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TopicsErdős unit distance conjecture Discrete geometry AI-assisted Erdős problem solving AI mathematical discovery

OpenAI's unit distance counterexample is a natural generalization of Erdős's lattice construction, introducing no new geometric tools

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Preliminary · from the Steward of OpenAI's AI-generated disproof of the unit distance conjecture meets a top mathematics journal's acceptance standard, recorded when this subclaim was created and pending this claim's own assessment.

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Bloom, in the companion paper, calls the construction with the benefit of hindsight a natural albeit highly non-trivial generalisation of Erdős's lattice construction, and says it introduces no powerful new geometric tools or structural results; Gowers's hint-sequence analysis reaches a similar view that the proof's complexity is fairly short; Timothy B. Lee's commentary echoes it. On the other side, Tsimerman warns that it is always tempting to call a completed proof obvious after the fact and stresses that the construction is intimidating to see through; Sawin explains a specific subtlety that hid the approach from experts; Shankar and Wood call the idea beautiful and ingenious. The lineage half of the proposition is not really disputed; the no-new-tools half is a matter of emphasis on which the verifiers differ.

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https://arxiv.org/html/2605.20695v1May 20, 2026 Untraced · nothing upstream recorded

this was dampened slightly when I learnt it was a construction of a counterexample, and still further when I learnt that nature of the construction, being (with the benefit of hindsight) a natural, albeit highly non-trivial, generalisation of the original lattice-based construction of Erdős.

https://arxiv.org/html/2605.20695v1extraction 0.85

Bloom's reflection in the companion paper; he adds that the construction does not introduce any powerful new geometric tools or hitherto unsuspected structural results, while predicting it will still have a major impact in discrete geometry.

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