OpenAI Astra Solves 10 Longstanding Math Problems Including Non-Sofic Group Discovery

OpenAI announces solutions to 10 longstanding maths problems | New Scientist

OpenAI has revealed solutions to 10 longstanding mathematical problems that were found by its prototype AI model Astra. The announcement is the latest in a string of mathematical discoveries made by AI that are threatening to change the field beyond recognition. In May, an OpenAI model cracked a decades-old conjecture by Paul Erdős, causing a stir in mathematical circles. Last month, the Claude Fable 5 AI found a counterexample to the Jacobian conjecture, which had stood for nearly a century. Hundreds of other AI-led discoveries have been made in recent months. The newest solutions from OpenAI tackle problems ranging from how densely spheres can be packed into spaces with more than three dimensions to quantum game theory. But the one receiving the most attention is the discovery of a non-sofic group. Soficity, first described in 1999, is the property of a group of operations that can be approximated by smaller, finite groups of permutations. You can think of it like a game played on an infinitely large chessboard being loosely approximated by games on a small one. Until now, all known groups had this property, so mathematicians proposed that all countable groups are sofic. Now OpenAI has discovered a single counterexample that disproves the claim. The news has sparked further hubbub among mathematicians who are having to adjust to a radical shake-up in their field. Elon Musk even said in a tweet that it was evidence we have reached the singularity - the point at which AI becomes self-improving and advances towards general intelligence at an accelerating pace. Francesco Fournier-Facio at the University of Cambridge has been studying the soficity problem since starting his PhD in 2020, and says he may well not have stayed in academia if AI had made this discovery back then. He is concerned that AI companies, particularly in this case, aren't being transparent about how their solutions rely on prior human work. Fournier-Facio thinks OpenAI's solution to the soficity problem relies heavily on papers by Andreas Thom and Gábor Kun that pushed the field ahead significantly, and that it is their work that should be celebrated more than this discovery of a counterexample. "In very broad strokes, the solution takes these two works from 2016 and 2019, pushes them forward and then does some clever tricks," says Fournier-Facio. We will never know if humans could have arrived at the counterexample, but AI certainly couldn't have if not for prior human work, he says. OpenAI's initial announcement claimed that all 10 solutions "have seen no progress on the main result for at least a decade", but Fournier-Facio complained this was incorrect and the company has since changed its statement. "Until AI shows that it can develop theory independently, it's hard to believe that it could have come up with this [counterexample] independently," says Fournier-Facio. He points out that many AI mathematical discoveries so far have focused on finding counterexamples, which can be easily and quickly checked, rather than developing new theory. "Developing theory, it's very much less clear [if it is correct]: there's no tick at the end. How do you know if you have developed the right theory, if you're going a step in the right direction?" asks Fournier-Facio. Abhishek Saha at Queen Mary University of London says human mathematicians had also made significant progress with high-dimensional sphere-packing, and Astra has built on that work to arrive at a solution. Nevertheless, the release of these 10 solutions is still the most impressive display of AI mathematical prowess to date, he says. "Any one of them would be a significant and impressive achievement," says Saha. "Some of them are not counterexamples; some of them are actually proofs, but they are all of the kind where you don't have to build a huge new amount of theory. Instead, you [have to] very technically put together things that have been done, in unusual ways, and in very technical ways, and do something that no one had done before." Saha is optimistic about the future of AI models in mathematics. "I would not be surprised if, in two or three years, they can actually build enough new theory to solve some of the deeper questions," he says. OpenAI did not respond to an interview request for this article.

OpenAI teases Astra, its next major AI model, after it solves 10 long-standing math problems

OpenAI has revealed Astra, an unreleased model designed to tackle complex, long-running tasks, after an internal version produced ten significant advances in mathematics and theoretical computer science. In a new research post, OpenAI described Astra as "our next major model" and said the problems had seen no progress on their central results for at least a decade, and in most cases, much longer. According to OpenAI, its internal research focused on a wide range of areas, including high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, quantum complexity, lattice cryptography, and extremal combinatorics. OpenAI also says it's advancing rapidly in science. Some examples include the existence of non-sofic groups, a disproof of Connes's rigidity conjecture, new bounds for high-dimensional sphere packing, and results resolving several problems posed by mathematician Paul Erdős. "The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates," OpenAI noted. Human researchers used the same model to prepare the arguments as manuscripts. Astra then formalized every argument as a Lean certificate, allowing the proofs to be checked using the mathematical verification system. Astra could launch as GPT-5.7 or GPT-6 The Information also independently confirmed that OpenAI is indeed working on Astra, a new model family built for long-running workloads. As per OpenAI, Astra is a powerful model that allows AI agents to collaborate on different parts of a larger problem. BleepingComputer understands that OpenAI has reportedly not decided whether the model will be released as GPT-5.7, GPT-6, or under another name. At this point, we know that Astra qualifies as a major breakthrough AI model, and it could be subject to Anthropic-like policies where one version is released to consumers, while a more powerful variant requires special approval.

