GPT-5.5 Pro-assisted construction lowers the smallest known Borsuk counterexample dimension from 64 to 63
In May 2026 Max Grinsztajn, assisted by OpenAI's GPT-5.5 Pro, built a 321-point set in R^63 that cannot be split into 64 parts of smaller diameter, so Borsuk's conjecture fails in dimension 63 (b(63) ≥ 65). The previous smallest known failing dimension, 64, had stood since 2013. A second, independent AI-generated version (GPT-5.6 Sol) was posted to arXiv in August and withdrawn because the result already existed.
Key facts
- Construction: 320-point Jenrich–Brouwer core from the G2(4) strongly regular graph in a codimension-2 subspace of R^63, plus one projected and rescaled point
- Result: 321 points, any subset of smaller diameter has at most 5 points, so at least 65 parts are needed (b(63) ≥ 65)
- Open range for Borsuk's conjecture moves from 4 ≤ n ≤ 63 to 4 ≤ n ≤ 62
- Repository README: 'The construction and proof were obtained with assistance from GPT-5.5 Pro'; exact verification script plus Sage-checkable certificates (no Lean proof)
- Recorded in Tao's optimization-constants table (constant 28a) as [Gri2026]
- arXiv 2608.12561 (Yibo Ji, 12 Aug 2026): same 321-point set 'generated entirely by ChatGPT using GPT 5.6 Sol'; withdrawn 14 Aug 2026 because the construction had already been published
Science result
- Field
- mathematics / discrete geometry
- Problem
- Borsuk's conjecture: smallest dimension where it fails (open since 1933)
- Result
- Counterexample in dimension 63 (321-point three-distance set needing ≥ 65 parts), improving the 2013 record of 64.
- AI system
- GPT-5.5 Pro
- Human role
- AI-assisted: Max Grinsztajn worked with GPT-5.5 Pro; exact computer verification
- Verification
- Exact finite computation with published certificates; not peer-reviewed
- Status
- confirmed
- Why surprising
- A 13-year-old record in a famous geometry problem moved by a single added point that a chatbot helped find.
What happened
Borsuk asked in 1933 whether every bounded set in R^n can be split into n+1 pieces of smaller diameter. Kahn and Kalai showed in 1993 that the answer is no in high dimensions, and later work pushed the smallest known failing dimension down to 64 (Jenrich, 2013, from Bondarenko's construction). In May 2026 Max Grinsztajn, working with GPT-5.5 Pro, added one carefully projected point to the 320-point Jenrich–Brouwer set and got a 63-dimensional counterexample. His repository ships an exact verification script and certificates.
The exact day is not known. Wikipedia dates the result to May 2026. In August 2026 Yibo Ji posted the same kind of 321-point construction to arXiv, saying it was "generated entirely by ChatGPT using GPT 5.6 Sol". He withdrew it two days later because the result was already published. Secondary sources also mention an independent find by "Konz", which we have not verified.
Why it matters
It is a clean, checkable improvement to a well-known geometry record. Two separate human+model pairs reached it within a few months, which suggests these gaps are now within easy reach of frontier models.
Changelog
- 2026-09-29: created. The lead had mixed up the model and date: the primary result is GPT-5.5 Pro (May 2026), and the August arXiv paper using GPT-5.6 Sol is a withdrawn independent rediscovery.
Related events
- GPT-5.5 Pro finds counterexample disproving McKean's 1966 conjecture and the Gaussian completely monotone conjecture ★★★
- Erdős–Szemerédi sum-product conjecture shown false over the reals; a GPT-5.5 Pro agent re-disproves it in 7 of 8 runs ★★★★
Sources (5)
- codeGitHub: maaxgrin/borsuk-63-counterexample (paper PDF + verifier)
- docsTao et al. optimization constants: constant 28a (Borsuk)
- paperarXiv 2608.12561: An AI Generated Counterexample to Borsuk Problem in Dimension 63 (withdrawn)
- discussionWikipedia: Borsuk's conjecture
- discussionWikipedia: List of mathematical discoveries by artificial intelligence
id: 2026-05-01-borsuk-conjecture-dimension-63 · updated 2026-09-29 · open in the interactive timeline