OpenAI and Anthropic say models disproved the exponential time hypothesis
Start here
On October 26, OpenAI announced that its internal models produced new results awaiting verification in harmonic analysis, complexity theory, and quantum field theory, and that it is piloting a release approach in which external experts lead verification and interpretation. In mathematics, the model gave a proof of the local smoothing conjecture in all dimensions, and claims that this resolves the Kakeya and Stein restriction conjectures in each dimension; the natural language argument and Lean formalization have been made public. In computer science, the paper claims to give an algorithm for exactly counting k-SAT solutions in subexponential time, which, if valid, would refute the exponential time hypothesis (ETH); Impagliazzo, Paturi, and Ryan Williams, who proposed ETH, participated in independent review and emphasized that participation does not imply endorsement. In physics, the model claims to give a proof of the existence of certain four-dimensional quantum field theories, including quantum chromodynamics, which has not yet been published. Hours later, Anthropic also released an independent disproof of ETH, stating that the core algorithm was independently found by its internal Claude research model on October 16 with no human input, and that the sole named author is Claude. Both companies state that they obtained the result with internal models on October 16, have provided Lean formal proofs, and that peer review is still pending.
Generated by AI from reports · Updated 1 days ago
Progress timeline
- WebThe Information· Media coverageEncryption Under Siege? OpenAI, Anthropic Disprove Key Cryptography Theorem(opens the original post in a new tab)
- XX @garyMarcus· DiscussionThe practical value of this new OpenAI result is immense!(opens the original post in a new tab)
- X@antiAIvo· DiscussionIn Chinese, as written今天被ETH刷屏，这不是以太坊 而是理论计算机科学中一个 27 年的猜想 OpenAI 和 Anthropic 的 AI 在同一天推翻了它 说到理论计算机 让我想来了看陈立杰的清华特奖答辩视频 巧的是，他在 MIT 读博时的导师 Rya…(opens the original post in a new tab)
- WebNext (France)· Media coverageOpenAI patauge encore en annonçant des « découvertes » scientifiques sans en être sûr(opens the original post in a new tab)
- WeChat · QbitAISelectedOpenAI and Anthropic announce that internal models overturned the Exponential Time Hypothesis; conclusions await peer review
OpenAI and Anthropic successively announced that on October 16 the two companies separately used internal models to obtain results overturning the Exponential Time Hypothesis (ETH), both having provided Lean formal proofs, with the conclusions still awaiting peer review. The ETH hypothesizes that no subexponential-time algorithm exists for the 3-SAT problem, and it had previously been widely treated by the academic community as a premise, with a large number of complexity conclusions in fields such as graph theory and computational biology built on it. OpenAI's model found an algorithm that exactly counts solutions to logical formulas in subexponential time and polynomial space, and invited ETH proposers Russell Impagliazzo and Ramamohan Paturi and MIT mathematician Ryan Williams to form a group responsible for verification, and made all raw data public at the mathematicians' request. Anthropic stated that its core algorithm was independently discovered by an internal Claude model, with human researchers participating in improving the presentation. The three mathematicians published an article stating that their participation does not constitute endorsement of the claim and that they are organizing an independent review; if the conclusion is ultimately verified as valid, a large number of computational hardness judgments relying on the ETH will need to be reexamined.
- X@tcimage3d· DiscussionIn Chinese, as writtenOpenAI宣稱AI解出ETH、Kakeya、量子場論三難題，Anthropic同日跟進。重點不在宣稱多猛，在都還沒驗證。審查者明說：參與不代表背書。這句話比宣稱本身更重要。(opens the original post in a new tab)
- X@guan_xujun· Discussionhttps://mp.weixin.qq.com/s/YAXCYdP-UNuhrHOygS01Dw(opens the original post in a new tab)
- WeChat · AI Era (Xinzhiyuan)SelectedOpenAI and Anthropic announce on the same day that internal models disproved the Exponential Time Hypothesis ETH
OpenAI and Anthropic announced on nearly the same day that their respective internal models refuted the Exponential Time Hypothesis (ETH) proposed in 1999, namely that no subexponential-time algorithm exists for 3SAT. An Anthropic paper states that the core algorithm was independently found by an internal Claude research model on October 16, consuming 14 million output tokens with no human input, and that the sole named author is Claude; both companies released Lean formal proof code. It is said that Impagliazzo and Paturi, the original proposers of ETH, are jointly reviewing both companies' code line by line with teams from MIT and other universities, and that Nvidia senior AI architect Shiva Kintali condensed OpenAI's 60-plus-page proof into a 22-page simplified version. Because hundreds of papers in the past assumed ETH holds, a large number of conditional research conclusions will need to be reassessed.
