{"schema":"postcutoff/event@1","as_of":"2026-10-10T23:43:00+02:00","url":"https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/","md":"https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-08-alphaproof-nexus-science-paper","date":"2026-10-08","date_precision":"day","short_title":"Google DeepMind's AlphaProof Nexus paper published in Science","deck":"LLM + Lean agent resolved 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures","takeaway":"On 8 Oct 2026 Science published Google DeepMind's AlphaProof Nexus paper (\"Advancing mathematics research with AI-driven formal proof search\", doi 10.1126/science.aej2213), first posted on arXiv on 21 May 2026.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"Science: Advancing mathematics research with AI-driven formal proof search (doi 10.1126/science.aej2213)","url":"https://www.science.org/doi/10.1126/science.aej2213","type":"paper","group":"primary","domain":"science.org"},{"n":2,"title":"Science Perspective: AI for research mathematics has arrived (Avigad & Ballard)","url":"https://www.science.org/doi/10.1126/science.ael6628","type":"paper","group":"primary","domain":"science.org"},{"n":3,"title":"arXiv 2605.22763: Advancing Mathematics Research with AI-Driven Formal Proof Search","url":"https://arxiv.org/abs/2605.22763","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"GitHub: google-deepmind/alphaproof-nexus-results (Lean proofs + prose proofs)","url":"https://github.com/google-deepmind/alphaproof-nexus-results","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"Science Expert Voices: AI has solved many math problems, but it has not solved math (Emily Riehl)","url":"https://www.science.org/doi/10.1126/science.ael8339","type":"press","group":"press","domain":"science.org"},{"n":6,"title":"EurekAlert (AAAS): Introducing AlphaProof Nexus (8 Oct 2026)","url":"https://www.eurekalert.org/news-releases/1146436","type":"press","group":"press","domain":"eurekalert.org"},{"n":7,"title":"Science Media Centre Spain: expert reactions (8 Oct 2026)","url":"https://sciencemediacentre.es/en/alphaproof-nexus-new-ai-tool-designed-tackle-mathematical-proofs","type":"press","group":"press","domain":"sciencemediacentre.es"},{"n":8,"title":"The Decoder: AlphaProof Nexus solves decades-old math problems for a few hundred dollars (May 2026)","url":"https://the-decoder.com/google-deepminds-alphaproof-nexus-solves-decades-old-math-problems-for-a-few-hundred-dollars/","type":"press","group":"press","domain":"the-decoder.com"},{"n":9,"title":"WION: Google AI solves decades-old maths problems but DeepMind CEO says AGI is still far away (May 2026)","url":"https://embed.wionews.com/technology/google-ai-solves-decades-old-maths-problems-but-deepmind-ceo-says-agi-is-still-far-away-1779695717629","type":"press","group":"press","domain":"embed.wionews.com"},{"n":10,"title":"Pushmeet Kohli on X: thread on DeepMind and mathematics, paper in Science today (8 Oct 2026)","url":"https://x.com/pushmeet/status/2108265638435119494","type":"discussion","group":"reactions","domain":"x.com"},{"n":11,"title":"Pushmeet Kohli on X: 'Our technical paper on AlphaProof Nexus appears in Science today'","url":"https://x.com/pushmeet/status/2108265646890815730","type":"discussion","group":"reactions","domain":"x.com"},{"n":12,"title":"Dr Singularity on X: viral summary (10 Oct 2026)","url":"https://x.com/Dr_Singularity/status/2108936442671992990","type":"discussion","group":"reactions","domain":"x.com"},{"n":13,"title":"erdosproblems.com #12 (DeepMind construction)","url":"https://www.erdosproblems.com/12","type":"discussion","group":"reactions","domain":"erdosproblems.com"},{"n":14,"title":"erdosproblems.com #846 (DeepMind and OpenAI, independently)","url":"https://www.erdosproblems.com/846","type":"discussion","group":"reactions","domain":"erdosproblems.com"},{"n":15,"title":"Anatol Wegner: critical analysis of the AlphaProof Nexus paper (31 May 