---
id: "2026-08-01-openai-astra-ten-advances"
url: "https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-08-01"
date_precision: day
category: science
importance: 5
confidence: medium
status: [Awaiting review]
verification: Lean-verified
sources: 14
editor: human editor
human_review: null
version: "2026-10-19"
---

As of: 2026-11-13 13:49 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/2026-08-01-openai-astra-ten-advances/

# OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs

On 1 Aug 2026 OpenAI published 'Ten advances in mathematics and theoretical computer science' by an internal model, Astra (released as GPT-6 Astra on 3 Sep). It came with a 249-page manuscript and Lean 4 proofs. The claims include the first explicit non-sofic group, a disproof of Connes's rigidity conjecture, the first improvement to the sphere-packing upper-bound exponent since 1978, and solutions to Erdős problems #146, #180 and #183.

## Key facts
- Claims: explicit non-sofic group (Gromov's question, ~1999); disproof of Connes's rigidity conjecture; quantum parallel repetition for general two-player entangled games
- Also: Ehrhart volume conjecture (partial per some sources); polynomial-factor NP-hardness of approximating the Closest Vector Problem; permanent circuit lower bound ~n⁴/log n
- Superexponential lower bound for multicolour Ramsey numbers (Erdős #183); Erdős #146 and #180; improved binary and spherical codes
- Sphere-packing upper-bound exponent ~0.5990558 → ~0.6044005, first improvement since Kabatiansky–Levenshtein (1978)
- Evidence: 249-page PDF, Lean 4 proofs (openai/ten-proofs); < $2,000 of tokens per solution at GPT-5.6 Sol prices; prompts not released
- Attribution dispute: Andreas Thom (11 Sep, guest post on Tao's blog) says the non-sofic proof relies crucially on his 2019 work with Gábor Kun (Prop. 2.3 of OpenAI's PDF) despite OpenAI's 'decade without progress' framing, and asks whether his own ChatGPT conversations about these techniques reached the model; Mark Sellke replied 'that did not happen'. Kun and Thom posted a follow-up, arXiv 2608.06222 (6 Aug)
- Independent audit (arXiv 2608.14673): 'No confirmed substantive mathematical error in a principal result remains'; one chapter needs major revisions, and some stronger results were not reproduced
- Follow-ups: Jihao Liu proved the equality case of Ehrhart's volume conjecture with GPT-5.6 Sol, Fable 5 and the Danus agent (arXiv 2608.01040); Raphael Steiner extended the superexponential multicolour Ramsey construction to odd cycles with a proof 'found autonomously by ChatGPT 5.6 Pro/Sol' (arXiv 2608.02537, later withdrawn and merged into arXiv 2608.02522)
- The Verge (10 Sep): Thom on Mastodon called Mark Sellke's categorical answer 'materially misleading; looking back it was plainly dishonest' and said 'De-identification may remove a name; it does not remove the intellectual content of a mathematical idea'. The Verge says OpenAI quietly amended its write-up to credit Kun and Thom without announcing the change
- Scientific American (Joseph Howlett, 6 Aug), 'OpenAI's latest math breakthroughs commit research misconduct, experts say': Stephen Miller (said the sphere-packing proof used his 2016 argument without proper citation): 'They are running roughshod over the work of others who came before them in a deliberate way. It seems completely systematic to me, and it points to research misconduct.' Francesco Fournier-Facio (Cambridge) on the soficity result: 'the big PR machine… does not care about being 100 percent accurate'. OpenAI: 'We take responsibility for the correctness of these results and are meeting the same standards generally expected of human mathematicians' and promised small updates

## What happened

OpenAI released, in one announcement, ten research results produced by an internal model a month before its launch. Most came with machine-checked proofs.

## Why it matters

It moved the frontier from individual AI-assisted results to a lab producing batches of significant theorems. An independent audit largely upheld them.

