Kusner's Conjecture on Equilateral Sets in
Kusner conjectured in 1983 that the maximum number of points in that are pairwise at -distance one is exactly for every , as in the Euclidean case. False: an explicit configuration of equilateral points exists for some exponent, placing the infimum of exponents at which the conjecture fails in . The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Discrete geometry
- Posed by
- Robert B. Kusner
- Year posed
- 1983
- Years open
- 43y
- Solved
- 2026-08-14
- Model
- GPT-5.6 Sol, Claude Fable 5
- Vendor
- OpenAI, Anthropic
- Collaborators
- Logan R. Chalmers
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The paper's disclosure: GPT-5.6 Sol assisted in implementing the computational search strategy in code, and Claude Fable 5 was used as a tool in drafting. No mathematical step is attributed to a model by name, but the search that produced the configuration is the load-bearing computation.
Verification
Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14013): the disclosure is verbatim as quoted, Kusner's 1983 attribution is in the introduction, and the data is deposited on Zenodo (10.5281/zenodo.21911503). The exact-arithmetic certificate was not re-run here. Days-old preprint, no independent review.