{"id":2207,"date":"2026-08-31T17:30:00","date_gmt":"2026-08-31T17:30:00","guid":{"rendered":"https:\/\/netguide.io\/news\/?p=2207"},"modified":"2026-08-29T00:25:51","modified_gmt":"2026-08-29T00:25:51","slug":"openai-astra-solves-decade-old-math-problems-lean-proofs","status":"publish","type":"post","link":"https:\/\/netguide.io\/news\/en\/2026\/08\/31\/openai-astra-solves-decade-old-math-problems-lean-proofs\/","title":{"rendered":"OpenAI: &#8220;Astra&#8221; Model Solves Ten Decade-Old Math Problems \u2013 With Machine-Checked Proofs"},"content":{"rendered":"<div id=\"netgu-2333368796\" class=\"netgu-before-content netgu-entity-placement\"><script async src=\"\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-6258556257245998\" crossorigin=\"anonymous\"><\/script><ins class=\"adsbygoogle\" style=\"display:block;\" data-ad-client=\"ca-pub-6258556257245998\" \ndata-ad-slot=\"3494115342\" \ndata-ad-format=\"auto\"><\/ins>\n<script> \n(adsbygoogle = window.adsbygoogle || []).push({}); \n<\/script>\n<\/div>\n<p class=\"wp-block-paragraph\"><strong>OpenAI has unveiled &#8220;Astra,&#8221; an as-yet-unreleased AI model said to have solved ten problems in mathematics and theoretical computer science that had been open for at least a decade &ndash; each backed by a machine-checkable formal proof.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Proofs anyone can verify<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For the ten results, OpenAI published a 249-page manuscript plus formal proof certificates in the Lean 4 proof assistant &ndash; freely on GitHub under an Apache 2.0 license. The number of open proof gaps (the &#8220;sorry&#8221; count) is zero: every step is fully verified. Anyone can run the proofs against the published repository, so correctness no longer hinges on slow peer review.<\/p><div id=\"netgu-4152481092\" class=\"netgu-content netgu-entity-placement\"><aside class=\"deals-top deals-top--compact\">\n\n\t\t\t<h3 class=\"deals-top__title\">Top-Deals<\/h3>\n\t\n\t\t\t<ul class=\"deals-top__list\">\n\t\t\t\t\t\t\t<li class=\"deals-top__item\">\n\t\t\t\t\t<a class=\"deals-top__link\" href=\"https:\/\/netguide.io\/deals\/de\/deals\/shoot-em-up-clive-owen\/\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<img class=\"deals-top__image\" src=\"https:\/\/netguide.io\/news\/wp-content\/uploads\/sites\/15\/2026\/09\/1200x675-1-150x150.jpg\" alt=\"\" width=\"52\" height=\"52\" loading=\"lazy\" \/>\n\t\t\t\t\t\t\n\t\t\t\t\t\t<span class=\"deals-top__body\">\n\t\t\t\t\t\t\t<span class=\"deals-top__name\">Shoot \u2018Em Up | Clive Owen<\/span>\n\n\t\t\t\t\t\t\t<span class=\"deals-top__meta\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"deals-top__price\">3.99 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src=\"https:\/\/netguide.io\/news\/wp-content\/uploads\/sites\/15\/2026\/09\/2833494_1-150x150.webp\" alt=\"\" width=\"52\" height=\"52\" loading=\"lazy\" \/>\n\t\t\t\t\t\t\n\t\t\t\t\t\t<span class=\"deals-top__body\">\n\t\t\t\t\t\t\t<span class=\"deals-top__name\">Crucial P510 SSD 1TB | PCIe 5.0 x4 f\u00fcr 164,99\u20ac<\/span>\n\n\t\t\t\t\t\t\t<span class=\"deals-top__meta\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"deals-top__price\">164.99 \u20ac<\/span>\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<s>199.00 \u20ac<\/s>\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"deals-discount\">-17%<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t<\/span>\n\n\t\t\t\t\t\t<span class=\"deals-top__temperature\">\n\t\t\t\t\t\t\t1\u00b0\n\t\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/li>\n\t\t\t\t\t<\/ul>\n\n\t\t\t\t\t<p class=\"deals-top__more\">\n\t\t\t\t<a href=\"https:\/\/netguide.io\/deals\/\">Alle Deals ansehen \u2192<\/a>\n\t\t\t<\/p>\n\t\t\t<\/aside>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">What Astra solved<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The results include an explicit construction of a &#8220;non-sofic&#8221; group &ndash; a question open since Mikhail Gromov defined soficity in 1999 &ndash; the disproof of Connes\u2019 rigidity conjecture, a proof of Ehrhart\u2019s volume conjecture and the resolution of three problems from Paul Erd\u0151s\u2019s catalog, including problem 183 on multicolor Ramsey numbers.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Context<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Importantly, none of the ten results has yet gone through regular peer review &ndash; the formal Lean verification stands in for it here. And the model itself is not yet publicly available. Still, the step is seen as a notable sign of how AI could seriously contribute at the research frontier.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Sources: <a href=\"https:\/\/qz.com\/openai-astra-model-math-problems-lean-proofs-080326\" target=\"_blank\" rel=\"noopener\">Quartz<\/a>, <a href=\"https:\/\/aiweekly.co\/alerts\/openai-releases-ten-astra-math-proofs-with-lean-certificates\" target=\"_blank\" rel=\"noopener\">AI Weekly<\/a>.<\/em><\/p>\n<div id=\"netgu-2279980317\" class=\"netgu-after-content netgu-entity-placement\"><script async src=\"\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-6258556257245998\" crossorigin=\"anonymous\"><\/script><ins class=\"adsbygoogle\" style=\"display:block;\" data-ad-client=\"ca-pub-6258556257245998\" \ndata-ad-slot=\"4559785002\" \ndata-ad-format=\"auto\"><\/ins>\n<script> \n(adsbygoogle = window.adsbygoogle || []).push({}); \n<\/script>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>OpenAI\u2019s unreleased &#8220;Astra&#8221; model solves ten 10+-year-old math\/CS problems \u2013 each with a formal Lean proof published on GitHub.<\/p>\n","protected":false},"author":1,"featured_media":1697,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[],"tags":[],"class_list":["post-2207","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry"],"brizy_media":[],"_links":{"self":[{"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/posts\/2207","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/comments?post=2207"}],"version-history":[{"count":1,"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/posts\/2207\/revisions"}],"predecessor-version":[{"id":2215,"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/posts\/2207\/revisions\/2215"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/media\/1697"}],"wp:attachment":[{"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/media?parent=2207"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/categories?post=2207"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/netguide.io\/news\/wp-json\/wp\/v2\/tags?post=2207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}