OpenAI says its next model, Astra, has solved ten open problems in mathematics

OpenAI says its unreleased Astra model solved ten open maths problems, shipping Lean proofs on GitHub for roughly $2,000 in compute OpenAI says an internal version of its next major model, called Astra, has produced ten new results in mathematics and theoretical computer science. Each of the problems had been open for at least a decade. The company published a 249-page manuscript alongside machine-checkable Lean 4 certificates for every result on GitHub. The headline result is the first-ever explicit construction of a non-sofic group, resolving a central question in group theory that has stood since Mikhail Gromov introduced the concept of soficity in 1999. No mathematician had managed to prove or disprove whether non-sofic groups exist in the 27 years since. The other results span several fields. Astra disproved Connes's rigidity conjecture on von Neumann algebras, proved Ehrhart's volume conjecture, and resolved three problems from Paul Erdos's famous catalogue, including problem number 183 on multicoloured Ramsey numbers. It also produced the first improvement to the general upper bound on high-dimensional sphere-packing density since 1978, proved a parallel repetition theorem for two-player quantum games, and established new lower bounds on the circuit complexity of computing the permanent. OpenAI's head of mathematics research, Sebastien Bubeck, confirmed the results on X, calling them "beautiful" and noting that each ships with a Lean certificate and a chain-of-thought walkthrough. The total compute cost for all ten solutions was roughly $2,000 at Sol API rates, according to OpenAI. The announcement lands against a backdrop of escalating tension between AI companies and the mathematics community. In June, mathematicians issued the Leiden Declaration, endorsed by the International Mathematical Union, warning that AI companies are using published research without consent, bypassing peer review, and threatening the integrity of proof and attribution. The Declaration specifically cited companies that announce results through press releases rather than peer-reviewed journals. OpenAI has form on this front. In May it announced that the same long-horizon model family disproved the Erdos unit distance conjecture, an 80-year-old problem in discrete geometry. Fields Medalist Tim Gowers said at the time that he would recommend that proof for publication in Annals of Mathematics without hesitation. Thomas Bloom, who runs the erdosproblems website, called the latest ten results "big news" on X, saying they are more significant than the unit distance counterexample. OpenAI has not said when Astra will be released publicly, describing it only as its "next major model." Some observers, including investor Mark Kretschmann, have speculated that Astra is the GPT-6 series. The company is also giving 100,000 academic researchers free access to its frontier models through 2027, a move that deepens its ties to the scientific community while concentrating research infrastructure on its own platform. The Lean certificates address a key objection that the mathematical community has raised about AI-generated proofs: that they are difficult to verify independently. Machine-checkable proofs can be validated by anyone with the Lean compiler, without trusting the model or its operators. Whether the broader mathematical community will accept results announced through a blog post rather than a peer-reviewed journal remains an open question, one the Leiden Declaration was written to answer.

OpenAI Smuggled the Announcement of Astra, Its Next AI Model, Into a Blog Post About Math

OpenAI announced its next major AI model Saturday, and it did so in the third paragraph of a blog post called "Ten advances in mathematics and theoretical computer science." The math results the post is touting, OpenAI writes, "were achieved by an internal version of Astra, our next major model." So there you go. It sounds like after GPT-5.6 Sol comes either GPT-5.6 Astra, or GPT-6 Astra, or -- who knows? -- just "Astra" and the whole GPT part gets scrapped? It's not spelled out. OpenAI's naming conventions suggest that this would be another GPT-5.6 release. There's a GPT-5.6 Terra, which, as corny Latin-knowers are well aware, means "earth," Luna which is Latin for "moon," and Sol, which is Latin for "sun." Astra means "the stars." According to an anonymously sourced story in the Information, Astra boasts the ability to do "long-running" work. CEO Sam Altman was, the report claims, in Washington, D.C. over the past week, demoing the model to federal officials. Gizmodo asked OpenAI on Saturday for the official name of the model. We also asked OpenAI about the relationship between this model and another OpenAI model only vaguely described in one of that company's blog posts. We did not receive a reply. The "unprecedented cyber incident" covered in that July 21 blog post is already infamous. An entity described as a "combination of OpenAI models -- including GPT‑5.6 Sol and an even more capable pre-release model, all with reduced cyber refusals for evaluation purposes" compromised the AI resource depository Hugging Face during a model evaluation exercise that was supposed to remain inside OpenAI. A later update to the blog post clarified that the unreleased and unnamed model involved in the incident was an "internal-only research prototype and was never intended for public release," and that it had been "deactivated, encrypted, and restricted." So to be clear, Astra is not the model that broke into Hugging Face. Gizmodo asked OpenAI on Saturday to more fully clarify the relationship or lack thereof between Astra and the never-to-be-released model. We did not receive responses in time for publication, but will update if we receive a clarifying answer. As for the blog post about math, it comes with a paper. There are ten proofs covered, covering such topics as the "asymptotic strength of the Cohn-Elkies linear program," for sphere-packing, which OpenAI purports to be "determined exactly." That sounds very cool, but I'm just the guy who blogs nights and weekends for Gizmodo. For what it's worth, OpenAI published a mathematical disproof back in May, purportedly solved by an unnamed OpenAI model. For the most part, math folks wrote about the model's work approvingly, but didn't seem completely knocked out. For instance, Harvard mathematician Melanie Matchett Wood wrote that OpenAI's proof was, "a beautiful application of number theory to a natural, concrete question," but she also said the problem, described by OpenAI as "a central conjecture in discrete geometry" was nothing she had ever heard of before. She added: "This result does not show us all the times AI has claimed to have a proof of something and been wrong. Without that context (which many of us have just from personal experience), it is also easy to draw incorrect conclusions about the current state of AI and research mathematics."

