Gale’s 1953 conjecture fails in 3D
A 101-point set found with LLM code evolution and GPT-6 Astra beats the sphere’s Borsuk partition ratio
Result confirmed
The takeaway
Vsevolod Voronov built a 101-point set in space whose every split into four parts leaves a part of diameter above 0.88928, more than the sphere’s 0.88807. That refutes Gale’s 1953 conjecture in dimension 3. The search algorithms were developed with GPT-6 Astra, some proposed by the AI, and the set is checkable in exact arithmetic.
Status
- Claim
Result confirmed
- Our reporting
- High confidence
- Verification
- Unverified claim
- Importance
- 3 of 5
- Last verified
- 22 October 2026
Your AI and this story
- GPT-6 Astra173 days after its cutoff
- Claude Opus 5.5112 days after its cutoff
- Gemini 4 Argon203 days after its cutoff
- Grok 4.7142 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 112 days before it.
Key facts
- Result: |V| = 101, β₄(V) > 0.88928 > β₄(S²) = 0.88807…; verification: compute diam V, take pairs at distance > 0.88928·D, check the distance graph has no proper 4-colouring
- Before: counterexamples to Gale’s conjecture came only from Borsuk counterexamples, none in dimension below 9 (OpenAI’s 9-dimensional Borsuk counterexample of 23 Sep 2026 was the lowest)
- Also: similar examples for dimensions 4 ≤ n ≤ 6; code and configurations at github.com/vsvor/Gale-conjecture
- AI disclosure: ‘In April 2026, we carried out extensive code-evolution experiments with the GigaEvo framework using DeepSeek 4, Qwen 3 Max, Gemini 3, and Claude Opus 4.6; the best ratio obtained in R3 was about 0.872. In September 2026, we returned to the problem and, with the help of GPT-6 Astra, developed new algorithms and heuristics, some of which were proposed by the AI itself. GPT-6 was also used to prepare the GitHub repository and to write the document describing the algorithms; Claude Opus 5.5 was used for proofreading and language editing.’
- Author’s caveat: ‘At present, there is no full formalization in Lean 4 or an independent review of this paper.’
What happened
Borsuk asked whether every bounded set in Rⁿ splits into n+1 parts of smaller diameter. The quantitative version asks how small the largest part can be made. In R³ with four parts, the sphere gives √((3+√3)/6) ≈ 0.88807, and David Gale conjectured in 1953 that no set does worse.
Voronov’s paper (arXiv 2610.288281) gives a 101-point set that does worse, 0.88928. The set was found numerically by global search from random starts and verified exactly, by checking that a distance graph has no proper 4-colouring. Earlier LLM code-evolution experiments with GigaEvo (April 2026) reached only about 0.872. The successful algorithms were developed in September with GPT-6 Astra, which also proposed some of them.
Why it matters
This is a small, explicit, machine-checkable counterexample to a 73-year-old conjecture, found by human–AI algorithm design. It follows AI-assisted progress on the Borsuk problem itself (dimension 63 in May, OpenAI’s dimension 9 in September).
Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 primary
Primary
- arXiv 2610.28828: Voronov, A counterexample to Gale’s conjecture in R³arxiv.org, paper
- GitHub: vsvor/Gale-conjecture (code and configurations)github.com, code
Changes
- Filed (PDF read in full text)