--- id: "2026-10-13-ulam-problem-lie-groups-countably-representable-gpt-6-astra" url: "https://postcutoff.com/e/2026-10-13-ulam-problem-lie-groups-countably-representable-gpt-6-astra/" as_of: "2026-10-15T10:09:00+02:00" date: "2026-10-13" date_precision: day category: science importance: 4 confidence: high status: [Event confirmed, Awaiting review] verification: Preprint only sources: 2 editor: human editor human_review: null version: "2026-10-14" --- As of: 2026-10-15 10:09 CEST. 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/2026-10-13-ulam-problem-lie-groups-countably-representable-gpt-6-astra/ # Ulam's question answered Full title: Ulam's question answered: every Lie group is a group of permutations of a countable set (Monod & Thom; key construction from GPT-6 Astra) arXiv 2610.18464 (13 Oct 2026), "Lie groups are countably representable" by Nicolas Monod and Andreas Thom, proves that every Lie group can be realised by permutations of a countable set, answering a question Ulam asked in 1960 (and hence that every locally compact second countable group is countably representable). The abstract says "the key deformation construction originated in an argument produced by OpenAI's GPT 6 Astra during extended mathematical discussions"; the authors then developed, checked and substantially rewrote it, and found that parts were already in the literature. Thom had disputed OpenAI's attribution of an earlier Astra result in September. ## Key facts - Paper: arXiv 2610.18464 [math.GR], 13 Oct 2026, by Nicolas Monod and Andreas Thom - Theorem A: every Lie group (second countable) is countably representable, i.e. embeds in Sym(N). Corollary B: every locally compact second countable group is countably representable - History in the paper: Schreier and Ulam showed in 1935 that R is countably representable; Ulam asked whether all Lie groups are [Ula60, II.7; Ula64, V.2]. Linear Lie groups were known (Thomas 1999; Kallman 2000; Ershov–Churkin 2004), and Conversano–Monod (J. Algebra 2024) reduced the problem to connected simple Lie groups. The remaining obstacle was non-linear central extensions - Key new ingredient (Theorem C): a countable-index subalgebra A of R with a continuous one-parameter family of ring homomorphisms from the inclusion back to a countable real-closed field, which makes the cohomological obstruction from central extensions vanish - AI disclosure: 'The key countable-index deformation was first constructed by OpenAI's GPT 6 Astra during extended mathematical discussions. The argument was subsequently developed, checked and substantially rewritten by the authors' - Acknowledgments: 'The proof emerged from three weeks of discussions with OpenAI's GPT, initially aimed at exploring SL3(R) with ring deformations and K-theoretic arguments. GPT Astra eventually produced the countable-index deformation underlying Theorem C. We then suggested extending the argument to SLn(R) and other simple Lie groups' - Prior art the authors found while checking: 'substantial parts of the argument already appear, notably the algebraic-analytic construction of Parusiński–Rond [PR20]'; the simultaneous Puiseux embedding with prescribed residues (Lemma 2.4) and its combination with countable-index gluing 'appear to be new and creative' - Context: in September Thom argued that OpenAI's announced non-sofic group result relied on his 2019 work with Gábor Kun and that OpenAI's attribution was misleading; this paper credits Astra openly ## What happened On 13 Oct 2026 Nicolas Monod and Andreas Thom posted [arXiv 2610.18464](https://arxiv.org/abs/2610.18464), proving that every Lie group can be realised as a group of permutations of a countable set. Schreier and Ulam had shown this for R in 1935, which led Ulam to ask whether it holds for all Lie groups. Linear Lie groups were known, and Conversano and Monod reduced the question to connected simple Lie groups in 2024. The remaining case was simple groups that are only central extensions of linear groups, where the cohomological obstruction lives on "wild" subgroups of countable index. The new tool is Theorem C: a subalgebra of R of countable index together with a continuous path of ring homomorphisms from a countable real-closed field to the inclusion, built from Puiseux series over fields of analytic functions and glued along countably many pieces. The continuity lets the authors trivialise the central extension on the corresponding subgroups. The authors give a detailed account of the AI's role. The proof "emerged from three weeks of discussions with OpenAI's GPT", which started on SL3(R). "GPT Astra eventually produced the countable-index deformation underlying Theorem C." The humans proposed the extension to SLn(R) and other simple groups, checked and rewrote everything, and found that a large part of the construction already appears in Parusiński–Rond (2020). They still call the Puiseux embedding with prescribed residues and its gluing "new and creative". Andreas Thom is the Dresden group theorist who, in September, criticised OpenAI's framing of Astra's non-sofic group result as ignoring his work with Gábor Kun (see [Astra's ten advances](/e/2026-08-01-openai-astra-ten-advances/)). His recent Mathstodon posts did not mention this paper (checked 14 Oct). No Hacker News thread was found (Algolia, 14 Oct). ## Why it matters It is a classical, well-known question answered with a clearly credited AI contribution to the decisive step, and with an unusually frank account of what was new and what turned out to be in the literature. It comes from a mathematician who had publicly challenged AI-maths attribution, which makes the credit given here more significant. ## What is disputed or not yet verified - Verification: Unrefereed preprint by two established group theorists; no formal proof ## Your AI and this story - GPT-6 Astra (training cutoff April 2026): 166 days after its cutoff - Claude Opus 5.5 (training cutoff June 2026): 105 days after its cutoff - Gemini 4 Argon (training cutoff March 2026): 196 days after its cutoff - Grok 4.7 (training cutoff May 2026): 135 days after its cutoff ## Sources 1. [arXiv 2610.18464: Lie groups are countably representable (Monod & Thom)](https://arxiv.org/abs/2610.18464) (arxiv.org, paper) 2. [Andreas Thom (guest post on Tao's blog, Sept 11): On the existence of non-sofic groups](https://terrytao.wordpress.com/2026/09/11/on-the-existence-of-non-sofic-groups/) (terrytao.wordpress.com, discussion) ## Changes - 2026-10-14 (filed): Created from the arXiv paper (arXiv AI-disclosure list) ## Related - 2026-10-14: [Mid-October 2026: arXiv results with disclosed AI help](https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/index.md) - 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md) - 2026-08-01: [OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs](https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/index.md) - People: [Terence Tao](https://postcutoff.com/person/terence-tao/)