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Benefit Ledger · provisional · Mathematics

An unreleased OpenAI model disproves Erdős’s planar unit-distance conjecture

On 20 May 2026 OpenAI reported that an internal reasoning model produced an infinite family of planar point sets with n^{1+δ} unit distances for some δ>0, disproving Erdős’s n^{1+o(1)} conjecture. External mathematicians checked the argument and later made the exponent explicit.

20 May 2026Tier 3 MajorMethodology 0.1

Current score

+2.10

10 base · Major (tier 3 of 5, 10 pts)
× 0.8500 attribution · Primary causal contribution
× 0.5500 evidence · Peer review or independent validation
× 0.6000 realization · Experimentally validated
× 0.7500 durability
Event-level product before credit split: 2.10

Disproving a 1946 Erdős conjecture in combinatorial geometry is major (tier 3), not Historic, until the argument is absorbed as a standard theorem. Attribution is to the internal model that found the construction (0.85). Evidence is a lab announcement plus an independent human writeup and an improved explicit bound (0.55). Realization is a public, checked proof (0.60).

What happened

Erdős asked in 1946 how many times the unit distance can appear among n points in the plane. The square-grid construction gave only a slightly superlinear lower bound, and the n^{1+o(1)} upper-bound heuristic was widely believed. OpenAI says an internal model, elicited by Lijie Chen and checked by Mark Sellke and Mehtaab Sawhney, produced a number-theoretic construction using algebraic number fields that yields a polynomial improvement for infinitely many n. Noga Alon, Thomas Bloom, Timothy Gowers, and others published a human-digested writeup (arXiv:2605.20695). Will Sawin then made an explicit exponent of about n^{1.014}. This is not the Clay Navier–Stokes claim and not public GPT-6 Astra; credit stays with the May 2026 internal system. The best upper bound remains O(n^{4/3}).

Model attribution

+2.10

GPT-6

Unreleased reasoning model OpenAI says produced the unit-distance construction in one shot; humans verified and later simplified it.

OpenAI and the Alon–Bloom–Gowers et al. remarks attribute the original construction to an internal OpenAI model, not to GPT-6 Astra.

Attribution 0.8500 · Credit share 100% · OpenAI

Claims

  • An OpenAI internal model produced a construction giving n^{1+δ} unit distances for infinitely many n, disproving Erdős’s n^{1+o(1)} conjecture.

    outcome · supported

  • GPT-6 Astra found the unit-distance disproof.

    attribution · disputed

Sources

primary sources

independent sources

Secondary domains: Computer Science

Revision history

  • 13 Sep 2026 · 0.00 → 2.10

    Imported events under methodology 0.1.

An unreleased OpenAI model disproves Erdős’s planar unit-distance conjecture · NetGoodIndex