2026-10-11 17:15 UTC

OpenAI reportedly claims a mathematical breakthrough on a Millennium Prize problem, potentially establishing substantive progress on a major open problem through AI-assisted research without yet establishing a complete solution.

state: acceleratingheat: mediumuncertainty: highconvergesscott: highai-assisted-mathematics research-agentsOpenAISam Altman
Surfaced 2026-10-07T05:38:19Z — OpenAI publishes 722 mathematical proofs & manuscripts — The Oct 6 artifact release has crossed from announcement-plus-Wired into an accelerating four-platform wave (WSJ entry, HN traction building, velocity up ~7x in three hours, 95.6 peer percentile, magnitude-valve spread), and the retrieved first-party README pins its true scope: a versioned pipeline of 722 manuscripts across 372 families with a Lean-verified subset, the 'quasi-Riemann' preprint confirmed as a Re(s) > 11/12 zero-free region and the Hodge claim as the CM abelian-varieties case — partial results, not full conjectures. Meaning shift: OpenAI's math program is now a standing artifact-shipping operation whose live question is what survives independent kernel runs and expert review; validity remains single-sourced, but attention is now worth pricing at high.

What is this?

OpenAI claims its unreleased internal model 'Bel' produced a proposed solution to the Navier–Stokes Millennium Prize problem (September 2024, 10k agents, 88 hours) and, on October 6, released 722 manuscripts across 372 problem families with a Lean-verified subset, a quasi-Riemann preprint establishing a zero-free region for Re(s) > 11/12, and a Hodge conjecture result for CM abelian varieties. The Clay Institute has not validated any result; the Millennium Prize rules require a two-year waiting period after publication. Mathematicians including Tristan Buckmaster dispute attribution, 24 Fields Medalists signed a letter criticizing methods and understanding, and New Scientist alleges a critical translation error in the Lean formalization. No independent kernel audit of the 722-manuscript Lean subset or expert verdict on the zeta/Hodge items has been published; early readers describe the manuscripts as extremely hard to parse and many results as incremental.

Why it matters to Scott

OpenAI's Millennium program is a live, high-stakes stress test of Scott's verification-loop and proof-carrying frameworks: it ships Lean-formalized artifacts (722 manuscripts, quasi-RH, Hodge-CM) but lacks the independent kernel audits, adversarial verification passes, and governed production evidence that proof-carrying-transformation, claim-bounded-adversarial-verification, and evaluation-driven-development require. The AGMAI advisory group instantiates deterministic-first-ai-council and decision-authority-infrastructure; the Bel model's unreleased status and release rumors test model-perishability and sovereign-software-assurance; the 722-manuscript corpus is a wiki-is-the-kernel / institutional-memory artifact awaiting knowledge-promotion gates. This is not merely an example of Scott's patterns — it is a consequential case that would change what he builds (verification harnesses, agent provenance stacks, validated-release boundaries) and argues (publication gating, proof-carrying receipts, formalisation-bottleneck economics).
ip:concept.verification-loopsip:concept.formalisation-bottleneckip:framework.proof-carrying-transformationip:concept.proof-carrying-receiptsip:concept.evaluation-driven-developmentip:dev:concept.claim-bounded-adversarial-verificationip:dev:concept.deterministic-first-ai-councilip:framework.agent-provenance-stackip:framework.decision-authority-infrastructureip:framework.attention-native-publishingip:framework.sovereign-software-assuranceip:concept.model-perishabilityip:framework.wiki-is-the-kernelip:concept.institutional-memoryradar:concept.ai-assisted-mathematicsradar:concept.research-agentsradar:concept.lean-formalizationradar:concept.formal-verificationradar:concept.reproducibilityradar:concept.automated-theorem-provingradar:anthropic-fermat-lean-formalizationradar:hilbert-smith-lean-formalizationradar:qinz1yang-poincare-lean-formalizationradar:proofatlas-collatz-formalizationradar:odin-komlos-proofradar:theo-35-year-math-solutionradar:kbr-ai-formalized-math-proofradar:lean-transformer-ai-proofsradar:concept.ai-safetyradar:openai-catastrophic-risk-team-disbandingradar:openai-safety-insider-resignationradar:anthropic-uk-testing-access-withheldradar:anthropic-model-2-risk-delayradar:pacing-frontier-employee-letterradar:safa-frontier-safety-authorityradar:concept.open-weight-modelsradar:concept.sovereign-airadar:openai-open-weight-policy-shiftradar:openai-anthropic-open-weight-policy-alignmentradar:reflection-open-weight-releaseradar:concept.agent-memoryradar:concept.knowledge-systemsradar:karpathy-llm-wiki-adoptionradar:wenlan-source-backed-living-wikiradar:concept.local-inferenceradar:concept.inference-economicsradar:concept.frontier-model-releasesradar:openai-gpt6-sol-luna-releaseradar:gpt-6-astra-arc-agi-3-scoreradar:c5r-astra-research-facility
queries asked of Scott's wikis
  • ai-assisted-mathematics verification lean formalization
  • research-agents independent reproducibility artifacts
  • open-weights strategy model sovereignty regulation
  • agent memory maintained wikis research knowledge systems
  • local inference economics frontier model release
  • safety asymmetry withholding results publication gating

