OpenAI and Anthropic say AI has disproved the exponential time hypothesis, and complexity theory is counting its dead

OpenAI and Anthropic each claimed this week that AI systems they built have disproved the exponential time hypothesis, the 27-year-old assumption first set out in 1999, publishing machine-checkable Lean proofs on GitHub that no researcher has independently verified. UC Santa Barbara professor Daniel Lokshtanov estimates that if the result is confirmed, around 50 of his published papers are "pointless" or need revisiting, while OpenAI has asked Russell Impagliazzo and Ramamohan Paturi of UC San Diego and MIT professor Ryan Williams to lead an independent examination that so far consists of a Google form and a conference OpenAI has committed to fund. OpenAI's repository correction log dated October 7 records that a sign error invalidated one manuscript's argument, three manuscripts were withdrawn, 14 were revised with proof repairs and 13 updated to cite the repaired editions, and that 300 of 719 top-line results in its 722-manuscript release on 400 problems carried Lean formalizations, about 42 percent.

Science & Frontier https://provenbrief.com/category/science October 29, 2026 OpenAI and Anthropic say AI has disproved the exponential time hypothesis, and complexity theory is counting its dead Separate AI-produced proofs published this week claim the exponential time hypothesis, a 27-year-old assumption underpinning decades of algorithm and complexity research, is false. Researchers have not verified the machine-checkable Lean proofs yet, but one estimates around 50 of his papers are now obsolete or need revisiting, cryptosystem designers are being told to be scared, and OpenAI is funding a conference to unpack the fallout. OpenAI and Anthropic each claimed this week that AI systems they built have disproved the exponential time hypothesis, a 27-year-old assumption first set out in 1999 that has anchored algorithm and complexity research ever since 1 ref-1 2 ref-2 . No researcher has independently verified either proof, or had time to absorb it 2 ref-2 . The accounting started anyway: UC Santa Barbara professor Daniel Lokshtanov estimates that if the result is confirmed, around 50 of his published papers are "pointless" or need revisiting 2 ref-2 . The claims are the headline. What is underneath is the checking apparatus. The reading apparatus is a Google form Both firms published their proofs on GitHub in natural language and in Lean, a language a machine can check step by step 2 ref-2 . OpenAI's claim arrives with a 60-odd-page paper that academics tell Wired is better written than the companies' past output 2 ref-2 . What the field has to read it with is smaller. OpenAI asked Russell Impagliazzo and Ramamohan Paturi of UC San Diego, part of the group that originally proposed the hypothesis in 1999, and MIT professor Ryan Williams, who has refuted a version of it, to lead an independent examination 1 ref-1 2 ref-2 . So far the effort consists of a Google form asking for volunteers and feedback, plus a conference OpenAI has committed to fund 2 ref-2 , and an open letter asking the community what norms should govern early access to AI results, how credit should recognize the work of checking, and how students should be supported as research directions shift 1 ref-1 . Lean is a checker, not a reviewer: formal verification confirms a proof follows its stated logical rules, and does not replace scrutiny of the assumptions, interpretation, and implications around it 3 ref-3 . The day before the release, a panel of math advisers OpenAI had assembled recommended that labs not designate chosen recipients for early access to results, and instead publish on scholarly repositories, with Lean proofs and funding for the people doing the understanding; OpenAI's release follows part of that recipe, and the company admits the approach could raise issues of its own 1 ref-1 . The repo keeps a correction log on itself Wired reports the proofs are on GitHub; eWeek reports both versions of the proof are public. Neither reports what else OpenAI's repository says about itself. Its history file carries an entry dated October 7 that is a correction log: a sign error invalidated one manuscript's argument, and three manuscripts were withdrawn in consequence, the sign-error paper itself plus two that depended on its construction; 14 more were revised with proof repairs, and 13 were updated to cite the repaired editions 4 ref-4 . The same entry tallies the program behind October's 722-manuscript release on 400 problems 2 ref-2 : 300 of 719 top-line results carried Lean formalizations, about 42 percent, which leaves 419 without one 4 ref-4 . OpenAI says its models produced the ETH proof on October 16 and formalized it in Lean the following day 3 ref-3 . The manuscript accompanying the Lean proof is dated October 25 in the repository's own scope note, and the public release landed October 26 5 ref-5 1 ref-1 . So the cadence runs: 722 manuscripts dumped October 6, repairs and withdrawals logged October 7, the next headline claim produced October 16, nine days after part of the dump was pulled. Generation runs on a changelog. Verification runs on a reading list. The researchers are counting their dead - Linköping University professor Peter Jonsson will have work presented at the IEEE Symposium on Foundations of Computer Science in two weeks and says much of it has been "dead" as of Monday; he expects talk openings like "ETH is false, half of my paper does not work" to be common at the conference, though he thinks his paper may be salvageable by another method 2 ref-2 . - Daniel Lokshtanov, who plans to join the verification effort, says that if the result holds it will have rendered him "galactically wrong" and calls it "a giant bomb" dropped into the middle of the field 2 ref-2 . - Thore Husfeldt, head of the algorithms research group at the IT University of Copenhagen, spent most of this year on one problem in his subfield and says that as far as he can tell it is now "all irrelevant"; his department has discussed hiring an occupational psychologist because some colleagues are depressed, even as he describes researchers as "overexcited bunnies" exploring suddenly illuminated ground 2 ref-2 . Mid-interview he received what he suspects is an AI-generated 17-page paper on the fallout, sent from a random Gmail address, and declined to read it because reading math is a large investment of cognitive time 2 ref-2 . - Victor Lagerkvist, an associate professor at Linköping, plans to return to problems he had abandoned on the assumption the hypothesis was accurate 2 ref-2 . If either proof holds, what changes in practice The hypothesis says, in effect, that no algorithm can solve every instance of the standard 3-SAT problem in subexponential time as variables grow, and researchers have leaned on it in hundreds of studies as a conditional limit on how efficiently other problems can be solved 3 ref-3 . If it falls, conclusions that assumed it need revisiting 3 ref-3 . On cryptography, the honest version is narrow: this particular result does not directly affect the cryptography underpinning web security, but as Lokshtanov puts it, "it should definitely scare some cryptosystem designers" 2 ref-2 . And nothing runs faster this week: OpenAI says finding practical algorithms "will require significant further research," and it has demonstrated no improvement to production software 3 ref-3 . Claims this week arrived at model speed. The checking runs on three professors, a Google form, and a conference budget, and it will take exactly as long as reading takes. That gap between the two speeds is the week's real result. References The Verge, Oct 26 2026 https://www.theverge.com/ai-artificial-intelligence/1019143/openai-eth-kaseya-science-math-physics theverge.com ↗ Wired, Oct 29 2026 https://www.wired.com/story/ai-made-decades-of-computer-science-research-irrelevant/ wired.com ↗ eWeek, Oct 27 2026 https://www.eweek.com/news/openai-ai-proof-exponential-time-hypothesis/ eweek.com ↗ GitHub, OpenAI math repository history https://github.com/openai/math/blob/main/history.md github.com ↗ GitHub https://github.com/openai/math/blob/main/lean/docs/378.md github.com ↗ Cite this story ProvenBrief 2026 . "OpenAI and Anthropic say AI has disproved the exponential time hypothesis, and complexity theory is counting its dead." ProvenBrief. https://provenbrief.com/story/openai-and-anthropic-say-ai-has-disproved-the-exponential-time-hypothesis-and-co Free to quote and link with attribution. 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