OpenAI bulk release of mathematical papers
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On 7 October 2026, American artificial intelligence company OpenAI released 722 manuscripts claiming to have solved a large number of open and unsolved major mathematical problems. The release was preceded by the Navier–Stokes priority controversy in which the company's claim to solving the Navier–Stokes equations was disputed by researchers who were about to release a similar breakthrough. Many months may be required for academics to fully assess the impacts of the event and quality of the breakthroughs.[1] Within a day, three of the manuscripts have been retracted and an additional 14 manuscripts were revised.
Background
In 2026, tens of thousands of AI generated mathematical solutions were released.[2][verification needed]
Navier–Stokes
On 8 September 2026, OpenAI claimed a solution to the Navier–Stokes existence and smoothness problem with a smooth external force.[3][4] However, approximately 12 hours before OpenAI's announcement, mathematician Tristan Buckmaster claimed that his and Levent Alpöge's recent advances on Euler equations[5] were leaked to OpenAI a few days earlier and may have inspired OpenAI's prompts for the agents in the process that led to the solution.[6][7] OpenAI stated they began large-scale work on all Millennium Prize Problems on 1 September, following rumors that Anthropic had solved two such problems.[8][9][10][11][12] In response, a declaration was published on 11 September and signed by 28 Fields Medalists, titled A Severe Misalignment of AI in Mathematics, warned against "detrimental" impact on mathematics from AI companies treating unsolved problems as benchmarks without enhancing human understanding.[13][14][15]
Expectation
In a September blog post, they claimed an internal frontier model had resolved 100 long-standing open problems in mathematics.[16] Thus, the results of these solutions had been expected for weeks.[17] OpenAI announced they were working with a new advisory board of mathematicians to publish research with best practices.[18] OpenAI spokesperson Lindsay McCallum claimed the company was "not aware of" any plan to release all the solutions at once.[2]
Release
On 7 October, OpenAI released 722 manuscripts organized into 372 topics.[17] The topics spanned algebra, number theory, theoretical computer science, mathematical logic and topology.[18] The model used for the research was internal, and not released publicly. OpenAI acknowledged that future results would have citations written descriptions of findings; the release did not include contextual information like prompts or run-time details.[19] The papers were released onto code-sharing platform GitHub.[20]
Validity
Verifying and confirming whether the research is valid will take a long time, given some pages are hundreds of pages long and with dense calculations.[1] OpenAI stated that not all results in the collection had been formalized in Lean and that unformalized results could contain errors.[21] Upon release, OpenAI researchers had not been able to read through all the papers.[19] Three papers were retracted within a day as well as an additional 14 papers were revised.[1][22] Only about half of the manuscripts were certified as correct by the automated logic platform Lean, which computationally verifies proofs.[19][22]
Claimed results
Examples of topics covered by the papers include:[23][24][25]
- A new result on zero-free regions of the Riemann zeta function, Dirichlet L-functions and Hecke L-functions for . Precisely, the papers claim that all nontrivial zeros should lie in the restricted critical strip: . If true, the new estimate would imply non-existence of Siegel zeros, leading to several results in analytic number theory.[26][27][28]
- Partial result on the Birch and Swinnerton-Dyer conjecture: elliptic curves over rational numbers whose Selmer group has corank 0 or 1. For such curves both the rank of curve and multiplicity of zero at of associated L-function are claimed to equal this corank and thus be equal to each other. The papers also include a proof of the conjectural formula for leading coefficient of the Taylor expansion at this zero, together with a proof of finiteness of the Tate-Shafarevich group for such elliptic curves.
- Partial result on the Hodge conjecture: for complex projective abelian varieties of CM type every class in Hodge cohomology is a rational linear combination of cohomology classes of algebraic subvarieties.[29]
- A proof of the Hilbert-Smith conjecture related to Hilbert's fifth problem: every locally compact topological group acting continuously and faithfully on a connected topological manifold is claimed to be a Lie group.
- Disproof of the Hadwiger conjecture: the paper claim to introduce a construction of loopless graphs that have no as a minors, but are not -colorable.[30]
Response
Academic responses were mixed.[vague][31] Some mathematicians focused on the progress of the papers. Melanie Wood, a member of the advisory group that OpenAI claimed it would work with, stated it was "the beginning, not the completion, of the process of human understanding and the incorporation of the work into mathematical knowledge".[18]
Scott Aaronson, chair of computer science at the University of Texas at Austin, stated:[23]
If you want some sense of what things feel like now in maths, imagine a hunter-gatherer who's spent his entire life learning to survive deep in an unforgiving rainforest, then a giant resort hotel springs up right next to him with a helipad and heated pools and Airbnbs.
Other mathematicians were more skeptical about the process of release. Tristan Buckmaster, a mathematician at New York University, stated "I don't think they've done their sort of due diligence at all" given so many results were released at once.[18] Andrew Sutherland, a mathematician at MIT, stated "Until and unless they release the model and people can replicate their results, I think you should treat any claims about one-shotting problems with a single agent as unverified".[19]