OpenAI says it has produced three major mathematical and computer-science results, while Anthropic says its own work also challenges a long-standing complexity conjecture. The claims could matter enormously if they withstand scrutiny, but the proofs and methods are not yet open to full public inspection.
Watch Desk analysis
What happened
Scientific American reports that OpenAI claims to have proved the Kakeya conjecture in every dimension, established the existence part of the Yang-Mills problem, and refuted the Exponential Time Hypothesis (ETH) with a faster-than-exponential-time algorithm for 3-SAT. The Yang-Mills mass-gap question remains open, so this is not a solution to the full Millennium Prize problem.
The same report says Anthropic has announced a result that also disproves ETH, based on work by an internal model. OpenAI’s ETH claim is being assessed by outside computer scientists, including researchers who first proposed the hypothesis. The report says OpenAI has not made its model, prompts, computing time or chain-of-thought outputs public.
Why it matters
ETH is treated by many computer scientists as a foundational assumption about how brutally the difficulty of certain problems grows. A genuine refutation would change that picture, with possible consequences for theory and for assumptions built on computational hardness. The reported 3-SAT result does not, by itself, settle P versus NP or establish a practical route to breaking encryption.
The distinction between a claimed proof and a checked proof is doing real work here. The reporting describes extraordinary results, but the underlying proofs and full methods are not available for public examination. Anthropic’s result is a competing claim, not independent confirmation that either company’s proof is sound.
Our read
This is a major scientific story, and also a reminder that a headline saying “proved” can arrive before the mathematical community has had time to do the proving-about-the-proof. The most concrete next step is independent scrutiny, not choosing a corporate winner. Keep the claims attributed until the papers, methods and expert review can support firmer conclusions.
What to watch
- Whether OpenAI publishes the proofs and enough detail for independent researchers to check them.
- What the outside investigators conclude about the ETH result, and when.
- Whether Anthropic publishes its proof and methods for independent examination.
- Whether any claimed result changes expert understanding after review.
Discussion spark: For claims this consequential, should AI companies publish the full proof and methods before announcing a breakthrough, or is early disclosure worthwhile even when independent checking comes later?
Sources and evidence
- Source update (27 October 2026, 14:30 UTC)
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