## What is disputed or not yet verified

- Verification: Lean 4 formalizations published for most of the ten results (openai/ten-proofs); results not refereed; a review of 18 chapter reviews (Sienicki & Sienicki, arXiv 2608.14673, unrefereed) found no remaining substantive error in principal results but one chapter needing major revision; experts disputed attribution and citation (Scientific American). Status varies by result

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 93 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 32 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 123 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 62 days after its cutoff

## Sources

1. [OpenAI: Ten advances in mathematics and theoretical computer science](https://openai.com/index/ten-advances-in-mathematics/) (openai.com, official)
2. [OpenAI: ten proofs manuscript (PDF)](https://cdn.openai.com/pdf/ten-proofs-oai.pdf) (cdn.openai.com, paper)
3. [Kun & Thom: Nonsofic wreath products of residually finite groups (arXiv 2608.06222)](https://arxiv.org/abs/2608.06222) (arxiv.org, paper)
4. [arXiv 2608.01040: The equality case of Ehrhart's volume conjecture (Liu)](https://arxiv.org/abs/2608.01040) (arxiv.org, paper)
5. [arXiv 2608.02522: Locally bipartite subgraphs via multicolor Ramsey numbers (Steiner)](https://arxiv.org/abs/2608.02522) (arxiv.org, paper)
6. [Scientific American: OpenAI's latest math breakthroughs commit research misconduct, experts say (6 Aug)](https://www.scientificamerican.com/article/openais-latest-math-breakthroughs-commit-research-misconduct-experts-say/) (scientificamerican.com, press)
7. [Quanta: Why the legendary Erdős problems are falling to AI](https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/) (quantamagazine.org, press)
8. [The Verge (11 Aug): The AI takeover of mathematics has begun](https://www.theverge.com/ai-artificial-intelligence/977273/the-ai-takeover-of-mathematics-has-begun) (theverge.com, press)
9. [The Verge Decoder (20 Aug): Welcome to the AI crisis in math (Robert Hart)](https://www.theverge.com/podcast/982434/ai-math-openai-astra-existential-crisis) (theverge.com, press)
10. [The Verge (10 Sep): Mathematicians want proof OpenAI didn't use their work (Thom)](https://www.theverge.com/ai-artificial-intelligence/993263/where-does-openai-get-mathematics-training-data) (theverge.com, press)
11. [A Human Audit of OpenAI's AI-Generated Mathematical Proofs (arXiv 2608.14673)](https://arxiv.org/abs/2608.14673) (arxiv.org, discussion)
12. [Simon Willison on the ten advances](https://simonwillison.net/2026/Aug/1/ten-advances-in-mathematics/) (simonwillison.net, discussion)
13. [Andreas Thom (guest post on Tao's blog): On the existence of non-sofic groups (attribution concerns)](https://terrytao.wordpress.com/2026/09/11/on-the-existence-of-non-sofic-groups/) (terrytao.wordpress.com, discussion)
14. [MathOverflow: key new ideas in the non-soficity proof](https://mathoverflow.net/questions/513866/what-are-the-key-new-ideas-in-the-proof-of-nonsoficity-of-groups-in-openai-s-con) (mathoverflow.net, discussion)

## Changes

- 2026-10-19: Status confirmed → pending (batch of ten unrefereed results, Lean for most but not all, one chapter needing major revision); the 'independent audit' is described as what it is (an unrefereed review of reviews)
- 2026-10-19: Added video(s) fireship-summer-math-fell-to-the-machines
- 2026-10-13: Related: added [October 2026: the arXiv stream of conjectures settled with disclosed AI help continues](https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/) (October catalogue, split out of the summer catalogue)
- 2026-10-09: Added Scientific American's 6 Aug misconduct article (Stephen Miller, Fournier-Facio, OpenAI response), missed until found while processing October coverage
- 2026-10-07: Science block: kind: batch (many results in one entry) (not a long-open problem; listed with other AI science results, not in the open-problems count)
- 2026-10-05: Added The Verge's Aug 11 feature, Aug 20 Decoder episode and Sept 10 Thom article (found via its Oct 5 storystream)
- 2026-09-30: Added AI-assisted follow-ups (Ehrhart equality case; odd-cycle Ramsey)
- 2026-09-29: Added Andreas Thom's attribution critique of the non-sofic result, Kun–Thom follow-up paper, MathOverflow thread
- 2026-09-29 (filed): Created

## Related

- 2026-10-13: [October 2026: the arXiv stream of conjectures settled with disclosed AI help continues](https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/index.md)
- 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md)
- 2026-09-21: [OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts)](https://postcutoff.com/e/2026-09-21-openai-100-open-problems-claim/index.md)
- 2026-09-03: [OpenAI releases GPT-6 Astra, its first GPT-6 model](https://postcutoff.com/e/2026-09-03-gpt-6-astra/index.md)
- People: [Terence Tao](https://postcutoff.com/person/terence-tao/), [Raphael Steiner](https://postcutoff.com/person/raphael-steiner/), [Simon Willison](https://postcutoff.com/person/simon-willison/)
- Archived: 2 posts and 1 video, listed in https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/index.json