Measured heat

now 0 pts/hpeak 254 pts/hcomments 0/hpeers p50momentum: steady4 platformsage 818h
points/hour across evidence · reading as of 2026-10-12 02:59:37.977291+11:00 · deterministic, not a model opinion

How the heat travelled

10-07 00:22⭐ origin directly observedOpenAI publishes 722 mathematical proofs & manuscripts
socialistshroom on r/singularity
—
09-07 14:00first on x (echo) · published · +-706.4hOpenAI’s original announcement said: “We’re sharing a solution to the Navier-Stokes Millennium Prize Problem,” produced by a group of agents
OpenAI
—
09-09 22:21first on hacker news · published · +-650.0hOpenAI claims maths breakthrough on a famed 'Millennium Problem'
Levitating
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09-09 23:20first on r/singularity · published · +-649.0hOpenAI challenged to solve a few more Millenium Problems. Sam Altman's response: "ok fine"
borowcy
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09-11 05:11first on r/OpenAI · published · +-619.2hA kernel run on the Navier-Stokes blowup Lean project
pvaa
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09-13 08:44first on r/artificial · published · +-567.6hOpenAI's Millennium Prize proof has turned into a credit dispute, and Fields Medalists are now getting involved
CiccioPixel
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09-21 12:00first on openai · published · +-372.4hAdvisory Group on Mathematics and Artificial Intelligence
OpenAI
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09-30 16:59first on r/MachineLearning · published · +-151.4hOpenAI’s Lean 4 Navier-Stokes proof compiles with zero errors, but the fluid vaporizes at 0.7 nm. What does this mean for Neuro-Symbolic AI? [D]
OrganizationTop9026
—
09-09 22:21amplified on hacker newshn.story.49635362
Levitating
peak 2 · 0 comments · 0% of case engagement
09-09 23:20amplified on r/singularityreddit.post.1wc1u24
borowcy
peak 18 · 9 comments · 0% of case engagement
09-10 01:25amplified on hacker newshn.story.49637103
encodedrose
peak 1 · 1 comments · 0% of case engagement
09-10 04:19amplified on hacker newshn.story.49638353
tamnd
peak 283 · 161 comments · 4% of case engagement
09-10 09:23amplified on r/singularityreddit.post.1wce3o4
socoolandawesome
peak 1100 · 367 comments · 8% of case engagement
09-10 10:00amplified on r/singularityreddit.post.1wces4x
kacoef
peak 1 · 0 comments · 0% of case engagement
49 more amplifiers in ainews.case_chain
09-09 23:20our radar first saw it · +-649.0hdiscovery anchor: hn.story.49635362—
09-21 22:42reached heat=high · +-361.7h · via ledger——
pace: p99 vs 519 stories at the 720h mark (now 818h old) — ahead of anthropic-fable-mythos-51-release (1.1x), behind fable-51-multimodal-coding-cost (0.9x)