OpenAI Astra: The mysterious new quantum math-solving model

Do you understand quantum parallel repetition? What about quantum complexity, lattice cryptography, or extremal combinatorics? The new OpenAI model Astra knows all about them. Recently, OpenAI confirmed the existence of Astra, calling it "our next major model." Over the weekend, the ChatGPT-maker revealed that "an internal version of Astra" solved 10 major open math problems, some of which have been unresolved for decades. This follows OpenAI's earlier AI-generated disproof of the Erdős unit distance conjecture, another major math accomplishment. Our big Guessing Game is back! Enter now for a chance to win an Apple Watch. What do we know about OpenAI Astra? Precious little, so far. Besides its internal name, Astra, and the fact that it's currently in testing to become OpenAI's next major model, the company has released few concrete details. At this point, it's unclear if Astra will be released as GPT-5.7, the beginning of GPT-6, or something else entirely. (The company has also so far declined to answer our questions about Astra, but we'll update this story if we receive more information.) However, we can make some deductions based on OpenAI's recent blog post, "Ten advances in mathematics and theoretical computer science." We also talked to a mathematician about what the company's results mean for the future of AI and science -- and what it doesn't mean -- which we will explore shortly. Bleeping Computer recently described Astra as "a powerful model that allows AI agents to collaborate on different parts of a larger problem." That suggests it's designed for agentic work and can "tackle complex, long-running tasks," also per Bleeping Computer. For now, it's also safe to say that OpenAI's new Astra model has made significant advances in science and mathematics. New Anthropic and OpenAI models are really good at coding and math Earlier this year, Anthropic announced that its unreleased Mythos model was so good at hacking that it was too dangerous to release to the public. At the time, we questioned this narrative, as the company's warnings effectively doubled as PR for itself. However, there's no denying that the latest frontier models from Anthropic and OpenAI have gotten remarkably good at cybersecurity coding and discovering zero-day bugs. More recently, OpenAI and Anthropic have shown progress on research-level mathematics as well. First, OpenAI released a disproof of the aforementioned Erdos unit-distance conjecture in May. Then, an Anthropic-linked mathematician casually announced that he used Fable 5 to disprove the Jacobian Conjecture, one of the infamous Smale's problems in mathematics. Now, OpenAI has announced that Astra solved 10 more open problems in mathematics. It's a potentially paradigm-shifting moment, though this is hardly proof that artificial general intelligence or the singularity is nigh. First, most of these math accomplishments take the form of disproofs and counterexamples, which aren't as impressive as positive proofs that greatly expand our understanding of the universe, like, say, Andrew Wiles' proof of Fermat's Theorem. When Fable 5 disproved the Jacobian Conjecture, I spoke to Columbia professor Andrew Blumberg, who is also on the board of directors of the First Proof project, which tested the capabilities of frontier large language models in solving research-level mathematics. At the time, he told me that Fable 5's feat "did not cause me to update my priors about what AI can and can't do." Blumberg added, "This is exactly the kind of thing I would expect AI to be able to do. If there was a counterexample that was concise and easy to state that people haven't found because it's a pain to search through all this stuff, AI will find it." Ultimately, Blumberg said that Fable 5's counterexample to the Jacobian Conjecture didn't necessarily teach us anything new or exciting about the world, even though it is very impressive. I went back to Blumberg to ask about OpenAI's latest work, and he said his priors still haven't changed -- AI is a very useful tool for advanced science, but this doesn't suggest AI is ready to replace human scientists and mathematicians. "If you look at what these are, they're pretty short. They are either counterexamples or they are small, extremely clever constructions that build on known things, and it's super cool, right? I want to be clear: If a person had done many of these things, that person would be justly lauded for their achievement, and so [this work is] great, but they don't change my priors." And, as many people have pointed out (including, most recently, data scientist Nate Silver), we have to put these results in perspective. While OpenAI was keen to point out that these problems were solved with just $2,000 worth of tokens, that doesn't account for the trillions spent on AI technology in recent years. "When you hear about the amount of money that's being poured into these machines, and you think about what it would be like if we spent a trillion dollars on hiring and training mathematicians and paying the best mathematicians football players' salaries to do nothing but solve these problems, I think we'd see a shitload of progress," Blumberg told Mashable. At the same time, the prospect of Astra-level models being in the hands of every scientist, physicist, and mathematician on Earth is exciting. When will GPT Astra be released? That's hard to say, but we can make an educated guess. First, the White House is reportedly close to finalizing a voluntary AI framework for testing new frontier models before they're released publicly. OpenAI is very likely to adhere to these guidelines. And if Astra is deemed to have dangerous capabilities like Claude Mythos, then a safe-for-public-consumption version would need to be developed as well. However, The Information recently reported that OpenAI is actively previewing Astra in Washington, D.C., so this process is likely already underway. However, let's look again at how OpenAI described Astra: "The results were achieved by an internal version of Astra, our next major model." (Emphasis added.) Major AI companies like OpenAI and Anthropic have been releasing major updates to their models every few months, with entirely new model families coming out every one to two years. GPT-5 was introduced in August of 2025, which means we're due for GPT-6 any time now. Presumably, if Astra is already generating major results in math, it's pretty far along in testing. I'm speculating, but I'd be surprised if we don't see Astra by the end of the year. Disclosure: Ziff Davis, Mashable's parent company, in April 2025 filed a lawsuit against OpenAI, alleging it infringed Ziff Davis copyrights in training and operating its AI systems.