- DigitalTodayOpenAI says its AI discovered an algorithm that may challenge the "Exponential Time Hypothesis," sparking heated discussion in crypto circles due to the shared abbreviation
OpenAI stated that its AI model discovered an algorithm that can count the satisfying solutions of 3-SAT problems faster than existing theoretical expectations, potentially challenging the Exponential Time Hypothesis (ETH) in computational theory, with the results still awaiting independent verification by the academic community. According to a report by blockchain media outlet Decrypto on the 27th local time, the result came from a 67-page paper; the AI discovered the result on October 16, and the next day the proof was formally verified with the help of Lean and made public on GitHub. Russell Impagliazzo, Ramamohan Paturi and MIT's Ryan Williams pointed out in an open letter that participating in review does not equal endorsement, and the conclusion still needs independent reproduction. Because ETH is also the ticker for the Ethereum token, the news caused confusion and jokes in the crypto community.
- The Verge AIResearchers Increasingly Tired of OpenAI Gating Its Breakthrough Results
OpenAI announced three major results in mathematics, computer science and physics, and said it is adjusting how it releases results, but several researchers interviewed expressed growing dissatisfaction with its gated disclosure. The mathematics result claims to resolve the Kakeya conjecture in all dimensions, and OpenAI has publicly released the AI-generated output; the computer science result claims to refute the exponential time hypothesis and will be explained by an external team; the physics result concerns quantum chromodynamics and Yang-Mills theory and has not yet been released, with OpenAI saying it will defer to external experts on how to release it. Brown University mathematics professor Brendan Hassett said the claims appear well-founded but that he has not yet checked the Lean formalization. AGMAI, one day before the announcement, advised AI companies not to select specific individuals to receive results first, and after the release urged OpenAI to publish the natural-language and Lean outputs of the physics result without delay.
- TechCrunch AISelectedFrontier AI labs say their models refuted the Exponential Time Hypothesis. What does this mean?
OpenAI announced solutions to three mathematical problems that have long troubled researchers, one of which is a refutation of the Exponential Time Hypothesis (ETH); several hours later, Anthropic also released its own refutation of ETH. ETH is part of the P vs NP problem and has long provided evidence for P ≠ NP; the two labs say their models found algorithms that solve the relevant problems in subexponential time, which, if valid, would shake this hypothesis. MIT professor Ryan Williams and ETH's original authors Russell Impagliazzo, Ramamohan Paturi and others participated in interpreting the results, and published an open letter soliciting feedback and discussing norms for AI participation in scientific research. Williams said he asked OpenAI science lead Dan Roberts whether this affects cryptographic systems; the latter said it had been checked and believed it would not lead to practical cracking.
- The DecoderSelectedOpenAI invited external experts, including Nobel laureates, to review proofs produced by its AI in mathematics, computer science and physics
OpenAI says its AI produced three major proofs in mathematics, computer science and physics, and asked external experts to serve as gatekeepers to review them. In computer science, OpenAI says the system proved that the Exponential Time Hypothesis (ETH) is false, which, if it holds, would shake the foundations of hundreds of papers; Ryan Williams of MIT and Impagliazzo and Paturi, the original proposers of ETH, will independently review it, and the three stressed in an open letter that participation does not mean endorsement; OpenAI has published the written proof and the Lean proof. In mathematics, OpenAI says the model proved that the local smoothing conjecture holds in all dimensions, and with it resolved the Kakeya conjecture and the Stein restriction conjecture; the relevant proofs have been published directly. In physics, OpenAI says the AI proved that certain quantum field theories, including Yang-Mills theory, exist mathematically in four dimensions, but the corresponding Millennium Prize problem is only resolved as to the existence part; a team led by David Gross, winner of the 2004 Nobel Prize in Physics, is not disclosing materials for now and plans to write up the results itself.
- News9liveSelectedOpenAI says it has solved the Kakeya conjecture, ETH and quantum field theory
OpenAI announced that its internal models produced three AI-assisted research results, covering mathematics, theoretical computer science and physics, all of which are currently research claims awaiting expert verification. In mathematics, OpenAI says the model proved the local smoothing conjecture in all spatial dimensions and would simultaneously resolve the Kakeya and Stein restriction conjectures in each dimension, with the output produced on October 18 and Lean formalization completed the next day, and the natural-language argument and Lean proof have been made public. In computer science, the system found a deterministic subexponential-time exact counting method for satisfying assignments of fixed-width Boolean formulas, which, if the 3-SAT case holds, would refute the exponential time hypothesis (ETH); a working group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will review the proof.