Evidence (58) — ⭐ canonical anchor

sourceobjectauthorscorecomments
🟧 hnOpenAI claims maths breakthrough on a famed 'Millennium Problem'Levitating20
🟠 redditOpenAI challenged to solve a few more Millenium Problems. Sam Altman's response: "ok fine"
singularity
borowcy189
🟧 echo.xOpenAI’s original announcement said: “We’re sharing a solution to the Navier-Stokes Millennium Prize Problem,” produced by a group of agentsOpenAI——
🟧 hnOpenAI has solved the Navier-Stokes Millennium problem using $15M of AI effortencodedrose11
🟧 hnOpenAI might have stolen another major prooftamnd28310
🟠 redditOpenAI’s Navier-Stokes Proof and Quantum Limit Reentrance ($ZrTe_5$) are Actually Connected.
singularity
kacoef10
🟠 redditSome more millennium prize problems possibly solved…
singularity
socoolandawesome1100366
🟠 redditIn its Wednesday night statement, OpenAI said: "In addition, since the completion of Navier-Stokes, we have made substantial progress on another Millennium Prize problem. We are working through how to share these results thoughtfully."
singularity
ilkamoi35692
🟧 hnOpenAI shares they have made substantial progress on another Millennium problemhelloplanets149
🟠 redditThe stolen millennium problem narrative is hilarious to me
singularity
Apollo18Teslaa42156
🟠 redditLooks like every Open AI maths breakthrough is going to be questioned by default
singularity
SGC-UNIT-555168214
🟠 redditRumors are spreading everywhere that OpenAI is close to verifying a proof of the Hodge conjecture, while either OpenAI or Anthropic may be nearing a solution to Birch–Swinnerton-Dyer. Is physics solved?
singularity
TheGoldenLeaper86104
🟠 redditBig news is that OpenAI is nearing solving another millennium prize, but OpenAI now saying categorically it’s impossible that Levent/Buckmaster’s recent codex usage influenced their model’s output
singularity
socoolandawesome21069
🟠 redditOpenAI is also using their new internal model to attempt the Riemann hypothesis and P versus NP, as well as some unnamed 'high impact problems'.
singularity
Neurogence5850
🟠 redditWhat do you think of mathematicians accusing AI companies of stealing their unpublished works?
singularity
Present_Award8001138
🟠 redditThe release of OpenAi's internal model 'Bel' is going to be the most hyped of all time.
singularity
imadade11298
🟧 hnAn NYU Mathematician Clashed with OpenAI over a $1M Proofalach1110
🟧 hnAsk HN: What Can We Do?silexia76
🟠 redditA kernel run on the Navier-Stokes blowup Lean project
OpenAI
pvaa021
🟧 hnOpenAI changed Navier-Stokes press release and Lean4 code on GHrawland10
🟧 hnTop mathematicians are outraged by OpenAI's methodssbulaev9420
🟧 hnA misalignment of AI in mathematicsmeredydd12421213
🟠 reddit24 Fields Medal winners sign letter titled "A Severe Misalignment of AI in Mathematics"
singularity
Charuru498495
🟠 redditUsed GPT-6 Astra to turn OpenAI’s Navier–Stokes solution into an animation
OpenAI
PossiblyAnAstronaut4224
🟠 redditOpenAI's feud with mathematicians is only escalating
singularity
joe4942168163
🟧 hn'Immature playground boasting': Mathematicians uneasy at OpenAI's latest scalpsbulaev40
🟧 hnAI Is Powerful Enough to Crack Our Hardest Math Problems–and Kill Us Alldiogenes_atx2338
🟧 hnOpenAI's Navier-Stokes claim splits the math communityledoncool30
🟠 redditOpenAI's Millennium Prize proof has turned into a credit dispute, and Fields Medalists are now getting involved
artificial
CiccioPixel10863
🟧 hnThe Navier–Stokes Millennium Prize ProblemEridanus221
🟠 redditScott Aaronson says that labs, "having been burned by the hostile response to the Navier-Stokes proof, are now sitting on solutions to some very major problems until they figure out a better way to handle things"
singularity
spryes633268
🟧 hnOpenAI breakthrough triggers 'existential crisis' in mathsbulaev40
🟠 redditScott Aaronson writes on his blog tonight that he has heard rumors that the AI companies have reached solutions to some very longstanding open problems in theoretical computer science, and are now sitting on multiple major announcements because of the Navier-Stokes firestorm.
singularity
TheGoldenLeaper377158
🟧 hnMath Can't Go on Like Thistaiwandongsuan31
🟠 redditOpenAI is getting close to solving another Millennium Prize problem, the Hodge Conjecture
singularity
Distinct-Question-16513209
🟧 hnOpenAI close to solving Millennium Prize problem Hodge Conjecturedude25071120
🟠 redditOpenAI breakthrough triggers ‘existential crisis’ in math
artificial
Fcking_Chuck04
🟧 hnAdvisory Group on Mathematics and Artificial Intelligencejs73js87767
🟠 redditOpenAI solved 100 open problems in math
singularity
filterdust1282505
🟧 openaiAdvisory Group on Mathematics and Artificial Intelligence
Retrieved article excerpt