OpenAI Astra model solves 10 open math problems for $2,000

OpenAI announced Saturday that Astra, its next major model still awaiting public release, had generated solutions to 10 longstanding problems across mathematics and theoretical computer science, each unsolved for ten or more years. Alongside the announcement, OpenAI released a 249-page manuscript and Lean 4 proof certificates on GitHub under an Apache 2.0 license; the repository's "sorry" count stands at zero, indicating that every step across all ten formalized proofs is fully verified. Chief among the findings is an explicit construction of a non-sofic group, settling a question that has gone unanswered since Mikhail Gromov laid out the concept of soficity in 1999. Astra also disproved Connes's rigidity conjecture on von Neumann algebras, proved Ehrhart's volume conjecture, and resolved three problems from Paul Erdős's catalog, including problem 183 on multicolor Ramsey numbers, according to SiliconAngle. Additional results span high-dimensional sphere packing, binary and spherical codes, arithmetic circuit complexity, quantum parallel repetition, and the hardness of the closest vector problem. OpenAI put the total compute cost for all 10 solutions at roughly $2,000 at GPT-5.6 Sol API rates. The company's head of mathematics research, Sebastien Bubeck, confirmed the results on X $TWTR, calling them "beautiful," according to The Next Web. Research scientist Noam Brown called the results "a major step for scientific reasoning" in a post on X. OpenAI credited Astra with the underlying mathematical reasoning, while human researchers worked the model's output into papers suitable for publication. Thomas Bloom, who curates the erdosproblems.com database, described the ten results as "big news" on X, placing their significance above the unit distance counterexample that an internal OpenAI model generated in May, according to The Next Web. Because Lean's kernel either accepts or rejects a proof outright, independent verification requires nothing more than running the certificates through the compiler. However, none of the 10 results has been through peer review, according to SiliconAngle. The announcement comes against a backdrop of friction between AI companies and the mathematics community. The International Mathematical Union's June endorsement of the Leiden Declaration formalized the community's concerns, with the declaration charging that AI companies exploit published research without permission, sidestep peer review, and erode standards around proof and credit. OpenAI has not given a release date for Astra. OpenAI recently cut prices on two models in its GPT-5.6 family after efficiency improvements during internal development, while Sol pricing -- the rate used to calculate the $2,000 Astra compute cost -- remained unchanged. OpenAI has yet to clarify whether Astra will carry a GPT-6 designation or slot into the existing GPT-5 line, and the model will need to clear the federal AI safety review before it can reach users.