- WeChat · Synced (Jiqizhixin)SelectedOpenAI claims to have disproved the exponential time hypothesis and solved the Kakeya conjecture in all dimensions
On October 26, OpenAI announced that its internal model achieved three results: proving the local smoothing conjecture in all dimensions, disproving the exponential time hypothesis (ETH), and establishing a rigorous existence proof for certain four-dimensional quantum field theories including quantum chromodynamics. ETH holds that no subexponential-time algorithm exists for solving all 3-SAT problems; OpenAI presented an algorithm with runtime 2^{O(n/√log n)}, which, if the proof holds, disproves this foundational hypothesis that has been used for nearly 30 years. The original proposers Impagliazzo and Paturi are organizing an independent review together with Ryan Williams of MIT. On the mathematics side, the model, with the help of a new auxiliary partial differential equation, advanced the local smoothing proof to all dimensions, and according to OpenAI's account thereby resolved the Kakeya conjecture and the Stein restriction conjecture in all dimensions; the Lean formalized proof has been made public. The physics part includes an existence proof for pure Yang–Mills theory; no resolution of the mass gap was announced, a working group led by David Gross is organizing it first, and the AI's original proof is not being made public for now.
- WeChat · AI Era (Xinzhiyuan)SelectedOpenAI announces that its internal AI disproved the exponential time hypothesis and gave a proof of the all-dimensional local smoothing conjecture
OpenAI announced that its internal AI independently completed three scientific discoveries: giving a proof of the local smoothing conjecture in all dimensions, disproving the exponential time hypothesis (ETH), and giving an existence proof for four-dimensional QCD and pure Yang-Mills theory; the results are being reviewed by external experts. The local smoothing conjecture is regarded as the mother conjecture of the Kakeya conjecture; on ETH, a 67-page manuscript proposes an algorithm for solving 3-SAT in subexponential time, and OpenAI explicitly states that refuting ETH is not equivalent to proving P=NP; the hypothesis's proposers Impagliazzo, Paturi and Ryan Williams participated in the review. The physics working group is led by Nobel laureate David Gross; the model gave an existence proof for Yang-Mills and QCD in September, and the positive mass gap required by the Millennium Prize problem has not been announced as solved.
- X @jiqizhixin (Synced)SelectedNew OpenAI paper claims the Exponential Time Hypothesis (ETH) is false
OpenAI has released a new paper claiming to give a deterministic subexponential-time algorithm that can exactly count the solutions to k-SAT; if the proof holds, it would overturn the Exponential Time Hypothesis (ETH). The paper bluntly titles itself "The Exponential Time Hypothesis Is False"; ETH is one of the core assumptions of modern fine-grained complexity theory and has long been used to argue that a large family of problems may not be substantially sped up. If the proof holds, a large number of conditional lower bounds built on ETH and SETH will need to be re-examined. The paper also claims this result has implications for problems such as #SAT, graph coloring, independent set, and perfect matching.
- Scientific AmericanSelectedOpenAI and Anthropic's head-to-head proofs shock scientists
OpenAI released three major proofs generated by internal large language models, involving mathematics, physics and computer science. The three results respectively solve the Kakeya problem in all dimensions, claim to prove the "existence" part of the Yang-Mills problem, and refute the exponential time hypothesis (ETH) with a 3-SAT algorithm slightly faster than exponential time. Hours later, Anthropic released independent results based on its internal model, likewise refuting ETH. The models that generated the proofs remain undisclosed; the prompts used, computation duration and chain-of-thought outputs were all not published; the related releases continue to conflict with the recommendations of the Mathematics and Artificial Intelligence Advisory Group (AGMAI) organized by OpenAI, which has called on OpenAI to publish an expert paper related to the Yang-Mills problem. Regarding the ETH result, OpenAI has invited its proposers Impagliazzo, Paturi and MIT's Williams and other external experts to independently verify it, and will sponsor a related computer science conference.
- The Verge AISelectedOpenAI announces three major scientific discoveries and a new way of collaborating with academia
OpenAI announced three major discoveries in mathematics, computer science, and physics, this time with external experts serving as "shepherds" to guide verification, less than a month after it mass-released hundreds of papers claiming to solve mathematical problems. The mathematics result is a proof of local smoothing for the Euclidean wave equation, which OpenAI says simultaneously resolves the Kakeya conjecture and the Stein restriction conjecture in all dimensions, and the model output has been published directly on GitHub; the computer science paper claims to refute the exponential time hypothesis (ETH) with a subexponential-time algorithm, and an independent working group consisting of Russell Impagliazzo, Ramamohan Paturi, and Ryan Williams will verify and explain the proof; the physics result has not been published in full, and OpenAI says it is a proof of the existence of four-dimensional quantum field theories including QCD, with David Gross to lead an expert panel to guide it.