Open article · Retrieved 2026-09-21T22:21:34.328733+00:00

September 21, 2026

[Company](https://openai.com/news/company-announcements/)

# Advisory Group on Mathematics and Artificial Intelligence

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On August 28, we began training a new internal model. In addition to [resolving the Navier–Stokes Millennium Prize problem⁠](https://openai.com/index/navier-stokes-solution/), this model has now resolved more than 100 long-standing open problems across most areas of mathematics. The [pace of its progress⁠](https://openai.com/index/navier-stokes-solution/#astra-internal-model-open-math-problems) in mathematics has surprised the mathematicians within OpenAI. This has led to internal discussions on the best way to inform the community of the rapid progress to prepare and adapt the field.

Mathematics is a fundamental science, and novel discoveries may result in a wide range of applications. That makes the promise of these capabilities substantial, and their responsible development and deployment important beyond mathematics itself. We recognize this and are working through how broader deployment of math-related AI capabilities can be done responsibly.

In a recent open letter “[A Severe Misalignment of AI in Mathematics⁠(opens in a new window)](https://mathandai.org/),” mathematicians raise concerns about the negative externalities of solving open problems as a benchmark for new AI systems.

Their criticisms highlight the need for thoughtful engagement of AI companies with the math community. To that end, we’re working with mathematicians who have established an independent mathematics advisory group. This group will serve as a bridge to the mathematical community and broader public, giving mathematicians a voice in how we move forward.

The group will advise on the review and communication of emerging results: they will help OpenAI assess their significance, advise on how to coordinate their dissemination, and advise on academic and professional standards of mathematical research. It will also advise on how our tools can support mathematical research and learning. We want to put capable tools in mathematicians’ hands so they can pursue the questions they know best and develop new ideas.

The group will operate independently from OpenAI. The group will have the freedom to offer advice we have not requested, comment on OpenAI’s impact on mathematics, and make its advice public. Its value depends on its members being able to exercise their own judgement and challenge ours. Its members will not be paid by OpenAI, and the group can change its membership as it sees fit. Importantly, the group will not be responsible for advising us on how to pace our internal progress on mathematics.

Working with this group is a first step. There are difficult questions ahead about how AI can support mathematical understanding and how the benefits of these capabilities can reach the wider community. We want mathematicians to be at the center of shaping the answers.

**Initial Members of the** [**Advisory Group on Mathematics and Artificial Intelligence**⁠(opens in a new window)](http://agmai.org)**, hosted at the** [**Institute for Advanced Study**⁠(opens in a new window)](https://www.ias.edu/nccr/projects#agmai)**:**

- François Charles (ENS-PSL)
- Camillo De Lellis (IAS, GSSI)
- Timothy Gowers (Collège de France, Cambridge)
- Martin Hairer (EPFL, Imperial College London)
- Nikhil Srivastava (Berkeley, Simons Institue)
- Ulrike Tillmann (Oxford, INI)
- Ravi Vakil (Stanford)
- Edward Witten (IAS)
- Melanie Matchett Wood (Harvard)

- [2026](https://openai.com/news/?tags=2026)

## Author

OpenAI

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🟠 redditErdos problem solver experienced sleepless nights over math internal OpenAI model has solved
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🟠 redditOpus 5.5 reportedly caught OpenAI off guard, compressing GPT-6.1 Astra’s timeline and indirectly bringing “Bel,” the model that solved Navier–Stokes, closer to release
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141_1337704153
🟧 openaiSam Altman’s remarks at the United Nations Security Council
Retrieved article excerpt

Open article · Retrieved 2026-09-28T13:21:26.006431+00:00

September 23, 2026

[Global Affairs](https://openai.com/news/global-affairs/)[Safety](https://openai.com/news/safety-alignment/)

# Sam Altman’s remarks at the United Nations Security Council

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Today, OpenAI CEO Sam Altman addressed the United Nations Security Council on artificial intelligence. He discussed AI’s potential to expand opportunity, the importance of keeping powerful systems under human control, and the need for international cooperation on AI safety.

