ResearchPapers, evidence & method
OpenAI and Anthropic say their AI models found subexponential 3-SAT algorithms, refuting the Exponential Time Hypothesis
OpenAI and Anthropic say internal models separately found subexponential-time algorithms for 3-SAT, disproving a 1999 complexity-theory assumption. Both labs released Lean-checked proofs that experts are now reviewing in public. The speedup has no practical use and leaves P vs NP open.
From the Two Minute Papers video of October 30, 2026 and cited sources: Both labs published papers with Lean formalizations. [2] [3] [4] [5]
SOURCES[2] Lance Fortnow, Computational Complexity: The Exponential… blog.computationalcomplexity.org[3] The Next Web: OpenAI says its AI model has disproved the ETH thenextweb.com[4] FourWeekMBA: Anthropic Says Claude Also Refuted ETH, Using… fourweekmba.com
AT Pulse research deskPublished · Updated · 5 min read · AI-assisted analysisAI-assisted, not human-edited. Methodology Corrections Who owns this site
- Original source
- Two Minute Papers video, independent perspective
- Brief by
- AT Pulse research desk AI-assisted; publisher and contact on the desk page
- When
- October 30, 2026 video published
- Covered
- Oct 31 edition
- Evidence
- 11 sources across 10 sites; transcript sourced
- 1OpenAI Claims It Just Did The Impossibleyoutube.com
- 2Lance Fortnow, Computational Complexity: The Exponential…blog.computationalcomplexity.org
- 3The Next Web: OpenAI says its AI model has disproved the ETHthenextweb.com
- 4FourWeekMBA: Anthropic Says Claude Also Refuted ETH, Using…fourweekmba.com
- 5anthropics/formal-math GitHub repositorygithub.com
- 6ETH Refutation Discussion Blogethdiscussion.blog
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YouTube video · 337K views · 6.9K likes
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Both labs published papers with Lean formalizations. OpenAI's is a deterministic, polynomial-space algorithm that also counts satisfying assignments. Anthropic's bound is stated on a word RAM. If the result holds, it refutes both ETH and its stronger form, SETH. Kintali and O'Donnell have written simpler human proofs. [2] [3] [4] [5] [10]
A Lean check confirms the logic is valid, but not that the formal statement matches the paper. OpenAI's Lean statement is reportedly weaker than its paper's bound, though still subexponential. No person has yet checked all 1,050 modules of Anthropic's formalization. [4] [9]
The result does not speed up real workloads. One analysis puts the crossover with brute force at inputs beyond 10^10,000 variables. Reductions to problems such as TSP or Subset Sum blow up the number of variables, so this is not a fast algorithm for all of NP. Standard cryptography is reportedly unaffected. [2] [4] [8]
Checking the work has become the bottleneck. ETH's originators are leading a public review and have said that taking part does not mean they endorse the claim. Community members have rerun the Lean proofs and posted no flaws so far. Only about 42% of the 721 manuscripts in OpenAI's repository include Lean files. [3] [6] [7]
two machine-checked proofs, plus simpler human proofs, against a long-held complexity assumption. Implication: frontier models may now produce real theory results.
Read the full assessment
That puts the pressure on checking formal statements against claims, not on solvers or cryptography.
Executive brief
OpenAI and Anthropic each say an internal AI model found an algorithm that solves 3-SAT in subexponential time. If correct, that disproves the Exponential Time Hypothesis (ETH), a 1999 assumption that underpins a large body of complexity-theory results (Fortnow). Both companies say the models found their algorithms on the same day, October 16, and both published Lean-checked proofs. ETH's original authors are now leading a public review and have said explicitly that taking part does not mean they certify the claim (TNW). The result has no practical effect: the speedup only appears at absurdly large input sizes. It also leaves P vs NP unsettled.
What changed and event timeline
Two models, same discovery date
OpenAI says its internal model found a natural-language disproof on this date. Anthropic says a Claude research model found its algorithm the same day without human input (;).
Lean formalization
OpenAI reports finishing a machine-checked Lean proof one day after the discovery ().
OpenAI announces
It released "The Exponential Time Hypothesis is False" alongside claimed proofs of the local smoothing conjecture and of Yang–Mills existence. Impagliazzo, Paturi and Williams agreed to review the computer-science claim ().
Anthropic follows
It published Claude's own ETH refutation, with a Lean 4 formalization checked by an independent kernel (;).
Expert framing
Lance Fortnow explained what the result covers and what it leaves out, and pointed to simplified proofs by Ryan O'Donnell and Shiva Kintali ().
Community replication
Contributors to the review site report rerunning the Lean proofs locally. No flaws have been posted so far ().
Capabilities and access
- OpenAI: an unnamed internal model. The openai/math repo holds 721 manuscripts with Lean files for about 42% of headline results.
- Anthropic: an internal Claude research model that used 14M output tokens (FourWeekMBA). Paper and Lean code are in anthropics/formal-math, built on Lean v4.33.1.
