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Background
Background
The local smoothing conjecture was proposed by Christopher Sogge in 1991 and sits at the top of a chain of harmonic-analysis conjectures: it implies the Bochner–Riesz conjecture, which implies the restriction conjecture, which implies the Kakeya conjecture. The Exponential Time Hypothesis posits that 3-SAT cannot be solved in sub-exponential time and is widely used to derive hardness results, including in parameterized complexity. Yang–Mills existence and mass gap is one of the seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000.
OpenAI announced on October 26, 2026 that its models produced natural-language proofs of three long-standing open results: local smoothing for the Kakeya conjecture, the falsity of the Exponential Time Hypothesis (ETH), and the existence of quantum chromodynamics and completions of the Standard Model. The Yang–Mills existence proof was generated on September 9, 2026 and the QCD proof on September 18, 2026, with the release held back until Lean formalization of the Yang–Mills result was completed on October 16, 2026.
OpenAI framed the release around how researchers can verify and build on the results, and said the volume and complexity of the work was why publication waited on Lean formalization. Commenters with complexity-theory background treated an ETH refutation as the most consequential item, since ETH underpins many hardness and lower-bound results.
The local smoothing claim is presented as settling Kakeya and Stein's restriction conjecture in every dimension, a problem open since 1991. No independent verification of the proofs has been reported; coverage at the time of the announcement described the results as still unverified.
hackernews · OpenAI Blog · · Discussion · 4 sources
Analysis and references
Impact analysis
The claims feed the AI-crypto narrative around verifiable computation and machine-assisted research, where formalization artifacts such as Lean proofs are treated as a trust layer; that channel moves sentiment for AI- and verifiable-compute-related tokens rather than any on-chain flow. The story's market relevance depends on independent verification, not on the announcement itself.
Discussion
Commenters with complexity-theory backgrounds debated what an ETH refutation implies for existing hardness proofs and whether it should shift priors toward P=NP; one noted that an author of the ETH paper was participating in the discussion. Others called the pace of results "mind blowing," while several said the headline was crafted to preempt the usual backlash and pointed to the announcement text listing the three results.
References
Tags
#openai#ai-for-science#mathematics#complexity-theory#quantum-field-theory