## Remarks as delivered

Thank you, Mr. President and your Excellencies. I’m honored to be here and to be able to speak with you all at this important time.

We have been talking about artificial intelligence for years, but it feels different in recent weeks and months. Rapid model progress has made the timeline feel more compressed, the upside more tangible, but also the stakes and the risks more immediate.

We have a choice in front of us. AI can either be more like a new Renaissance of creativity and discovery, or more like a new Industrial Revolution of upheaval and disarray.

The Renaissance was a time when new tools, new ideas, and new institutions expanded what people believed they could do. AI can play that role for our time. I believe that people are much more capable than they imagine themselves to be, but have been held back by limited technology that was meant to empower but instead distracts and often steals attention.

The best version of AI is not about making people cogs in a giant machine, or optimizing every part of life until the human parts disappear. It is about giving people more agency in an ever more complex world: more ability to learn, to create, discover, build, participate. There are many things that AI cannot—and should not—automate.

AI should be built for people, not simply to turn the crank of the machines faster for their own sake. Some of the things people working to build AI have said are so dystopian that they sound like the plot of bad science fiction movies.

Anything this powerful is also intimidating. This moment is complicated as we feel both tremendous potential and very understandable anxiety at the same time.

These models can help people discover new knowledge: new medicines, better health care, stronger education, and build more productive economies. They can be new tools for people to pursue dreams and ambition that today feel out of reach. The complexity of life has increased so dramatically for the average person. AI can be a defensive tool against this complexity as much as it can be this tool of unprecedented discovery and innovation.

On the other hand, as AI systems become more capable and more autonomous, they could move faster than our institutions, concentrate power in too few hands, or make decisions that people no longer understand or control.

That concern becomes especially important as we approach systems that can improve themselves and future versions of themselves, often called recursive self-improvement. As the process of building AI becomes more automated, the pace of AI progress could accelerate rapidly. This moment calls for extreme care.

I largely agree with Professor Bengio. Although I’ll say two instead of three—I believe there are two ways that AI progress could go very badly that we have to avoid.

First, we could lose control of the future to AI. The risk is that it moves so fast that people can no longer follow what’s happening or intervene when needed. This would obviously be terrible.

The industry must not accept too much technological risk just because the benefits are too great and that they feel too important to slow down. Beating companies in a competitive pace is not a reason to make rash decisions. Nor do we believe we are locked in a race where we are unable to do that. We have unilaterally slowed down in the past. We will do so in the future. We do not want to build systems—we think it’s important that everyone avoids building systems—where we cannot make the alignment, monitorability, and safety guarantees that we must.

We need to understand what these systems are doing and have strong evidence that they will do what people intend, even as they get very, very smart. Actually especially as they get very very smart. It doesn’t matter whether people put the risk of catastrophe at 10%, or 1%, or 12%, or .1%. None of these levels are remotely acceptable. And we should not train models that we cannot make an extremely strong case that we will be able to keep under human control.

So that’s one way things can go wrong. In the other direction, these systems could concentrate too much power in too few hands. No one person or company or country should be able to use the most powerful AI models to impose their worldview on everyone else. A company or country that believes only it can be trusted with this technology can use that belief to justify almost anything else. We have to reject that logic, even when it’s convenient for any one actor.

I believe there is a pragmatic path through both of these. At OpenAI, we approach these questions with a clear goal: AI should give people more power over their own lives. More freedom to live lives the way they want. More ability to solve problems, big or small, and to improve their communities. We don’t want to fall into the trap of blind optimism. We also don’t want to fall into the trap of doomerism either. Getting pulled into either extreme will not help us successfully figure out the challenges in front of us, especially at a time when the world and its people face such enormous challenges that this technology can help solve. So we try to walk the middle path between these two poles.

For us, that means three principles.

First, as AI becomes more capable, people must remain at the center of AI decision-making. Alignment is not an abstract research question. It is the work of ensuring that these systems reliably remain under human control, reflect human values, and help people guide the next stages of development. We are not trying to, and must not, automate human judgment, or human values.

