Right-angled hyperbolic polytopes exist only up to dimension 8 (claim)
Awaiting review
The takeaway
Finite-volume right-angled hyperbolic polytopes were known in dimensions up to 8 (Potyagailo–Vinberg), and Dufour had shown they cannot exist above 12. For polytopes with all vertices at infinity the gap was 4 to 6. Alexander Kolpakov’s preprint (arXiv 2610.38900, 29 Oct 2026) claims both gaps are closed: at most 8 in general and at most 4 when all vertices are ideal. He used GPT-6 Astra and GPT-6.1 Sol ‘to investigate proof ideas, help strengthen the results’, and Lean modules check the internal deductions. Not refereed.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- 3 of 5
- Last verified
- 31 October 2026
Your AI and this story
- GPT-6 Astra183 days after its cutoff
- Claude Opus 5.5122 days after its cutoff
- Gemini 4 Argon213 days after its cutoff
- Grok 4.7152 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 122 days before it.
Key facts
- Claim (abstract): ‘We determine the largest dimensions in which right-angled hyperbolic polytopes of finite volume can exist. The dimension is at most eight in general, and at most four when all vertices lie at infinity. Both bounds are attained, improving the previously known upper bounds of twelve and six.’
- AI statement (verbatim): ‘The author used OpenAI’s GPT-6-astra and GPT-6.1-sol as research and coding assistants to investigate proof ideas, help strengthen the results, improve the exposition, and generate and audit the Python and Lean 4 code accompanying this paper.’
- History (paper, §1): Potyagailo and Vinberg constructed finite-volume examples through dimension eight; Dufour proved dimension ≤ 12; for entirely ideal polytopes the regular right-angled 24-cell gives dimension 4 and the upper bound 6 was known; the compact case already had the sharp bound 4
- Method: repeated projection to a facet of maximal unit-normal height gives a 6-dimensional right-angled face whose facet normals form a spherical code with inner products below 1/4; a cubic sum of tensor squares bounds it by 26 facets, contradicting the known minimum of 27 facets in dimension 6
- Lean: ‘Selected Lean 4 modules verify the internal logical deductions for the projection, the cubic bound and the dimension bounds. The geometric statements, face inheritance and cited theorems are supplied as explicit premises.’
- Verification status: single-author arXiv preprint, not refereed; partial Lean check with geometric premises as hypotheses; no reaction found on 31 Oct
What happened
Right-angled hyperbolic polytopes are the fundamental domains of right-angled reflection groups. Examples of finite volume exist up to dimension 8, but the best upper bound was 12. On 29 Oct Alexander Kolpakov posted a proof that 8 is the maximum, and that 4 is the maximum when every vertex is ideal. He credits GPT-6 Astra and GPT-6.1 Sol with help in finding and strengthening the arguments.
Why it matters
It would close a gap that had stood since Dufour’s bound, using only elementary geometry and spherical codes. Specialists still need to check it.
What is disputed or not yet verified
| Verification | Unrefereed preprint; Lean checks the logical deductions with geometric inputs as premises |
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Sources
2 sources from 1 site. Numbers match the chips in the text.
2 sources: 2 primary
Primary
- arXiv 2610.38900: Sharp dimension bounds for right-angled hyperbolic polytopes (Kolpakov)arxiv.org, paper
- PDF with the AI statementarxiv.org, paper
Changes
- Filed (science agent; arXiv abstract, introduction and acknowledgments read)