---
id: "2026-10-30-right-angled-hyperbolic-polytopes-dimension-bounds-kolpakov-gpt-6-astra"
url: "https://postcutoff.com/e/2026-10-30-right-angled-hyperbolic-polytopes-dimension-bounds-kolpakov-gpt-6-astra/"
as_of: "2026-11-09T21:13:00+01:00"
date: "2026-10-30"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Awaiting review]
verification: Lean-verified
sources: 2
editor: human editor
human_review: null
version: "2026-10-31"
---

As of: 2026-11-09 21:13 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-30-right-angled-hyperbolic-polytopes-dimension-bounds-kolpakov-gpt-6-astra/

# Right-angled hyperbolic polytopes exist only up to dimension 8 (claim)

Full title: Right-angled hyperbolic polytopes: Alexander Kolpakov claims the sharp dimension bounds 8 (finite volume) and 4 (all vertices ideal), with GPT-6 Astra and GPT-6.1 Sol

arXiv 2610.38900 ("Sharp dimension bounds for right-angled hyperbolic polytopes", Alexander Kolpakov) proves that a finite-volume right-angled hyperbolic polytope has dimension at most 8, and at most 4 if all its vertices are ideal. Both bounds are attained (the Potyagailo–Vinberg examples and the ideal right-angled 24-cell), improving the earlier upper bounds 12 (Dufour) and 6. The proof projects to low-dimensional faces and bounds spherical codes; selected Lean 4 modules verify the internal logic. GPT-6 Astra and GPT-6.1 Sol helped find and strengthen the proofs.

## 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 183 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 122 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 213 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 152 days after its cutoff

## Sources

1. [arXiv 2610.38900: Sharp dimension bounds for right-angled hyperbolic polytopes (Kolpakov)](https://arxiv.org/abs/2610.38900) (arxiv.org, paper)
2. [PDF with the AI statement](https://arxiv.org/pdf/2610.38900) (arxiv.org, paper)

## Changes

- 2026-10-31 (filed): Created (science agent; arXiv abstract, introduction and acknowledgments read)

## Related

- 2026-10-31: [AI-assisted results, 31 Oct: arXiv of 30 Oct and Hexagon of 29–31 Oct](https://postcutoff.com/e/2026-10-31-ai-assisted-results-october-31-listing/index.md)