--- title: AI just disproved the 87-year-old Jacobian conjecture description: Anthropic's Fable 5 AI model, working with Harvard mathematician Levent Alpöge, has disproved the 87-year-old Jacobian conjecture by finding a 216-character... canonical: https://daily.dev/posts/ai-just-disproved-the-87-year-old-jacobian-conjecture-ve8b2wkpc twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: AI just disproved the 87-year-old Jacobian conjecture | daily.dev og:description: Anthropic's Fable 5 AI model, working with Harvard mathematician Levent Alpöge, has disproved the 87-year-old Jacobian conjecture by finding a 216-character... og:url: https://daily.dev/posts/ai-just-disproved-the-87-year-old-jacobian-conjecture-ve8b2wkpc og:image: https://api.daily.dev/og/posts/VE8b2WKpc.png og:image:alt: AI just disproved the 87-year-old Jacobian conjecture og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # AI just disproved the 87-year-old Jacobian conjecture **[The Next Web](https://daily.dev/sources/tnw)** · 5 min read · 0 upvotes · 0 comments ## Summary Anthropic's Fable 5 AI model, working with Harvard mathematician Levent Alpöge, has disproved the 87-year-old Jacobian conjecture by finding a 216-character counterexample — a polynomial map from 3D space to itself with a constant Jacobian determinant that cannot be inverted. The result is easily verifiable but was extremely hard to find, which is where the AI excelled. Fields medallist Timothy Gowers called it the first AI solution to a major problem outside his own field. However, some mathematicians like Andrew Blumberg are unimpressed, arguing a counterexample is just a search problem for machines and reveals little mathematical insight compared to a proof. OpenAI's Codex reportedly found the same counterexample independently. Alpöge hints a positive result may be hiding inside the counterexample, and a full write-up is forthcoming. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Questions this post answers ### What is the Jacobian conjecture and how was it disproved? The Jacobian conjecture, proposed by Ott-Heinrich Keller in 1939, claimed that any polynomial map with a non-zero constant Jacobian determinant must be reversible with a polynomial inverse. It was disproved using a 216-character counterexample: a polynomial map from three-dimensional space to itself that meets the determinant condition but cannot be reversed, found with help from Anthropic's Fable 5 model. _Developers tracking how AI models tackle hard search problems can follow frontier model capabilities on daily.dev._ ### Did OpenAI also find a counterexample to the Jacobian conjecture? Yes, an OpenAI researcher named Aaron Lou said an internal version of the company's Codex model independently found essentially the same counterexample to the Jacobian conjecture around the same time Anthropic's Fable 5 produced its result with mathematician Levent Alpöge. _Anyone comparing Anthropic and OpenAI model capabilities can track these developments on daily.dev._ ### Why do some mathematicians say the AI-found counterexample to the Jacobian conjecture doesn't matter much? Columbia mathematician Andrew Blumberg argues a counterexample just ends an argument without explaining why something is true, unlike a proof. He contrasted it with OpenAI's May disproof of a discrete geometry conjecture, which mathematicians could unpack and build on, calling the Jacobian result something machines are simply better suited to brute-force search than humans. _Developers weighing what AI-assisted discovery actually delivers can follow this debate on daily.dev._ ## Similar posts on daily.dev - [Human mathematicians are being outcounterexampled](https://daily.dev/posts/human-mathematicians-are-being-outcounterexampled-ocacgpf22) · Hacker News · 3 upvotes · 0 comments - [A Simplified View of the Jacobian Conjecture](https://daily.dev/posts/a-simplified-view-of-the-jacobian-conjecture-31xgiqpq6) · Towards Data Science · 0 upvotes · 0 comments --- Tags: [#ai](https://daily.dev/tags/ai), [#llm](https://daily.dev/tags/llm), [#openai](https://daily.dev/tags/openai), [#anthropic](https://daily.dev/tags/anthropic), [#math](https://daily.dev/tags/math) [View this post on daily.dev](https://daily.dev/posts/ai-just-disproved-the-87-year-old-jacobian-conjecture-ve8b2wkpc) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"AI just disproved the 87-year-old Jacobian conjecture","url":"https://daily.dev/posts/ai-just-disproved-the-87-year-old-jacobian-conjecture-ve8b2wkpc","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/ai-just-disproved-the-87-year-old-jacobian-conjecture-ve8b2wkpc"},"datePublished":"2026-07-21T13:35:07.986Z","dateModified":"2026-09-13T19:41:18.440Z","description":"Anthropic's Fable 5 AI model, working with Harvard mathematician Levent Alpöge, has disproved the 87-year-old Jacobian conjecture by finding a 216-character...","image":"https://media.daily.dev/image/upload/f_auto,q_auto/v1/posts/6b154535b4a27af72d83d45ff7f58147?_a=AQAEuop","thumbnailUrl":"https://media.daily.dev/image/upload/f_auto,q_auto/v1/posts/6b154535b4a27af72d83d45ff7f58147?_a=AQAEuop","isAccessibleForFree":true,"articleSection":"The Next Web","inLanguage":"en","publisher":{"@type":"Organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180}},"author":{"@type":"Organization","name":"The Next Web","logo":"https://media.daily.dev/image/upload/t_logo,f_auto/v1/logos/tnw","url":"https://daily.dev/sources/tnw"},"commentCount":0,"discussionUrl":"https://daily.dev/posts/ai-just-disproved-the-87-year-old-jacobian-conjecture-ve8b2wkpc","interactionStatistic":[{"@type":"InteractionCounter","interactionType":{"@type":"LikeAction"},"userInteractionCount":0},{"@type":"InteractionCounter","interactionType":{"@type":"CommentAction"},"userInteractionCount":0}],"keywords":"ai,llm,openai,anthropic,math","timeRequired":"PT5M"} {"@context":"https://schema.org","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://daily.dev"},{"@type":"ListItem","position":2,"name":"The Next Web","item":"https://daily.dev/sources/tnw"},{"@type":"ListItem","position":3,"name":"AI just disproved the 87-year-old Jacobian conjecture"}]} {"@context":"https://schema.org","@type":"FAQPage","@id":"https://daily.dev/posts/ai-just-disproved-the-87-year-old-jacobian-conjecture-ve8b2wkpc#faq","mainEntity":[{"@type":"Question","name":"What is the Jacobian conjecture and how was it disproved?","acceptedAnswer":{"@type":"Answer","text":"The Jacobian conjecture, proposed by Ott-Heinrich Keller in 1939, claimed that any polynomial map with a non-zero constant Jacobian determinant must be reversible with a polynomial inverse. It was disproved using a 216-character counterexample: a polynomial map from three-dimensional space to itself that meets the determinant condition but cannot be reversed, found with help from Anthropic's Fable 5 model. Developers tracking how AI models tackle hard search problems can follow frontier model capabilities on daily.dev."}},{"@type":"Question","name":"Did OpenAI also find a counterexample to the Jacobian conjecture?","acceptedAnswer":{"@type":"Answer","text":"Yes, an OpenAI researcher named Aaron Lou said an internal version of the company's Codex model independently found essentially the same counterexample to the Jacobian conjecture around the same time Anthropic's Fable 5 produced its result with mathematician Levent Alpöge. Anyone comparing Anthropic and OpenAI model capabilities can track these developments on daily.dev."}},{"@type":"Question","name":"Why do some mathematicians say the AI-found counterexample to the Jacobian conjecture doesn't matter much?","acceptedAnswer":{"@type":"Answer","text":"Columbia mathematician Andrew Blumberg argues a counterexample just ends an argument without explaining why something is true, unlike a proof. He contrasted it with OpenAI's May disproof of a discrete geometry conjecture, which mathematicians could unpack and build on, calling the Jacobian result something machines are simply better suited to brute-force search than humans. Developers weighing what AI-assisted discovery actually delivers can follow this debate on daily.dev."}}]} ```