{"schema":"postcutoff/event@1","as_of":"2026-10-22T13:31:00+02:00","url":"https://postcutoff.com/e/2026-10-20-gale-conjecture-r3-counterexample-gpt-6-astra/","md":"https://postcutoff.com/e/2026-10-20-gale-conjecture-r3-counterexample-gpt-6-astra/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-20-gale-conjecture-r3-counterexample-gpt-6-astra","date":"2026-10-20","date_precision":"day","short_title":"Gale's 1953 conjecture fails in 3D","deck":"A 101-point set found with LLM code evolution and GPT-6 Astra beats the sphere's Borsuk partition ratio","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.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv 2610.28828: Voronov, A counterexample to Gale's conjecture in R³","url":"https://arxiv.org/abs/2610.28828","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: vsvor/Gale-conjecture (code and configurations)","url":"https://github.com/vsvor/Gale-conjecture","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","Anthropic"],"title":"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","summary":"Gale (1953) conjectured that among sets of unit diameter in R³, the sphere is the hardest to cut into four parts of small diameter, with ratio √((3+√3)/6) ≈ 0.88807. Vsevolod Voronov (arXiv 2610.28828, 20 Oct 2026) gives a 101-point set V in which every partition into four parts has a part of diameter greater than 0.88928·diam V, verified in exact integer arithmetic. It is the first counterexample in dimension below 9. The search algorithms were developed \"with the help of GPT-6 Astra … some of which were proposed by the AI itself\", after earlier GigaEvo LLM code-evolution runs fell short.","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.'"],"key_numbers":[],"tags":["math","discrete-geometry","borsuk-problem","gale-conjecture","counterexample","gpt-6-astra","gigaevo","code-evolution"],"science":{"field":"mathematics","subfield":"discrete geometry (Borsuk partition problem)","problem":"Gale's conjecture (1953) on partitions of 3D sets into four parts of smaller diameter","result":"A 101-point set in R³ whose every 4-partition has a part of diameter > 0.88928·diam, exceeding the sphere's 0.88807; first counterexample below dimension 9.","open_since":"1953","ai_system":["GPT-6 Astra","GigaEvo (DeepSeek 4","Qwen 3 Max","Gemini 3","Claude Opus 4.6)"],"human_role":"Human-led search; GPT-6 Astra co-developed the algorithms and heuristics (some proposed by the AI) that found the set","verification":{"key":"none","label":"Unverified claim","text":"Explicit finite object checkable in exact integer arithmetic (coordinates in Appendix A, verification scripts in the repository); unrefereed, no Lean"},"status":"confirmed","shock":""},"body_md":"## What happened\n\nBorsuk 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.\n\nVoronov's paper ([arXiv 2610.28828](https://arxiv.org/abs/2610.28828)) 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.\n\n## Why it matters\n\nThis 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).","disputed":[],"related":[{"id":"2026-05-01-borsuk-conjecture-dimension-63","url":"https://postcutoff.com/e/2026-05-01-borsuk-conjecture-dimension-63/","date":"2026-05-01","date_precision":"month","short_title":"GPT-5.5 Pro-assisted construction lowers the smallest known Borsuk counterexample dimension from 64 to 63","deck":null,"takeaway":"It is a clean, checkable improvement to a well-known geometry record.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":3,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-22","type":"filed","text":"Created (PDF read in full text)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-22","run":null,"sources_read":"PDF read in full text","updated":"2026-10-22","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":173,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":112,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":203,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":142,"in_training_data":false}],"short_url":null}