Second, the benefits of scientific progress and economic growth must be by people and for people. AI should help researchers make discoveries, doctors treat patients, teachers teach students, entrepreneurs start companies, communities solve problems that felt out of reach, and much more. The continuous story of human progress is not that tools replace human agency. It is that the right tools make people more capable, and I believe this is going to go much further than we believe possible.

Third, this technology must empower people individually. A great future will come from people realizing their potential and building value for themselves, their communities, their countries, and the broader world. We might be one of the builders of this technology, but we are not the heroes of this story. Our role is to enable people all over the world and to trust in the magic of humanity’s skill and faculties.

We now have models capable of discovering knowledge that we have never had before. Consider the progress we’ve seen in math. Three summers ago, our models were okay at grade-school math. Two summers ago, they were pretty good at high-school math competitions. Last summer, we reached a gold-medal level in the most prestigious international math competition. And just a few weeks ago this summer, one of our models solved one of the Millennium Prize Problems, the Navier-Stokes equations.

These equations are used for aircraft design, weather forecasting, and the study of blood flow. We are not solving math problems for their own sake, but to empower people to discover new knowledge and to use it to improve healthcare, raise standards of living, and expand everyone’s potential around the world. There remains so much to be discovered.

But it’s very important that companies not substitute for the democratic process.

If AI is to be democratic, the most important decisions cannot be made by labs in San Francisco alone. They must be shaped through democratic processes, and by governments accountable to the people that they serve.

At the international level, this will require cooperation. In our history, there have been times where countries who compete and don’t always like each other very much still come together for shared interests and the collective good in the face of a powerful new technology. We believe this must be one of those times.

We need a mechanism for complementary national and international frontier AI standards: standards for measuring capabilities, assessing risks, determining whether safeguards are sufficient, and preserving meaningful human oversight as systems become more autonomous.

We need common standards so countries can compare evidence, verify compliance, and have a shared language and understanding about what is happening.

We need accurate and speedy incident reporting, classification and reporting protocols, so the world can learn from failures before they become catastrophes.

And we need secure channels among governments, critical infrastructure operators, and technical experts to share emerging vulnerabilities and new threats.

Obviously, these standards should not lock in incumbents or favor one business model over another. They must support open and closed model developers, new entrants and established labs, national authority and international cooperation. Each government should decide how to incorporate standards into its own legal system. But we will all be better off if we can agree on what good evidence, good safeguards, and good oversight look like on the global stage.

We do not have all the answers and we are very open to any better ideas. We are ready to work with you all and try our hardest to figure out the path forward.

We are at a crossroads. One path leads to a world where power is concentrated, decisions are opaque, and people feel that the most important technology of their lifetime is something being done to them. The other path leads to a world where AI works for people because people, through democratic institutions, help decide how it should develop and guide it; where scientific progress improves real lives; where more people can participate in the future; and where this technology continues to give individuals and communities greater power to live life as they choose. This is a future of realized potential, excitement, and wonder.

And it is the future we believe is worth building. Thank you very much.

- [Global Affairs](https://openai.com/news/?tags=global-affairs)

## Author

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OpenAI——
🟠 redditWhat happened to solving the Hodge Conjecture?
singularity
Strict_Cucumber91177831
🟠 redditOpenAI’s Lean 4 Navier-Stokes proof compiles with zero errors, but the fluid vaporizes at 0.7 nm. What does this mean for Neuro-Symbolic AI? [D]
MachineLearning
OrganizationTop9026021
🟠 redditOpenAI Is Pissing Off a Bunch of Mathematicians—Again
OpenAI
wiredmagazine89206
🟠 redditafter navier-stokes last month, their internal model now claims pieces of two more millennium problems
OpenAI
physiopeng16179
🟧 hnThe Quasi-Riemann Hypothesis [pdf]akalin30
🟧 hnOpenAI releases 722 math manuscriptsjumploops91
🟧 hnOpenAI just dropped 700 preprints of mathematical proofs and counterexamplestootie402
🟧 hnWSJ: OAI Just Cracked Hundreds More Math Problemstalon863522
🟠 reddit ⭐OpenAI publishes 722 mathematical proofs & manuscripts
singularity
Retrieved article excerpt

Open article · Retrieved 2026-10-07T03:33:21.092766+00:00

# Readme

This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model.

As part of model development, we evaluate our models on open research problems. We expanded these evaluations after performance on our existing mathematical evaluations saturated. Some outputs build upon earlier results produced by the models.