OpenAI's Astra solves 10 long-open math problems and publishes the proofs

OpenAI Group PBC revealed Saturday that an internal version of Astra, the model family it calls its next major release, produced new results for 10 problems in mathematics and theoretical computer science that had been open for at least a decade, and it published machine-checkable proofs alongside the claim. The company posted a 249-page manuscript collection, model-written reasoning walkthroughs and Lean 4 certificates for all 10 results. The certificates sit on GitHub under an Apache 2.0 license, and the repository reports a "sorry" count of zero, meaning no step in any of the formalized proofs has been left unproven. The headline result is an explicit construction of a non-sofic group, a question left open since Mikhail Gromov introduced soficity in 1999. Astra also disproved Connes's rigidity conjecture, constructing infinitely many non-isomorphic groups with property (T) that share the same von Neumann algebra, and it proved Ehrhart's volume conjecture. Three problems from Paul Erdős's catalog fell as well, including problem 183 on multicolor Ramsey numbers. Stripped of the terminology, a group is the mathematical description of a set of symmetries, and a sofic group is one whose structure can be approximated by shuffling a finite deck of cards. Every group anyone had examined turned out to be sofic, and no one could prove that all of them are. Astra built the exception. Connes's conjecture, posed in 1980, held that for one rigid class of groups, a related algebraic object acts as a unique fingerprint, pinning down the group it came from. Astra produced infinitely many distinct groups sharing a single fingerprint. Erdős problem 183 is about Ramsey numbers. Color the links in a network with a fixed number of colors and past a certain size you cannot avoid a triangle whose three links match. The Ramsey number is the size at which that becomes true. The remainder of the list runs across high-dimensional sphere packing, binary and spherical codes, arithmetic circuit complexity, quantum parallel repetition and the hardness of the closest vector problem, the last of which bears on lattice cryptography. Astra also produced counterexamples in extremal graph theory, resolving two more Erdős problems. The Lean certificates are what give the announcement its weight. Lean's kernel returns a binary verdict, either the proof compiles or it does not, which takes trust in the model out of the equation. What it does not take out is the need for a mathematician to confirm that each formal statement says what the open problem actually asks and to judge whether the result matters. None of the 10 has been through peer review. OpenAI has been here before. The company's then vice president of science, Kevin Weil, claimed in October 2025 that GPT-5 had solved 10 previously unsolved Erdős problems. Thomas Bloom, who maintains the erdosproblems.com database, called that "a dramatic misrepresentation." The model had found papers in the literature that Bloom was personally unaware of. Weil deleted the post, and Google DeepMind Chief Executive Demis Hassabis called the episode embarrassing. Bloom called the Astra results "big news" and rated them ahead of the Erdős unit distance counterexample an internal OpenAI model produced in May, a paper he helped verify. Astra itself remains unreleased. OpenAI describes it as a model family built to run long tasks by coordinating multiple agents over extended periods, an extension of the test-time reasoning work associated with research scientist Noam Brown, who called the results "a major step for scientific reasoning" in a post on X. Human researchers turned the model's output into publishable papers, though OpenAI said the mathematical arguments themselves came from Astra. The compute bill was modest. OpenAI put the token cost for all 10 solutions at roughly $2,000 at GPT-5.6 Sol application programming interface rates. Chief Executive Sam Altman demonstrated Astra to policymakers in Washington in recent days. The company has not given a release date, pricing or a decision on whether the model ships as GPT-6 or as another GPT-5 variant, and any launch will run through the federal AI safety review process that already staggered the GPT-5.6 rollout. The timing is awkward for a mathematics community that has been pushing back. In June, the International Mathematical Union endorsed the Leiden Declaration, which warns that AI companies are "using published research without consent, bypassing peer review, and threatening the integrity of proof and attribution." Fernando Borretti, a software engineer who writes on technology, argued in a blog post responding to the release that the usual defenses of human mathematicians no longer hold and that the frontier of the field will recede past the point where anyone can follow it. "We will live in a demon-haunted world, full of marvelous devices whose operation we will not understand," he wrote.

OpenAI Astra: OpenAI's Astra solves 10 longstanding math problems, igniting new debate

Apart from praise, the news, published on August 1, also kickstarted a debate on social media over whether the achievement is Fields medal-worthy -- and what these honours mean in the artificial intelligence (AI) era. The Fields medal is awarded by the International Mathematical Union (IMU), and is often called the "Nobel Prize of Mathematics". It is awarded every four years to young mathematicians. ChatGPT-maker OpenAI has revealed that its unreleased model family Astra, pegged as its next big release, has solved century-old open problems in mathematics and theoretical computer science. Apart from praise, the news, published on August 1, also kickstarted a debate on social media over whether the achievement is Fields medal-worthy -- and what these honours mean in the artificial intelligence (AI) era. The Fields medal is awarded by the International Mathematical Union (IMU), and is often called the "Nobel Prize of Mathematics". It is awarded every four years to young mathematicians. What has Astra solved? The results span group theory, high-dimensional geometry, coding theory and quantum complexity, compiled into a 249-page manuscript. The main result is the construction of the first known non-sofic group, resolving a question that had stood open since mathematician Mikhail Gromov introduced the concept in 1999. Astra has also disproved the Connes Rigidity Conjecture, posed by Fields Medalist Alain Connes in 1980, and cleared three problems from Paul Erdős's catalogue of open questions. Reactions Fields Medalist Tim Gowers said he would have recommended the proof for publication in a top mathematics journal. He has coauthored a companion paper with mathematician Noga Alon to help translate the proof for human readers. "There is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics: if a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation," he wrote. The timing has amplified the debate. Jacob Tsimerman, who won the Fields Medal just last week, announced hours later that he is taking leave from the University of Toronto to join OpenAI to work on AI safety. Nicolas Bustamante, a Microsoft executive, questioned the validity of accolades like the Fields medal when hard problems are solved with just a prompt. "What happens to Nobel Prizes when AI designs the experiments and robotic labs run millions of them at scale? Who gets the credit: the person asking the question, the team building the model, the owner of the lab, or the AI itself? Probably not the guy who typed the prompt but who knows," he wrote in a post on X. A user replied to his post saying, "I don't think these prizes remain. They are kind of senseless." Another said, "It's all going to go away. Post AGI we play by different rules." Some also pointed out that these ten problems were selected by OpenAI, rather than posed by an independent panel, which raises questions about how the results should be benchmarked against traditional achievement in the field.