2026)","url":"https://buttondown.com/anatol/archive/deepminds-alphaproof-nexus/","type":"discussion","group":"reactions","domain":"buttondown.com"},{"n":16,"title":"Hacker News discussion (May 2026)","url":"https://news.ycombinator.com/item?id=48248173","type":"discussion","group":"reactions","domain":"news.ycombinator.com"}],"official":4,"filed":"2026-10-10","updated":"2026-10-10","orgs":["Google DeepMind"],"title":"Google DeepMind's AlphaProof Nexus paper published in Science: LLM + Lean agent resolved 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures","summary":"On 8 Oct 2026 Science published Google DeepMind's AlphaProof Nexus paper (\"Advancing mathematics research with AI-driven formal proof search\", doi 10.1126/science.aej2213), first posted on arXiv on 21 May 2026. Agents built on Gemini 3.1 Pro search for Lean proofs with compiler feedback. The full agent autonomously resolved 9 of 353 open Erdős problems (two questions open since 1970) at a few hundred dollars per problem and proved 44 of 492 open OEIS conjectures, all Lean-verified. A basic generate-and-check loop solved the same 9 Erdős problems at higher cost.","key_facts":["Science, 8 Oct 2026, doi 10.1126/science.aej2213 (received 27 May, accepted 17 Aug 2026, per press reports); preprint arXiv 2605.22763 (v1 21 May 2026); 21 authors led by George Tsoukalas, Anton Kovsharov, Sergey Shirobokov; correspondence Pushmeet Kohli and Swarat Chaudhuri","Erdős problems: 9 of 353 attempted (~2.5%), counted by formal sub-question: #12(i), #12(ii), #125 (variant), #138 (variant), #152, #26 (Tenenbaum variant), #741(i), #741(ii), #846 (Lean files in github.com/google-deepmind/alphaproof-nexus-results)","The 'two questions open for 56 years' are Erdős #12(i) and #12(ii), posed by Erdős and Sárközy in 1970 (paper, Sec. 3)","OEIS: Gemini autoformalized 492 open OEIS conjectures; the agent proved 44 that manual review found correctly formalized and previously unproven. Test lemmas checking the first terms of each sequence guarded against misformalization","Comparison: the later OEIS Open benchmark (arXiv 2608.11941) reports 147 of 492 formalised open OEIS conjectures resolved by newer models at $50/attempt; OpenAI's 6 Oct 2026 release lists 722 manuscripts, mostly not formally verified (see related entries)","Deployment: proved an exact O(1/t) last-iterate rate for Anchored Gradient Descent-Ascent in min-max convex-concave optimization while also discovering a new step-size schedule; solved 2 of 4 open Hilbert-function problems with Gergely Bérczi, incl. Zanello's log-concavity conjecture for pure O-sequences of codimension 3 and type 2; helped on a Ben Green list problem; quantum optics problems with Mario Krenn","Agents: a 'basic' agent (prover subagents + Lean compiler feedback) and a 'full-featured' agent with evolutionary coordination, Elo-rated sketches (Gemini 3.0 Flash raters) and AlphaProof as a tool; the basic agent solved all 9 Erdős problems but cost more on the hardest","Science vol. 394, issue 6820, pp. 234-239 (Research Article). Same issue: Perspective 'AI for research mathematics has arrived' by Jeremy Avigad (CMU) and Matthew Ballard (U. South Carolina), pp. 166-167 (per EurekAlert: even unsuccessful AI proof attempts can help mathematicians understand problems), and an 'Expert Voices' piece by Emily Riehl, 'AI has solved many math problems, but it has not solved math'","erdosproblems.com credits DeepMind on #12, #26, #125, #138, #152, #741, #846; #741 and #846 were found independently by an internal OpenAI model too; for #846 the page notes a counterexample also follows from earlier work of Reiher, Rödl et al.","Viral 2nd wave: @Dr_Singularity post on 10 Oct 2026 (~45.6k views at fetch time) presented the result as new"],"key_numbers":[],"tags":["math","lean","formal-verification","erdos-problems","oeis","agents","gemini","peer-reviewed"],"science":{"field":"mathematics","subfield":"combinatorics / number theory / algebraic geometry / optimization","problem":"Open Erdős problems (erdosproblems.com), open OEIS conjectures, Zanello's log-concavity