{"id":105893,"date":"2026-09-10T07:25:18","date_gmt":"2026-09-10T11:25:18","guid":{"rendered":"https:\/\/www.adviews.info\/news\/?p=105893"},"modified":"2026-09-10T07:25:18","modified_gmt":"2026-09-10T11:25:18","slug":"ai-may-have-solved-one-of-maths-biggest-puzzles-raising-controversy","status":"publish","type":"post","link":"https:\/\/www.adviews.info\/news\/?p=105893","title":{"rendered":"AI may have solved one of math\u2019s biggest puzzles, raising controversy"},"content":{"rendered":"<p class=\"wp-block-paragraph\">The motion of fluids seems simple, but the equations behind it are the subject of one of the most famous puzzles in mathematics. That puzzle now appears solved, thanks to AI, even as the feat came with a flood of controversy.<\/p>\n<p class=\"wp-block-paragraph\">OpenAI announced September 8 a solution to the Navier\u2013Stokes existence and smoothness problem, which is so monumental that a $1 million prize has been offered for its resolution. \u201cIt\u2019s one of the guiding problems for the field. It is a huge deal to know the answer,\u201d says mathematician Dallas Albritton of the University of Wisconsin\u2013Madison, who was not involved with the new work.<\/p>\n<aside class=\"sn-conversion rich-text rich-text--with-sidebar\">\n<p>We summarize the week&#8217;s scientific breakthroughs every Thursday.<\/p>\n<p>                                input[type=&#8221;radio&#8221;].svelte-wecxzz{display:none}.payment-option-wrapper.svelte-wecxzz.svelte-wecxzz{display:flex;flex:1 0 28%;width:100%}.payment-option-wrapper.svelte-wecxzz.svelte-wecxzz:only-child{max-width:325px}.box-container.svelte-wecxzz.svelte-wecxzz{display:flex;flex-direction:column;flex:1 1 0;width:0;text-align:center}.inner-box.svelte-wecxzz.svelte-wecxzz{border:var(&#8211;zf-payment-options-border);border-radius:var(&#8211;zf-payment-options-borderRadius);background-color:var(&#8211;zf-payment-options-bgColor);height:100%;display:flex;flex-direction:column}.box-container.highlighted.svelte-wecxzz 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}}.zephr-form-disabled-div{pointer-events:none;opacity:0.8}.zephr-form-relative-container{position:relative}.zephr-form-flex-container{display:flex}.zephr-form-input-inner-button{cursor:pointer;position:absolute;top:50%;transform:translateY(-50%);right:5px;padding:10px 5px}.zephr-form-input-inner-text{color:var(&#8211;zephr-color-text-main);font-size:var(&#8211;zf-subtext-fontSize);font-weight:bold;font-family:var(&#8211;zf-root-fontFamily)}<br \/>\n]]&gt;<\/p>\n<\/aside>\n<p class=\"wp-block-paragraph\">The day before, a pair of mathematicians had reported an advance on a closely related problem. To do so, Tristan Buckmaster of New York University and Levent Alp\u00f6ge of the AI company Anthropic relied on assistance from various AI models, including OpenAI\u2019s. The relationship between the two advances \u2014 and whether OpenAI had access to the duo\u2019s progress \u2014 has been subject to intense speculation.<\/p>\n<p class=\"wp-block-paragraph\">The Navier-Stokes equations describe the relationships between pressure, density and velocity of a fluid as they change over time. The equations are a foundation of science and engineering, relevant for weather forecasting, studying <a href=\"https:\/\/www.sciencenews.org\/article\/melissa-omand-oceanographer-sn-10-scientists-watch\">ocean currents<\/a> and designing aircraft, pumps, turbines and more. \u201cIt\u2019s an incredibly important, ubiquitous set of equations,\u201d Albritton says.<\/p>\n<p class=\"wp-block-paragraph\">Despite dating to the 19th century, the equations are not fully understood \u2014 in particular, whether they are always well-behaved has not been clear. There might be situations in which the equations break down, producing physically impossible situations like fluid flowing at infinite speed, known as \u201cblowups.\u201d Determining whether the equations are naughty or nice is one of seven challenges selected in 2000 by the Clay Mathematics Institute as the Millenium Prize Problems, each of which has a $1 million bounty.<\/p>\n<p class=\"wp-block-paragraph\">Buckmaster and Alp\u00f6ge focused on a set of equations called the Euler (pronounced like \u201coiler\u201d) equations. While the Navier-Stokes equations include viscosity \u2014 a term describing a fluid\u2019s resistance to flow \u2014 the Euler equations don\u2019t, making them a stepping stone to the more famous Navier-Stokes problem. The pair found a blowup when an outside force pushes on the fluid, solving what\u2019s called the forced Euler problem.<\/p>\n<p>As the pair were finessing their results, rumors began flying about their work. Spurred by these vague rumors, OpenAI researchers started tackling the various Millenium Prize Problems, and quickly focused on Navier-Stokes, using around 10,000 AI agents at a time.