OpenAI Astra Solves 10 Math Problems with Logical Accuracy

OpenAI's latest AI model, Astra, has gained attention for solving ten significant mathematical problems, including breakthroughs in sphere packing and geometric conjectures. These areas have traditionally required deep human expertise, making Astra's achievements noteworthy. According to The Stack, OpenAI utilized Lean 4, a formal proof language, to ensure the logical accuracy of Astra's solutions. This approach not only reinforces trust in the model's outputs but also highlights the increasing role of AI in addressing abstract and complex challenges. Explore Astra's mathematical contributions, including how its solutions were independently verified by labs such as Anthropic. Gain insight into the model's limitations when dealing with open-ended problems and the ongoing importance of human-AI collaboration. Additionally, understand the ethical considerations surrounding credit attribution in AI-driven discoveries and their implications for future research. Breaking New Ground in Mathematics Astra's ability to tackle complex mathematical problems represents a major milestone in AI research. Among the ten problems it solved, several date back to pivotal moments in mathematical history, such as 1946, 1978 and 1999. These problems, including intricate sphere packing challenges and geometric conjectures, have long resisted resolution by human mathematicians. Astra's success in addressing these issues highlights the growing potential of AI to engage with abstract, highly complex problems that were once considered the exclusive domain of human intellect. This achievement not only demonstrates Astra's computational power but also underscores the evolving relationship between AI and mathematics. By solving problems that have stymied experts for decades, Astra has opened new avenues for exploration, offering tools that could redefine the boundaries of mathematical research. Making sure Accuracy Through Verification To ensure the accuracy of its solutions, OpenAI employed Lean 4, a formal proof language designed for automated logical verification. This rigorous approach guarantees that Astra's proofs are free from logical errors, as evidenced by the "zero sorry" count, a metric indicating the absence of unresolved gaps in the proofs. Lean 4's precision allows mathematicians and researchers to carefully examine Astra's work, fostering trust in its results. The use of Lean 4 also reflects a broader commitment to transparency in AI research. By providing a clear and verifiable framework, OpenAI has made it possible for the mathematical community to scrutinize Astra's contributions, making sure that its results meet the highest standards of academic rigor. This approach not only strengthens the credibility of Astra's work but also sets a precedent for future AI-driven research in mathematics. Advance your skills in Artificial General Intelligence (AGI) by reading more of our detailed content. Independent Validation Strengthens Credibility Astra's results have undergone independent validation by rival AI labs, including Anthropic, which successfully reproduced some of its solutions. This external scrutiny adds a layer of credibility to OpenAI's claims and demonstrates the robustness of Astra's problem-solving capabilities. However, critics have raised concerns about the framing of the problems presented to Astra. The way a problem is posed can significantly influence an AI's ability to solve it, leading to questions about whether Astra's success reflects genuine problem-solving ability or carefully curated inputs. These concerns highlight the importance of transparency in AI research. By openly addressing questions about problem framing and methodology, researchers can ensure that Astra's achievements are evaluated fairly and accurately. This transparency is essential for building trust in AI systems and for advancing the field of AI-driven mathematics. Limitations Highlight the Need for Collaboration Despite its impressive achievements, Astra is not without limitations. Its performance on uncurated, open-ended problems has been inconsistent, revealing gaps in its ability to handle challenges that lack clear framing or require creative reasoning. These limitations underscore the ongoing need for collaboration between AI systems and human mathematicians. Human expertise remains crucial in interpreting and refining Astra's outputs. While Astra can generate solutions to complex problems, it often relies on human mathematicians to provide context, validate results and explore the broader implications of its findings. This collaborative approach not only enhances the quality of Astra's contributions but also ensures that its work aligns with the goals and priorities of the mathematical community. Attribution and Ethical Debates Astra's accomplishments have sparked debates about how to attribute credit for AI contributions in mathematics. The Leiden Declaration, a recent academic initiative, advocates for clear guidelines on this issue, emphasizing the need for transparency and fairness in recognizing the contributions of AI systems. While Astra's results demonstrate the potential of AI to advance mathematical research, they also highlight the challenges of integrating AI into traditional academic frameworks. These debates are particularly relevant in the context of AGI. While Astra's achievements represent a significant step forward, they fall short of the capabilities required for true AGI. This distinction between advanced AI models and AGI remains a central topic of discussion within the academic and AI communities, shaping the future of AI research and its role in society. Efficiency and Accessibility One of Astra's most notable features is its efficiency. The compute cost for generating its solutions was approximately $2,000, a relatively low figure given the complexity of the problems it solved. This efficiency reflects significant advancements in AI compute systems and has important implications for the accessibility of AI-driven mathematics. Lower compute costs could provide widespread access to access to advanced AI tools, allowing a wider range of researchers and institutions to engage with innovative mathematical research. However, this accessibility also raises ethical questions about how such technology should be deployed. As AI systems like Astra become more widely available, it will be essential to establish guidelines for their use, making sure that they are applied responsibly and equitably. Looking Ahead The mathematical community is currently peer-reviewing Astra's proofs to confirm the validity of its solutions. While Astra's achievements represent a significant leap forward, they also highlight the challenges that remain in the pursuit of AGI. The rapid advancements in AI's mathematical capabilities suggest a promising future, but the journey toward AGI is far from complete. As researchers continue to explore the potential of models like Astra, the interplay between human ingenuity and AI innovation will remain a critical area of focus. By fostering collaboration and addressing the ethical and practical challenges of AI research, the mathematical and AI communities can work together to unlock new possibilities and shape the future of mathematics. Media Credit: The Stack Disclosure: Some of our articles include affiliate links. If you buy something through one of these links, Geeky Gadgets may earn an affiliate commission. Learn about our Disclosure Policy.