conjecture, last-iterate rate of anchored GDA","result":"Lean-verified resolutions of 9 of 353 attempted open Erdős (sub-)problems, 44 of 492 open OEIS conjectures, 2 of 4 open Hilbert-function problems, and an exact O(1/t) rate for anchored GDA.","open_since":"1970","ai_system":["AlphaProof Nexus","Gemini 3.1 Pro","AlphaProof"],"human_role":"Autonomous on the Erdős and OEIS evaluation (humans chose and formalized problems, reviewed formalizations); AI-assisted collaboration in the deployment results","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Formal proofs in Lean (public repo); peer-reviewed in Science"},"status":"confirmed","shock":"A plain LLM-plus-Lean loop matched the elaborate agent on every Erdős success, suggesting raw model capability now drives formal discovery."},"body_md":"## What happened\n\nOn 8 October 2026 *Science* published Google DeepMind's paper on **AlphaProof Nexus**, a framework of LLM agents that search\nfor formal proofs in Lean. The preprint (arXiv 2605.22763) had appeared on 21 May 2026 and was covered then by the\ntech press. Publication in Science, after peer review, brought a second wave: an EurekAlert release, expert reactions\nfrom the Spanish Science Media Centre, a Science Perspective by Jeremy Avigad and Matthew Ballard, an \"Expert Voices\"\npiece by Emily Riehl in the same issue, a Pushmeet Kohli\nthread on X, and on 10 October a viral summary by @Dr_Singularity.\n\nThe system pairs Gemini 3.1 Pro prover subagents with the Lean compiler. Every proposed proof is machine-checked, and\nfailed attempts feed back into the next try. The paper compares a **basic agent** (independent subagents plus compiler\nfeedback) with a **full-featured agent**, which coordinates subagents with an evolutionary algorithm, ranks proof\nsketches with Gemini 3.0 Flash raters and can call AlphaProof (the 2024 RL prover) as a tool.\n\nResults (from the paper):\n- **Erdős problems:** 9 of 353 attempted, autonomously, at a few hundred dollars of inference per problem. The 9 are\n  formal sub-questions: #12(i), #12(ii), #125, #138, #152, #26, #741(i), #741(ii) and #846, several in variant forms.\n  The two \"open for 56 years\" are #12(i) and (ii), posed by Erdős and Sárközy in 1970. In a post-hoc ablation, the basic\n  agent also solved all 9, at higher cost on the hardest ones.\n- **OEIS:** Gemini autoformalized 492 open conjectures from the Online Encyclopedia of Integer Sequences. The agent\n  proved 44 that manual review judged correctly formalized and previously unproven.\n- **Research deployments:** an exact O(1/t) last-iterate rate for Anchored Gradient Descent-Ascent in min-max\n  convex-concave optimization, found together with a new step-size schedule. With Gergely Bérczi (Aarhus), it solved 2 of 4\n  open problems on Hilbert functions, including Zanello's log-concavity conjecture for pure O-sequences in the\n  (codimension 3, type 2) case. It also helped resolve a problem from Ben Green's list, found misformalizations in\n  the literature and supported quantum optics work with Mario Krenn.\n\n## Why it matters\n\nThis is a peer-reviewed, fully formal account of LLM agents solving open research problems at scale and at low cost.\nIts main finding, that a simple generate-and-check loop does about as well as a complex agent, supports the view that\nmodel capability, not agent engineering, now drives AI mathematical discovery.","disputed":[],"related":[{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":71,"official":13,"filed":"2026-10-07","updated":"2026-10-10","orgs":["OpenAI"]},{"id":"2026-09-21-openai-100-open-problems-claim","url":"https://postcutoff.com/e/2026-09-21-openai-100-open-problems-claim/","date":"2026-09-21","date_precision":"day","short_title":"OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts)","deck":null,"takeaway":"If substantiated, it would mean open problems are being resolved at industrial scale.","category":"science","category_label":"Science & math","importance":3,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-29","updated":"2026-10-10","orgs":["OpenAI"]},{"id":"2026-09-10-astra-leanopenproblems-september-results","url":"https://postcutoff.com/e/2026-09-10-astra-leanopenproblems-september-results/","date":"2026-09-10","date_precision":"day","short_title":"GPT-6 Astra's Epoch AI run adds more Lean-checked results","deck":"Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved","takeaway":"Autonomous formal proof search now turns out a steady stream of mid-level resolved conjectures, not one-off headlines.