Kusner's conjecture is false for every
An equilateral set in is a set of points at equal pairwise distance. Kusner conjectured in 1983 that the largest such set has exactly points for every , as in the Euclidean case. Swanepoel disproved it for and Ge, Xu and Zhou proved it for , leaving open, where the catalog's earlier entry had found a single failing exponent and placed the infimum of failing exponents in . Does the conjecture fail for every ?
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Discrete geometry; equilateral sets
- Posed by
- Robert Kusner (1983)
- Year posed
- 1983
- Years open
- 43y
- Solved
- 2026-09-13
- Model
- GPT-6 Astra
- Vendor
- OpenAI
- Collaborators
- Nathan Xiong
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
False for every . This is strictly stronger than the catalog's August entry (arXiv:2608.14013, significance 30), which exhibited equilateral points for a single exponent and only located the infimum of failing exponents in . Together with Swanepoel for and Ge, Xu and Zhou for , Kusner's conjecture is now settled for all : true exactly on .
What the AI did
From the acknowledgements: the construction was found by OpenAI's GPT-6 Astra model, with the author taking responsibility for correctness.
Verification
Checked here on 22 September 2026 against arXiv:2609.14794. The abstract gives the construction explicitly - for every an equilateral set of points in for an depending on - and states that combining it with Swanepoel and with Ge, Xu and Zhou resolves Kusner's conjecture across the whole range . The attribution of the conjecture to Kusner was confirmed in the paper's introduction. The mathematics was not checked here; nine days old, no referee.