- An explicit lower bound for the unit distance problemS3 · Independent scholarly · Paper · accessed 2026-09-10
- AI-generated proof manuscriptArtifact · Proof artifact
- Human-verified companion remarksArtifact · Supplement
Verification history
- 2026-05-20 · Partially verified · E3 · Inspectable artifact
The Chronicle records this event from the cited primary or official source; the status does not imply independent Chronicle verification.
- 2026-05-20 · Partially verified · E3 · Inspectable artifact
OpenAI released the AI-generated proof together with a companion paper by nine external mathematicians describing a human-verified, digested version of the argument. Will Sawin independently published a stronger explicit lower bound building on the new construction. These sources materially strengthen the mathematical evidence while the Chronicle retains the human editorial gate for `independently_verified`.
- 2026-09-10 · Independently corroborated · E3 · Inspectable artifact
Under the Chronicle's event-level verification semantics, the nine-mathematician human-verified companion paper and Will Sawin's independent follow-up work substantively corroborate the core event description. This label records independent corroboration of the event and its public evidence trail rather than a Chronicle adjudication of ultimate mathematical correctness.
Corrections
No corrections recorded.