OpenAI Astra Solves 10 Unsolved Math and Computing Problems

OpenAI's Astra has achieved a significant milestone by solving ten previously unsolved problems in mathematics and theoretical computer science. This achievement highlights Astra's advanced reasoning capabilities, supported by its unique sub-agent architecture designed for tackling complex, multi-faceted challenges. Better Stack notes that OpenAI has gone a step further by verifying Astra's results using Lean, a formal proof-checking language. This rigorous validation process ensures that Astra's contributions are both credible and reproducible, setting a high standard for transparency in AI research. Explore how Astra's modular architecture allows it to break down intricate problems into manageable components, allowing sustained focus over extended problem-solving sessions. Gain insight into its potential applications, from automating research workflows to accelerating advancements in mathematical proofs. Additionally, understand the challenges Astra faces, such as coordination overhead and limited accessibility and what these mean for its broader adoption. This breakdown provides a comprehensive look at Astra's capabilities and the questions it raises for the future of AI in research. Breakthrough Achievements Astra's most remarkable accomplishment lies in solving ten longstanding problems in mathematics and computer science. These breakthroughs are not merely academic milestones but also evidence of the model's advanced reasoning capabilities. OpenAI has emphasized transparency by verifying Astra's results through Lean, a formal proof-checking language widely respected in the mathematical community. This verification process enables independent experts to validate Astra's reasoning, fostering trust in its outputs and making sure that its contributions are both credible and reproducible. Beyond solving specific problems, Astra's achievements highlight its ability to engage with abstract concepts and deliver concrete solutions. This positions the model as a valuable tool for researchers tackling some of the most intricate challenges in their fields. Innovative Architecture and Functionality Astra's architecture is purpose-built for tasks that demand sustained focus and iterative problem-solving. At its core, Astra operates through a root agent that coordinates multiple specialized sub-agents. Each sub-agent focuses on a specific aspect of a problem, allowing the model to decompose complex challenges into manageable components. This modular approach ensures coherence and efficiency, particularly during extended problem-solving sessions. Unlike AI models optimized for short, discrete tasks, Astra excels in scenarios requiring long-term engagement and refinement. Its ability to maintain focus over extended durations makes it uniquely suited for addressing problems that demand deep analysis and sustained effort. However, this sophisticated architecture introduces certain challenges, such as coordination overhead, which can impact performance on tasks requiring seamless integration across components. Take a look at other insightful guides from our broad collection that might capture your interest in OpenAI. Transparency and Accountability OpenAI has set a new standard for transparency with Astra by releasing a comprehensive 249-page paper detailing the model's methodologies, findings and reasoning processes. Notably, 62 pages are dedicated to explaining Astra's reasoning, offering an in-depth view of how the model arrives at its conclusions. This level of detail provides researchers and practitioners with valuable insights into Astra's inner workings, enhancing its credibility and fostering trust. The use of Lean proof language for verification further underscores OpenAI's commitment to accountability. By allowing independent experts to scrutinize and confirm Astra's results, OpenAI ensures that the model's contributions are both reliable and verifiable. This openness is a rare but essential practice in AI research, particularly for models with the potential to influence critical fields like mathematics and computer science. Applications Across Domains Astra's capabilities extend well beyond theoretical research, offering potential applications across various domains. These include: * Automating Research: Astra can analyze and synthesize large datasets, allowing researchers to tackle tasks that would otherwise be impractical or time-consuming. * Software Engineering: The model has the potential to transform complex code refactoring, allowing developers to optimize and maintain codebases with unprecedented efficiency. * Mathematical Proofs: Astra's ability to construct and verify proofs could accelerate advancements in mathematical research, providing researchers with new tools to explore uncharted territories. These applications illustrate Astra's versatility and its potential to drive innovation across multiple fields. By automating routine tasks and enhancing human capabilities, Astra could serve as a fantastic option for progress in both academic and practical contexts. Challenges and Limitations Despite its impressive achievements, Astra is not without limitations. The sub-agent architecture, while effective for breaking down complex problems, introduces coordination overhead that can hinder performance on tasks requiring seamless integration. This trade-off highlights the need for further optimization to enhance the model's efficiency and scalability. Additionally, Astra is currently accessible only to OpenAI's internal teams, limiting external validation of its performance and cost-effectiveness. This restricted access raises questions about the model's broader applicability and the feasibility of deploying it in real-world scenarios. Addressing these challenges will be crucial as Astra continues to evolve and expand its reach. Addressing Ethical Concerns The introduction of Astra has sparked concerns about its potential to replace human mathematicians and researchers. However, experts argue that such fears are largely unfounded. Astra relies heavily on existing mathematical theories and human input, positioning it as a tool to augment, rather than replace, human expertise. By enhancing researchers' efficiency and allowing them to tackle more ambitious projects, Astra aims to complement human capabilities and expand the boundaries of what is possible in mathematics and computer science. The Road Ahead Astra represents a significant advancement in the development of AI models designed for long-duration tasks. Its ability to deliver publicly verifiable results and tackle complex challenges positions it as a valuable asset in both research and practical applications. However, its true potential will only be realized once it is made available for external use, allowing a broader community of researchers and practitioners to explore its capabilities. As Astra continues to evolve, it holds the promise of reshaping how we approach complex problems in mathematics, computer science and beyond. By bridging the gap between theoretical research and practical application, Astra could pave the way for new discoveries and innovations that were previously thought to be out of reach. Media Credit: Better Stack Disclosure: Some of our articles include affiliate links. If you buy something through one of these links, Geeky Gadgets may earn an affiliate commission. Learn about our Disclosure Policy.

