{"slug": "openai-says-its-models-proved-the-local-smoothing-conjecture-in-every-dimension", "title": "OpenAI says its models proved the local smoothing conjecture in every dimension", "summary": "OpenAI announced on October 26, 2026 that its AI models produced a proof of the local smoothing conjecture in every dimension, a problem open since Christopher Sogge proposed it in 1991, with the proof generated on October 18, 2026 and formalized in the Lean theorem prover one day later. OpenAI says the result also resolves the Kakeya conjecture and Stein's restriction conjecture in all dimensions, and it published the formal proof on GitHub alongside a natural language explanation. The proof, built on a newly introduced auxiliary partial differential equation, is part of a broader release of over 700 AI-generated mathematical preprints, and the Lean formalization shifts scrutiny to whether the formal statement faithfully captures Sogge's conjecture.", "body_md": "OpenAI official logo (public domain, Wikimedia Commons) — CryptoBriefing brand treatment\n\n# OpenAI says its models proved the local smoothing conjecture in every dimension\n\nThe claimed proof would also settle the Kakeya and Stein restriction conjectures, and it comes with a machine-checked Lean formalization\n\n[OpenAI](https://cryptobriefing.com/markets/openai/) says its AI models have proved the local smoothing conjecture in every dimension. The problem has been open since 1991.\n\nAccording to OpenAI, the result also resolves the Kakeya conjecture and Stein’s restriction conjecture across all dimensions.\n\n## What OpenAI is claiming\n\nThe announcement came on October 26, 2026. OpenAI says its models produced the proof on October 18, 2026.\n\nOne day later, on October 19, the proof was formalized in Lean. Lean is a theorem prover, software that checks a proof step by step and refuses to accept any step it cannot verify.\n\nOpenAI published the formal proof on GitHub. It also released a natural language explanation, so human mathematicians do not have to read raw Lean code to follow the argument.\n\nThe key ingredient, per OpenAI, is a newly introduced auxiliary partial differential equation. A partial differential equation describes how a quantity changes across several variables at once, such as space and time.\n\n## Why these three conjectures matter\n\nThe local smoothing conjecture was proposed by Christopher Sogge in 1991. It belongs to harmonic analysis and concerns how solutions to wave equations behave when you average them over time.\n\n### AI, tech, and the markets they move—in one daily briefing.\n\nDaily. Free. Join 34,000+ readers across crypto, finance, and policy.\n\nThe Kakeya conjecture asks whether a set in Euclidean space containing a unit line segment in every direction must possess full Hausdorff dimension, even if its Lebesgue measure is zero.\n\nStein’s restriction conjecture deals with how functions behave when their frequency information is restricted to curved surfaces. It sits at the heart of modern Fourier analysis.\n\nThese problems are linked. OpenAI’s position is that proving local smoothing in every dimension implies both of the others in every dimension too.\n\nPrior progress came piece by piece. The two-dimensional case of local smoothing was settled in 2019, and the three-dimensional Kakeya conjecture was resolved in 2025.\n\n## Part of a much larger release\n\nThe local smoothing proof did not arrive alone. OpenAI says it is part of a broader release that includes over 700 AI-generated mathematical preprints.\n\n## What this means\n\nThe most important detail may be the Lean formalization. A machine-checked proof changes the conversation: the question shifts from whether the AI made an error to whether the formal statement in Lean matches the conjecture mathematicians actually care about.\n\nFormalization only guarantees that the stated theorem follows from its assumptions. Experts will want to confirm the formal statement captures Sogge’s conjecture faithfully, with no quiet simplifications.\n\n**Disclosure:** This article was edited by Diego Almada Lopez. For more information on how we create and review content, see our\n\n[Editorial Policy](https://cryptobriefing.com/editorial-policy/).", "url": "https://wpnews.pro/news/openai-says-its-models-proved-the-local-smoothing-conjecture-in-every-dimension", "canonical_source": "https://cryptobriefing.com/openai-proves-local-smoothing-conjecture/", "published_at": "2026-10-26 21:49:28+00:00", "updated_at": "2026-10-26 22:13:59.836728+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "machine-learning"], "entities": ["OpenAI", "Christopher Sogge", "Lean", "GitHub", "Kakeya conjecture", "Stein's restriction conjecture", "local smoothing conjecture"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/openai-says-its-models-proved-the-local-smoothing-conjecture-in-every-dimension", "markdown": "https://wpnews.pro/news/openai-says-its-models-proved-the-local-smoothing-conjecture-in-every-dimension.md", "text": "https://wpnews.pro/news/openai-says-its-models-proved-the-local-smoothing-conjecture-in-every-dimension.txt", "jsonld": "https://wpnews.pro/news/openai-says-its-models-proved-the-local-smoothing-conjecture-in-every-dimension.jsonld"}}