{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-10-30-right-angled-hyperbolic-polytopes-dimension-bounds-kolpakov-gpt-6-astra/","md":"https://postcutoff.com/e/2026-10-30-right-angled-hyperbolic-polytopes-dimension-bounds-kolpakov-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":"2026-10-30-right-angled-hyperbolic-polytopes-dimension-bounds-kolpakov-gpt-6-astra","date":"2026-10-30","date_precision":"day","short_title":"Right-angled hyperbolic polytopes exist only up to dimension 8 (claim)","deck":null,"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.38900: Sharp dimension bounds for right-angled hyperbolic polytopes (Kolpakov)","url":"https://arxiv.org/abs/2610.38900","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"PDF with the AI statement","url":"https://arxiv.org/pdf/2610.38900","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI"],"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","summary":"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"],"key_numbers":[],"tags":["math","geometry","hyperbolic-geometry","reflection-groups","discrete-geometry","gpt-6-astra","gpt-6-1-sol","lean","preprint"],"science":{"field":"mathematics","subfield":"hyperbolic geometry / reflection groups","problem":"Maximal dimension of finite-volume (and of entirely ideal) right-angled hyperbolic polytopes","result":"Claimed sharp bounds: 8 for finite volume (was ≤ 12), 4 for entirely ideal (was ≤ 6)","open_since":"","ai_system":["GPT-6 Astra","GPT-6.1 Sol"],"human_role":"AI-assisted: models investigated proof ideas, helped strengthen the results and wrote and audited the Python and Lean code; the author wrote the paper","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed preprint; Lean checks the logical deductions with geometric inputs as premises"},"status":"pending","shock":""},"body_md":"## What happened\n\nRight-angled hyperbolic polytopes are the fundamental domains of right-angled reflection groups. Examples of finite volume exist up to\ndimension 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\nwhen every vertex is ideal. He credits GPT-6 Astra and GPT-6.1 Sol with help in finding and strengthening the arguments.\n\n## Why it matters\n\nIt would close a gap that had stood since Dufour's bound, using only elementary geometry and spherical codes. Specialists still need to\ncheck it.","disputed":[{"label":"Verification","text":"Unrefereed preprint; Lean checks the logical deductions with geometric inputs as premises","sources":[]}],"related":[{"id":"2026-10-31-ai-assisted-results-october-31-listing","url":"https://postcutoff.com/e/2026-10-31-ai-assisted-results-october-31-listing/","date":"2026-10-31","date_precision":"day","short_title":"AI-assisted results, 31 Oct: arXiv of 30 Oct and Hexagon of 29–31 Oct","deck":null,"takeaway":"The 31 Oct sweep listed 344 arXiv papers that mention AI, 75 conjecture papers with no AI statement and 48 uncited Hexagon works. We skimmed every AI statement and read about 70 PDFs or Hexagon records. Seven results got their own entries. Two come from senior authors: Panchenko's Mézard–Parisi formula ('found by ChatGPT Astra') and Bhargava's positive-density cyclicity of elliptic curves, written with GPT-6. The other five are Tianchi Yang's count of C₄-free graphs, Kolpakov's sharp dimension bounds for right-angled hyperbolic polytopes, the generalized Lax conjecture (Kummer–Netzer's AI-found generic case and OpenAI's counterexample), Diening–Storn's AFEM optimality for every bulk parameter, and an unverified GPT-6 Astra claim on Onsager's hard rods. The rest are listed here, among them Pongrácz's R(5,6) ≤ 84 certificate with GPT-6 Astra and Bowler–Pitz's negative answers to Halin-type questions, where Claude found key lemmas.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":44,"official":42,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Anthropic","Google","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-31","type":"filed","text":"Created (science agent; arXiv abstract, introduction and acknowledgments read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-31","run":null,"sources_read":"Science agent; arXiv abstract, introduction and acknowledgments read","updated":"2026-10-31","human_review":null,"version":{"date":"2026-10-31"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":183,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":122,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":213,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":152,"in_training_data":false}],"short_url":null}