Mathematician posts an unchecked ChatGPT Astra proof of the planar Mumford–Shah conjecture (1989), citing OpenAI's '100 open problems' claim
On 22 Sep 2026 Francesco Deangelis (University of Münster) posted a claimed proof of the planar Mumford–Shah conjecture (1989), a central open problem in the calculus of variations and image segmentation. He says ChatGPT Astra produced it on 9 Sep. He calls it an 'AI-generated draft' that he is still checking. He released it early to document priority after OpenAI's 21 Sep claim that an internal model had solved 100+ open problems.
Key facts
- arXiv 2609.26732 (22 Sep 2026): 'On September 9, 2026, I obtained a proof of the Mumford–Shah conjecture, formulated in 1989, using ChatGPT Astra'
- Timeline in the author's note: 5 Sep, Astra in Work mode ran 6 hours without a complete proof; 6 Sep, it proved the author's Dirichlet-boundary conjecture in 1 h 20 min, building on his unpublished April 2026 work; 9 Sep, given that result, it produced a complete proof after 2 h 30 min (37 pages); 11 Sep, a shorter version, which is the one posted
- Author's caveat: 'This prompted me to make the current AI-generated draft publicly available now, to document the chronology of this work. I will continue checking the mathematical details'
- Claimed method: classify planar generalized global minimizers with a second-variation argument (two perpendicular translations of a compact crack portion, a Hodge identity and holomorphic rigidity). The known equivalence between that classification and interior regularity then gives the conjecture
- Status: unrefereed, not formalised, and not publicly checked by independent experts as of 30 Sep 2026
Science result
- Field
- mathematics / calculus of variations / free-discontinuity problems
- Problem
- Planar Mumford–Shah conjecture (regularity of minimizing discontinuity sets) (open since 1989)
- Result
- Claimed classification of planar Mumford–Shah generalized global minimizers, which would imply the conjecture.
- AI system
- GPT-6 Astra (ChatGPT, Work mode)
- Human role
- The human supplied his thesis, unpublished results and a research protocol; the model produced the proof text; checking by the author is ongoing
- Verification
- Unverified claim (preprint by the prompting author, still under his own review)
- Status
- pending
- Why surprising
- A major variational problem may have been solved in a 2.5-hour chat session, and the author says he published it mainly to establish priority.
What happened
Deangelis, who worked on Mumford–Shah regularity in his PhD, gave ChatGPT Astra his thesis, reference books and unpublished notes. First it proved a boundary-regularity conjecture of his. Then, given that result, it produced a full claimed proof of the interior conjecture. He posted the model's shortened manuscript almost unchanged, with a note on its provenance.
Why it matters
If correct, it would settle one of the best-known open problems in the calculus of variations. The case also shows OpenAI's unverified "100 open problems" announcement leading independent researchers to rush their own AI-generated claims onto arXiv before checking them.
Changelog
- 2026-09-30: created
Related events
- OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training; no list released ★★★
- OpenAI releases GPT-6 Astra, its first GPT-6 model ★★★★★
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
Sources (1)
id: 2026-09-22-mumford-shah-conjecture-astra-claim · updated 2026-09-30 · open in the interactive timeline