{"url":"https://newsforagent.com/news/2026-10-26/openai-local-smoothing-conjecture-proof","markdown_url":"https://newsforagent.com/news/2026-10-26/openai-local-smoothing-conjecture-proof.md","title":"OpenAI says models proved local smoothing conjecture","headline":"OpenAI says its models proved the local smoothing conjecture in every dimension","description":"OpenAI says its models proved the local smoothing conjecture in every dimension, a result it says also settles the Kakeya and Stein restriction conjectures.","section":"Research & Clinical Trials","published_at":"2026-10-26T22:24:43+00:00","modified_at":"2026-10-26T22:24:43+00:00","author":"News Desk","key_takeaways":["OpenAI says its models produced a proof of the local smoothing conjecture on October 18, 2026, and formalized it in Lean on October 19.","The local smoothing conjecture was proposed by Christopher Sogge in 1991 and belongs to harmonic analysis.","OpenAI says the proof is part of a broader release that includes over 700 AI-generated mathematical preprints."],"companies":[{"name":"OpenAI","is_primary":true,"profile_url":"https://akta.pro/company-profiles/0000008/openai","website":"https://openai.com"}],"source":{"publisher":"Crypto Briefing","title":"OpenAI says its models proved the local smoothing conjecture in every dimension","url":"https://cryptobriefing.com/openai-proves-local-smoothing-conjecture/"},"other_sources":[{"publisher":"OpenAI","url":"https://openai.com/index/building-scientific-understanding-with-academic-communities"}],"tags":["Research & Clinical Trials","OpenAI"],"body_markdown":"[OpenAI](https://akta.pro/company-profiles/0000008/openai) says its AI models have proved the local smoothing conjecture in every dimension, a problem open since 1991. The company says the result also resolves the Kakeya conjecture and Stein's restriction conjecture across all dimensions. [Crypto Briefing](https://cryptobriefing.com/openai-proves-local-smoothing-conjecture/) reported the announcement on October 26, 2026.\n\n## How OpenAI Says the Proof Was Produced\n\nOpenAI says its models produced the proof on October 18, 2026. One day later, on October 19, the proof was formalized in Lean, a theorem prover that checks a proof step by step and refuses any step it cannot verify. OpenAI published the formal proof on GitHub and released a natural language explanation so mathematicians do not have to read raw Lean code. The key ingredient, per OpenAI, is a newly introduced auxiliary partial differential equation, which describes how a quantity changes across several variables at once.\n\nThe local smoothing conjecture was proposed by Christopher Sogge in 1991 and concerns how solutions to wave equations behave when averaged over time. The Kakeya conjecture asks whether a set in Euclidean space containing a unit line segment in every direction must have full Hausdorff dimension even if its Lebesgue measure is zero. Stein's restriction conjecture deals with how functions behave when their frequency information is restricted to curved surfaces. OpenAI's position is that proving local smoothing in every dimension implies both of the others in every dimension too. The two-dimensional case of local smoothing was settled in 2019, and the three-dimensional Kakeya conjecture was resolved in 2025.\n\n[OpenAI](https://openai.com/index/building-scientific-understanding-with-academic-communities) said it announced three AI-generated proofs: the local smoothing conjecture proof, a disproof of the Exponential Time Hypothesis, and a rigorous existence proof for quantum chromodynamics and the Standard Model. The company said it will release AI outputs and Lean formalizations, with external expert groups to shepherd explanations, and pledged $5 million to the Simons Foundation for supporting research.\n\n## OpenAI's Scale Behind the Research Push\n\nOpenAI is a San Francisco-based AI research and deployment company that builds and commercializes frontier foundation models, according to [akta.pro company records](https://akta.pro/company-data). The same records show a $122 billion funding round closed in April 2026, the largest private market raise in history, led by Amazon, Nvidia and SoftBank, with Microsoft, BlackRock, Blackstone, Sequoia and Temasek participating. The records list a valuation of $852 billion after that round and monthly revenue of approximately $2 billion, up from $1 billion per quarter at the end of 2024, with enterprise sales comprising 40% of revenue.\n\n## Earlier OpenAI Announcements This Month\n\nOn October 26, 2026, OpenAI's Ari Weinstein and James Sun argued that user interfaces will persist despite AI agents outperforming humans in software tasks, Crypto Briefing reported. The same day, OpenAI researcher Roon clarified on X that Codex does not prohibit reverse engineering and that blocked requests to remove device software were accidental, with the blocks being fixed, Crypto Briefing reported. Also on October 26, the New York Post reported that OpenAI and Anthropic are instructing employees to hide company logos in public, citing threats from spies and protesters, with OpenAI warning staff of social engineering and espionage.\n\n## What Comes Next for the Proof\n\nOpenAI says it will release AI outputs and Lean formalizations, with external expert groups to shepherd explanations. The company says the local smoothing proof is part of a broader release that includes over 700 AI-generated mathematical preprints.\n","publisher":{"name":"News for Agent","url":"https://newsforagent.com"}}