- X @matthew_d_greenOpenAI announces three new discoveries in mathematics, computer science and physics; commentary says none involve practical cryptanalysis
OpenAI published a post announcing three new discoveries in mathematics, computer science and physics, involving harmonic analysis, complexity theory and quantum field theory, respectively. X user @matthew_d_green, citing the announcement, said none of these results is related to practical cryptanalysis.
- OpenAISelectedOpenAI collaborates with the academic community to build understanding of the model's scientific results
OpenAI announced that its model has produced new results in three directions: harmonic analysis, complexity theory, and quantum field theory, and is piloting a release approach in which external experts lead the organization of verification and interpretation. In mathematics, the model gave a proof of the local smoothing conjecture in all dimensions, thereby resolving the Kakeya and Stein restriction conjectures; the AI-written proof and Lean formalization have been made public. In computer science, it claims to prove that the exponential time hypothesis (ETH) is false; a group led by Russell Impagliazzo, Ramamohan Paturi, and Ryan Williams is responsible for verification, and OpenAI has committed to funding a conference to discuss its implications. In physics, it gives a proof of the existence of a four-dimensional interacting quantum field theory including gauge fields and fermions, covering the unification completion of QCD and the Standard Model; a group led by David Gross decided to temporarily withhold the AI-produced materials and write its own.
Related discussion
- X@RythemRemix · RythemRemix【P≠NP予想は解けたのか？】指数時間仮説（ETH）の否定で前提が崩壊！？【Anthropic・OpenAIが発表】 https://youtu.be/PFntJuOC1JU?si=wFUG0YjikB5vUtL8 @YouTubeより(opens the original post in a new tab)
- WebThe InformationEncryption Under Siege? OpenAI, Anthropic Disprove Key Cryptography Theorem(opens the original post in a new tab)
- X@HoodieFade_ · Hoodiethis is pure f*cking insane OpenAI published a paper titled "The Exponential Time Hypothesis Is False", and the machin…(opens the original post in a new tab)
29 likes · 10 comments·Oct 30 02:33Collect
- X@AIBoticssq · AIBotics⚡ OpenAI and Anthropic just independently shattered a 27 year computational complexity barrier Both labs revealed thei…(opens the original post in a new tab)
10 likes · 2 comments·Oct 30 02:33Collect
- XX @garyMarcusThe practical value of this new OpenAI result is immense!(opens the original post in a new tab)
- X@ZhihaoJu77910 · Judo### 1/5 OpenAI played three cards today: AI proved the Kakeya conjecture in all dimensions, overturned the "exponential time hypothesis," and gave an existence proof for quantum chromodynamics/Yang-Mills (Millennium Prize level). We used to say AI is a tool. Now it has begun to discover and prove on its own.👇 ###…(opens the original post in a new tab)
- X@zugzwangworld · zugzwangThe Zugzwang Report : "OpenAI refutes ETH?" (Exponential Time Hypothesis not Ethereum) 29th October, 2026(opens the original post in a new tab)
1 likes·Oct 30 02:33Collect
- X@qihanblmg · Wigan | AI SignalThere's new progress on the math front today: Anthropic also published a proof that the exponential time hypothesis does not hold, matching OpenAI's conclusion. Yesterday it was "one party's claim awaiting review"; today it's "two parties independently reaching the same conclusion." Two independent proofs matching does not mean the proof is correct, but the starting point for review has changed: from "believe it or not..."(opens the original post in a new tab)
- X@TarasMarks35G · MarksA 1999 conjecture that a lot of computer science quietly depends on might be dead. OpenAI says an internal model dispro…(opens the original post in a new tab)
- X@grok · GrokOn October 26, OpenAI released an internal model's counterexample to ETH; the paper and Lean formalization have been made public, and Anthropic independently gave a similar result the next day. ETH proposers Impagliazzo, Paturi and Williams are leading an open review, and made clear that their participation does not equal certification. It is still pending…(opens the original post in a new tab)
- X@0xTomcat0x · 汤姆猫的小黑屋How to view the ETH (exponential time hypothesis) conjecture being disproved by OpenAI? @grok(opens the original post in a new tab)
1 comments·Oct 30 04:50Collect
- X@kyny010020 · kyny計算機科学の根本が覆る｜ETHをOpenAIとAnthropicが同じ日に反証【これ1本で全てがわかる】 https://youtu.be/pnVVWoQcUhw?si=J8Oc4Y_TqYbRJwVv @YouTubeより(opens the original post in a new tab)
Attention trend
New sources per day
- Oct 26: 1 new sources (1 outlets, 0 accounts)
- Oct 27: 308 new sources (12 outlets, 296 accounts)
- Oct 28: 130 new sources (8 outlets, 122 accounts)
- Oct 29: 14 new sources (0 outlets, 14 accounts)
- Oct 30: 1 new sources (0 outlets, 1 accounts)