This collection includes results at different stages of verification. Not all have accompanying Lean formalizations. We will continue to update this repository with Lean formalizations as we obtain them.

Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly.
We are also exploring community-hosted repositories for these materials.

## Navigating the collection

The current catalogue contains 722 manuscripts organized into 372 families. A family groups related papers, which may include a principal result, companion arguments, consequences, or alternative proofs. Each family is classified by mathematical discipline.

- Start with the [overview](https://github.com/openai/math/blob/main/overview.pdf) for descriptions of the families.
- Use the [manuscript map](https://github.com/openai/math/blob/main/CONTENTS.md) to find individual papers and their supporting materials.
- The [`preprints/`](https://github.com/openai/math/blob/main/preprints) directory contains PDFs, source files, and manuscript-specific citation and build instructions.
- The [Lean library](https://github.com/openai/math/blob/main/lean/README.md) and [formalization catalogue](https://github.com/openai/math/blob/main/lean/formalization.yaml) describe the available formal proofs, their associated papers, and verification configurations. See the [Comparator instructions](https://github.com/openai/math/blob/main/lean/ComparatorChallenges/README.md) for additional checking instructions. Many, but not all, of the manuscripts have been formalized.

### Reasoning summaries

We are also releasing abridged summaries of the model's reasoning, covering the following results:

| Family | Subject |
| --- | --- |
| 007 | [Ordinary two-point correlations of multiplicative functions](https://github.com/openai/math/blob/main/reasoning_traces/ordinary-two-point-correlations.pdf) |
| 017 | [The irrationality exponent of π](https://github.com/openai/math/blob/main/reasoning_traces/irrationality-exponent-of-pi.pdf) |
| 087 | [Symmetric and general Mahler conjectures](https://github.com/openai/math/blob/main/reasoning_traces/symmetric-and-general-mahler-conjectures.pdf) |
| 102 | [Ordinary NP-hardness at the basic semidefinite threshold](https://github.com/openai/math/blob/main/reasoning_traces/basic-semidefinite-threshold-np-hardness.pdf) |
| 159 | [Quasipolynomial bounds for arithmetic progressions](https://github.com/openai/math/blob/main/reasoning_traces/quasipolynomial-arithmetic-progressions.pdf) |
| 197 | [Kaplansky's direct-finiteness conjecture in characteristic two](https://github.com/openai/math/blob/main/reasoning_traces/kaplansky-direct-finiteness-characteristic-two.pdf) |
| 221 | [The Mézard–Parisi formula for diluted spin glasses](https://github.com/openai/math/blob/main/reasoning_traces/mezard-parisi-formula.pdf) |
| 271 | [Spontaneous magnetization in the quantum Heisenberg ferromagnet](https://github.com/openai/math/blob/main/reasoning_traces/spontaneous-magnetization-quantum-heisenberg-ferromagnet.pdf) |
| 287 | [Isomorphism of free group factors](https://github.com/openai/math/blob/main/reasoning_traces/free-group-factor-isomorphism.pdf) |
| 362 | [The three-dimensional relativistic Vlasov–Maxwell system](https://github.com/openai/math/blob/main/reasoning_traces/relativistic-vlasov-maxwell.pdf) |

## How the results were produced

The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model. On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above.

Exceptions to this fixed procedure include work on a zero-free region for the Riemann zeta function and proof of the Hodge Conjecture for CM abelian varieties. Additionally, the writeup for the Re(s) > 11/12 zero-free region for the Riemann zeta function was human edited for readability.

## Versions and citations

We will preserve the public release history of this collection. Corrections and revisions will be recorded as new versions, with previously released versions remaining accessible.

To cite the individual manuscript, use the BibTeX block in its directory.
socialistshroom663123
🟧 hnOpenAI Mathematics Manuscript Collectionaltro30
🟧 hnAI Solved One Math Problem and Everyone Freaked Out. It Just Cracked 100s Morefortran7733
🟠 redditThe Mathocalypse
singularity
TMWNN16533
🟧 hnThe Mathocalypse6bitquant347362
🟠 redditProf. Tristan Buckmaster on the controversy over OpenAI's math findings
OpenAI
CuTe_M0nitor138
🟧 hnOpenAI mistranslated mathematics into code for its Navier-Stokes proofdanielmorozoff555

Interpretation history

Decision trace