- Neither model is publicly available.
Read the full section
- OpenAI: an unnamed internal model. The openai/math repo holds 721 manuscripts with Lean files for about 42% of headline results. OpenAI says each result used about three hours of ChatGPT Pro-level thinking compute on average, and calls the material "raw research data."
- Anthropic: an internal Claude research model that used 14M output tokens (FourWeekMBA). Paper and Lean code are in anthropics/formal-math, built on Lean v4.33.1.
- Neither model is publicly available.
Technical analysis for researchers and developers
- OpenAI bound: a deterministic algorithm with polynomial space.
- Anthropic bound: 2^O(min{k²n log log n/√log n, km/√log m}) on a word RAM (formal-math).
- Methods: one secondary write-up describes OpenAI's approach as sparse parity checks plus polynomial interpolation, and Anthropic's as tensor-network contraction (Lilting).
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- OpenAI bound: a deterministic algorithm with polynomial space. It counts satisfying assignments of k-CNF formulas (n variables, m clauses) in (n+m)^O(1)·2^O(n log log n/√log n) time (Fortnow).
- Anthropic bound: 2^O(min{k²n log log n/√log n, km/√log m}) on a word RAM (formal-math).
- Methods: one secondary write-up describes OpenAI's approach as sparse parity checks plus polynomial interpolation, and Anthropic's as tensor-network contraction (Lilting).
- Verification setup: both proofs use only Lean's standard axioms and the Comparator tool. Anthropic also checked its proof with the nanoda kernel.
- Formal statement vs paper: OpenAI's Lean statement is reportedly weaker than its paper's claim, with (log log n)² in place of log log n (John D. Cook). The formal result is still subexponential.
Claims and evidence
Read the full section
- "ETH is refuted." The evidence is two vendor papers with Lean proofs, plus simplified human-written proofs by Kintali and O'Donnell (Fortnow). A Lean check shows the proof is logically valid. It does not show that the formal statement says what the paper claims, and Anthropic's paper notes no person has yet checked its full 1,050-module formalization (FourWeekMBA).
- The algorithm also counts solutions. This is in both the video (06:00) and OpenAI's paper as described by Fortnow.
- "No human input" and the October 16 discovery date come only from the companies.
Context and prior work
Impagliazzo and Paturi proposed ETH in 1999 (Forklog). Many conditional lower bounds assume ETH or its stronger form, SETH, and Fortnow notes the new result refutes both. Before this, the best deterministic 3-SAT algorithm ran in about O(1.328^n) time (FourWeekMBA). The video's own explanation of ETH and 3-SAT runs from 00:57 to 03:40.
Limitations, safety and contested findings
- No practical speedup. Anthropic says its algorithm says nothing about solving formulas in practice (FourWeekMBA).
- It argues that Karp reductions carry the speedup to routing, frequency assignment and similar problems (04:34).
- P vs NP is untouched, as the video itself says (04:18).
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- No practical speedup. Anthropic says its algorithm says nothing about solving formulas in practice (FourWeekMBA). One analysis says it would need n > 10^10,000 to beat brute force on sparse formulas (Lilting).
- The video overstates the reach. It argues that Karp reductions carry the speedup to routing, frequency assignment and similar problems (04:34). Fortnow says Subset Sum and TSP gain nothing, because reductions blow up the number of variables. A subexponential algorithm for all of NP would need 2^(n^o(1)) time, which this result "doesn't get close to" (Fortnow).
- P vs NP is untouched, as the video itself says (04:18).
- Review is incomplete, which the video also acknowledges (03:59).
Business and practitioner implications
- Optimization stacks are unaffected. SAT/SMT solvers, scheduling and routing workloads see no real-world speedup.
- Cryptography is unaffected. Reviewer Ryan Williams reportedly said standard cryptosystems remain secure (Lilting).
- The real signal is AI as a research tool. Two labs independently arrived at frontier theory results with machine-checked proofs.
Read the full section
- Optimization stacks are unaffected. SAT/SMT solvers, scheduling and routing workloads see no real-world speedup.
- Cryptography is unaffected. Reviewer Ryan Williams reportedly said standard cryptosystems remain secure (Lilting).
- The real signal is AI as a research tool. Two labs independently arrived at frontier theory results with machine-checked proofs.
- A review template is emerging: reviewers working without NDAs, public repositories, and community re-checking of the Lean proofs (TNW; ETH discussion blog).
- Verification is now the bottleneck. With 721 OpenAI manuscripts and only about 42% formalized, practitioners should read "Lean-verified" claims carefully and check that the formal statement matches the paper's claim.
The source trail.
Sources (11)
OpenAI Claims It Just Did The Impossible
Transcript retrieved via youtube_auto_captions; language en. Timestamped text, not direct audiovisual review. Source: https://www.youtube.com/watch?v=CWdTBLsbaQs. Automatic captions/transcription may contain errors.
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