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI","Epoch AI"]},{"id":"2026-05-20-ai-disproves-erdos-unit-distance-conjecture","url":"https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/","date":"2026-05-20","date_precision":"day","short_title":"OpenAI model disproves Erdős's 80-year-old unit distance conjecture","deck":null,"takeaway":"This is the moment AI crossed from solving competition problems to settling a famous open research conjecture, reshaping debate about AI's role in mathematics.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":8,"official":2,"filed":"2026-09-29","updated":"2026-10-10","orgs":["OpenAI"]},{"id":"2026-05-09-deepmind-ai-co-mathematician","url":"https://postcutoff.com/e/2026-05-09-deepmind-ai-co-mathematician/","date":"2026-05-09","date_precision":"day","short_title":"Google DeepMind's 'AI co-mathematician' sets FrontierMath Tier 4 record and helps resolve a Kourovka Notebook problem","deck":null,"takeaway":"FrontierMath Tier 4, built to resist AI for years, was nearly half solved 18 months after launch.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Google DeepMind","University of Oxford"]},{"id":"2024-07-25-alphaproof-imo-silver","url":"https://postcutoff.com/e/2024-07-25-alphaproof-imo-silver/","date":"2024-07-25","date_precision":"day","short_title":"AlphaProof and AlphaGeometry 2 reach IMO silver-medal standard","deck":null,"takeaway":"First AI to reach medal level at the IMO; a year later, natural-language LLMs reached gold.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Google DeepMind"]}],"people":[{"id":"demis-hassabis","name":"Demis Hassabis","url":"https://postcutoff.com/person/demis-hassabis/"},{"id":"pushmeet-kohli","name":"Pushmeet Kohli","url":"https://postcutoff.com/person/pushmeet-kohli/"},{"id":"george-tsoukalas","name":"George Tsoukalas","url":"https://postcutoff.com/person/george-tsoukalas/"},{"id":"swarat-chaudhuri","name":"Swarat Chaudhuri","url":"https://postcutoff.com/person/swarat-chaudhuri/"}],"posts":[{"id":"2026-10-08-pushmeet-alphaproof-nexus-science-thread","title":"Pushmeet Kohli thread: DeepMind and mathematics; AlphaProof Nexus paper appears in Science today","url":"https://postcutoff.com/p/2026-10-08-pushmeet-alphaproof-nexus-science-thread/"},{"id":"2026-10-10-dr-singularity-alphaproof-nexus","title":"\"Another big AI math drop\": viral summary of AlphaProof Nexus","url":"https://postcutoff.com/p/2026-10-10-dr-singularity-alphaproof-nexus/"}],"videos":[{"id":"two-minute-papers-deepmind-alphaproof-nexus-strange-new-way","title":"DeepMind's New AI Found A Strange New Way To Think","url":"https://postcutoff.com/v/two-minute-papers-deepmind-alphaproof-nexus-strange-new-way/"}],"models":[],"changes":[{"date":"2026-10-10","type":"filed","text":"Created (Science publication 8 Oct; the May arXiv preprint had only been mentioned in passing in [OpenAI model disproves Erdős's 80-year-old unit distance conjecture](https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/)). The viral X post of 10 Oct was the trigger."}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-10","run":null,"sources_read":null,"updated":"2026-10-10","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":161,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":100,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":191,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":130,"in_training_data":false}],"short_url":"https://postcutoff.com/s/google-deepmind-alphaproof-nexus"}