OpenAI's 'Astra' AI Model Makes Breakthroughs in 10 Long-Standing Math Problems

OpenAI announced that its unreleased Astra model family made major progress on 10 long-standing mathematical problems. Independent verification would confirm a major evolution, positioning AI as a legitimate contributor to scientific discovery. According to OpenAI, the unreleased Astra model worked across several advanced mathematical fields, including high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. The company said the AI generated the mathematical arguments, while researchers later converted them into formal manuscripts and verified every logical step using the proof assistant Lean.

OpenAI unveils Astra: its next major model family for harder problems

Inside OpenAI, Astra is now the name attached to what appears to be the company's next big model family. The goal is to take on harder problems and stay with them much longer than today's chat-style tools can. It uses multiple agents that can work together for hours, even days. Think less one-shot chatbot, more research team: something that can make a plan, run tests, revise the work, and keep pushing through deep research, coding, or other multi-step analysis without needing much from you. Sam Altman has already shown Astra to policymakers in Washington, which suggests government conversations are starting early. As the industry moves toward more autonomous systems, questions about oversight, reliability, and responsibility are coming up in the US, the EU, and elsewhere. OpenAI says an internal version of Astra solved 10 long-standing open problems in math and theoretical computer science. That included problems in group theory, coding theory, and quantum complexity. Humans checked the results, the work was formalized in Lean, and OpenAI estimates the cost at about 2,000 tokens at Sol API rates. It didn't solve everything. Astra also came up short on other major open questions, including Millennium Prize Problems. So while it looks unusually strong on some reasoning tasks, it's still far from general-purpose. If you use AI for research or coding, this is something to keep an eye on. You can't download Astra yet, and OpenAI still hasn't decided whether it will ship as GPT-6 or as a new GPT-5 variant.

OpenAI's Astra Model solved 10 open math problems: This is how we know it's true

OpenAI is known to drop research bombs with no great announcement, and the newest one requires a closer look than just the small paper where it is published. The internal version of Astra, the next big model to be released by OpenAI, has come up with solutions to ten mathematical problems and problems in theoretical computer science which have been unsolved for at least a decade, and in many cases for much longer. The topics include high-dimensional geometry, coding theory, group theory, quantum complexity and lattice cryptography. The two last items in the list solve the remaining two problems from the famous problem list of Paul Erdős. One refutes a conjecture from operator algebras made in the seventies. So, what is the right answer? How can you trust the AI when it claims it solved a math problem? Also read: Best tablets for designers in India in 2026: Five picks for illustrators This is the story that is worth telling, and it is a lot more interesting than what meets the eye from the headline figure. OpenAI did not simply provide a list of the ten counterexamples and ask mathematicians to trust it because of the claim of mathematicians. All the ten counterexamples were made into a proof certificate using Lean prior to their being published. A proof certificate is a tool called a proof assistant which is basically a programming language capable of encoding mathematical arguments in a format that could be understood by the computer and could be checked logically. That's an essential distinction at this point in time. Big language models are well-known to make confident, fluent, and sometimes totally made-up assertions, which is a problem that research labs haven't figured out how to overcome yet. An assertion from a big language model that it has proved something is basically meaningless. An assertion from a big language model that its proof has passed mechanical verification to align with formal logic is an assertion of an entirely different order. OpenAI clearly recognized that difference. "OpenAI is responsible for the accuracy of the manuscripts and the Lean formalizations. The mathematical proofs come from the system." Also read: India's real chip opportunity is supply chain, says Lam Research's Rangesh Raghavan One should note what lean verification does and doesn't do. It verifies the logical integrity of the reasoning process assuming a set of premises is provided. It doesn't mean anything regarding elegance and profundity of a proof, or that it can be called "illuminating" by a professional mathematician. Mathematicians have discussed the idea that a proof is just as much about the understanding of a statement as its verification for a long time. A machine verified proof avoids that discussion altogether. That is precisely the issue that is of primary importance for the future credibility of AI-based science. Given that the applications of these increasingly powerful models will move on from solving toy problems to tackling ever-harder challenges in physics, chemistry, and engineering, a process of formal verification of some sort may well become the norm, not the exception. The expanding application of Lean to this end is no accident. It is a pattern. If the AI community wishes for its scientific statements to be taken seriously and not simply viewed as marketing slogans, formal verification will be the requirement. OpenAI appears to have got the memo. Whether the rest of the AI industry has followed suit is an open question.