<\/p>\n<p class=\"wp-block-paragraph\">The agents discovered a solution to the forced version of the Navier-Stokes problem: a vortex that gets skinnier and faster until its speed goes to infinity. That means that the Navier-Stokes equations are not always well-behaved, answering the longstanding question. The feat of running so many AI agents cost millions of dollars, OpenAI researchers estimated. (OpenAI stated in a blog post that they will not claim the Millenium Prize.)<\/p>\n<p class=\"wp-block-paragraph\">OpenAI\u2019s solution has been confirmed by Lean, a tool that allows for the verification of complicated mathematical proofs. But, as the paper reporting the result is 166 pages long, mathematicians are still digesting its contents. \u201cI don\u2019t think anyone has completely verified the proof yet, certainly not on the human side,\u201d says mathematical physicist Gregory Eyink of Johns Hopkins University.<\/p>\n<p class=\"wp-block-paragraph\">The new achievements are part of a wave of AI results deluging mathematics. In the past year, AI has enabled major leaps, leaving mathematicians grappling with rapid changes to their field. The new result, on one of mathematics\u2019 most important problems, is \u201cthe spectacular culmination of the arc we have seen over the last 12 months,\u201d OpenAI researcher S\u00e9bastien Bubeck said during a September 8 news conference.<\/p>\n<p class=\"wp-block-paragraph\">In a statement published on his website on September 7 along with the forced Euler solution, Buckmaster said that his team\u2019s results mark a \u201cDeep Blue-Kasparov moment,\u201d referencing the 1990\u2019s milestone when a supercomputer beat the best human chess player. \u201cThe community needs to have serious and unhurried discussion about where to go from here,\u201d he said.<\/p>\n<p class=\"wp-block-paragraph\">Mathematicians are certainly taking notice. Albritton got wind of Buckmaster and Alp\u00f6ge\u2019s result around 1 a.m., awake with his newborn baby. He stayed up until 6 a.m. discussing it with colleagues.<\/p>\n<p class=\"wp-block-paragraph\">Despite the problem\u2019s mathematical clout, its solution won\u2019t have significant practical implications, Eyink says. The Navier-Stokes equations describe fluid as a continuum, but real-world fluids are made of individual molecules and atoms, so it\u2019s already known that there\u2019s a cutoff where the equations no longer apply. It\u2019s more about prestige, Eyink says. \u201cThere\u2019s a huge mathematical celebrity associated with these equations.\u201d<\/p>\n<aside class=\"sn-conversion rich-text rich-text--with-sidebar\">\n<p class=\"has-text-align-center\">Sponsor Message<\/p>\n<p><!-- Tag ID: sciencenews-org_leaderboard_incontent --><br \/>\n<\/aside>\n<p class=\"wp-block-paragraph\">That also raises concerns about how credit for a discovery is doled out. After initial rumors began circulating, Buckmaster says he had a series of discussions with OpenAI researchers about how to present the two results. Those discussions, he alleged, involved a request to exclude his coauthor Alp\u00f6ge, who works for OpenAI competitor Anthropic. Buckmaster\u2019s narrative also raised questions about whether the AI agents had access to Buckmaster and Alp\u00f6ge\u2019s progress. OpenAI denies that its AI agents had direct access, but says, \u201cwhile unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models\u2060.\u201d<\/p>\n<p class=\"wp-block-paragraph\">The biggest implication of the advance may be the issues it raises regarding the difficulty of attributing credit when AI is involved, Eyink says. \u201cThis is, for me, the really serious ongoing problem and issue that has to be resolved.\u201d<\/p>\n<p class=\"wp-block-paragraph\" \/>\n","protected":false},"excerpt":{"rendered":"<p>The motion of fluids seems simple, but the equations behind it are the subject of one of the most famous puzzles in mathematics. That puzzle now appears solved, thanks to AI, even as the feat came with a flood of controversy. OpenAI announced September 8 a solution to the Navier\u2013Stokes existence and smoothness problem, which [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[10],"tags":[],"_links":{"self":[{"href":"https:\/\/www.adviews.info\/news\/index.php?rest_route=\/wp\/v2\/posts\/105893"}],"collection":[{"href":"https:\/\/www.adviews.info\/news\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.adviews.info\/news\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.adviews.info\/news\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.adviews.info\/news\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=105893"}],"version-history":[{"count":0,"href":"https:\/\/www.adviews.info\/news\/index.php?rest_route=\/wp\/v2\/posts\/105893\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.adviews.info\/news\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=105893"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.adviews.info\/news\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=105893"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.adviews.info\/news\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=105893"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}