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OpenAI mathematics proofs controversy

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OpenAI and Anthropic claim their internal models have independently refuted the Exponential Time Hypothesis.From: OpenAI and Anthropic AI Models Independently Refute the Exponential Time Hypothesis

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On October 29 by 00:53 UTC, OpenAI and Anthropic claimed their internal artificial intelligence models independently discovered a subexponential-time algorithm for the 3-SAT problem. The machine-generated proofs refute the Exponential Time Hypothesis, a 25-year-old foundational assumption in theoretical computer science. Both companies released machine-checked Lean 4 proofs, though human understanding of the highly technical mathematics remains incomplete. The algorithm is asymptotically faster but will not speed up commercial software or resolve the ultimate P versus NP question.

Earlier, on October 27 by 09:08 UTC, OpenAI claimed its models had also solved the Kakeya conjecture and generated an existence proof for quantum chromodynamics. The company verified the logic using the Lean theorem prover and released the outputs as raw algorithmic data. Independent groups of scientists, including the original authors of the Exponential Time Hypothesis, are reviewing the machine-generated proofs without non-disclosure agreements to translate the data into human scientific understanding.

How the story developed

On August 2, OpenAI announced its unreleased Astra model autonomously solved 10 decadal open problems in mathematics, including the 27-year-old non-sofic group mystery. The artificial intelligence generated the arguments for approximately $2,000 in compute tokens. To prove the results and eliminate hallucination concerns, OpenAI published machine-checkable Lean 4 certificates for every solution.

By September 9, OpenAI claimed an unreleased model solved a version of the Navier-Stokes Millennium Prize problem in 88 hours. The 165-page proof demonstrated that fluid equations can develop a singularity in finite time. Independent mathematicians alleged OpenAI rushed to replicate their unpublished approach after hearing rumors of their progress, and suggested the company may have leveraged unpublished data stored in Codex. OpenAI denied accessing private user data but admitted de-identified product usage data may have improved its models. Mathematicians criticized the proof on September 26 as incomprehensible and lacking physical insight, noting the solution relies on a specific loophole involving an external force that leaves the unforced problem unsolved. OpenAI stated it will not claim the $1 million prize attached to the challenge.

The new algorithm solves 3-SAT in subexponential time, beating the exponential curve.From: OpenAI and Anthropic AI Models Independently Refute the Exponential Time Hypothesis ·

Following criticism from 25 Fields Medalists regarding the use of open math problems as benchmarks, OpenAI announced on September 28 that a new internal model had resolved over 100 additional long-standing mathematical problems. The company established a nine-member independent advisory group hosted at the Institute for Advanced Study to review emerging results and coordinate their release, though the panel cannot dictate OpenAI's research pace.

On October 7, OpenAI uploaded a repository of 722 formal mathematical manuscripts to GitHub, generated by an unreleased model for approximately $2,000 in compute time. The company withheld the model itself, preventing independent researchers from testing the system's failure modes. On October 20, mathematicians dissected a 199-page proof from the data dump that solved the quasi-Riemann hypothesis, establishing a strict zero-free boundary at 7/8. The model synthesized existing, overlooked formulas using a novel "zeta shortcut," resolving scores of secondary theorems in number theory and computer science. In total, the data dump solved 372 open mathematical problems.

The algorithm lands 3-SAT in a middle zone, faster than exponential but still profoundly slow.From: OpenAI and Anthropic AI Models Independently Refute the Exponential Time Hypothesis ·

What to watch

  • Certification of the Navier-Stokes result by the Clay Mathematics Institute remains pending.
  • Independent groups of scientists are working to translate the raw algorithmic data refuting the Exponential Time Hypothesis into human scientific understanding.

Key numbers

10
Unsolved math problems resolved
$2,000
Estimated compute cost via API
249
Pages in the published manuscript
27 years
Time the non-sofic group problem stood unsolved

From: OpenAI's Astra Solves 10 Unsolved Math Problems, Publishing Formal Proofs for $2,000 in Compute

Charts, maps and images

The timeline of OpenAI's internal model generating and verifying the scientific proofs.From: OpenAI Models Claim Disproof of Exponential Time Hypothesis Alongside Math and Physics Breakthroughs ·
OpenAI is testing three distinct strategies to translate its raw algorithmic data into scientific understanding.From: OpenAI Models Claim Disproof of Exponential Time Hypothesis Alongside Math and Physics Breakthroughs ·
The AI's proof establishes a fixed 7/8 boundary, guaranteeing a zero-free corridor for Dirichlet L-functions.From: Mathematicians Dissect AI-Generated Proof of the Quasi-Riemann Hypothesis ·
The October 7 data dump included hundreds of solved problems, bypassing traditional academic peer review.From: Mathematicians Dissect AI-Generated Proof of the Quasi-Riemann Hypothesis ·
The unreleased model successfully generated formal proofs for roughly 18 percent of the 4,000 problems it attempted.From: OpenAI Publishes 722 AI-Generated Math Manuscripts Following Mathematician Backlash ·
The advisory group is tasked with reviewing and coordinating the release of AI-generated mathematical proofs.From: OpenAI Forms Independent Math Advisory Group After Claiming AI Solved 100+ Open Problems ·
Show 13 more
The proof required 88 hours of continuous computation by 10,000 AI agents.From: OpenAI Claims AI Solved Navier-Stokes Millennium Prize Problem in 88 Hours Amid Credit Dispute ·
The total computational cost to generate the 10 mathematical proofs was roughly $2,000 at current API rates.From: OpenAI's Astra Solves 10 Unsolved Math Problems, Publishing Formal Proofs for $2,000 in Compute ·
The hybrid human-AI pipeline used to generate and formally verify the mathematical discoveries.From: OpenAI's Astra Solves 10 Unsolved Math Problems, Publishing Formal Proofs for $2,000 in Compute ·
Illustration: External academics are organizing working groups to verify the machine-generated proofs.From: OpenAI and Anthropic AI Models Independently Refute the Exponential Time Hypothesis ·
OpenAI also claimed a rigorous existence proof for pure Yang-Mills theory, a Millennium Prize Problem.From: OpenAI and Anthropic AI Models Independently Refute the Exponential Time Hypothesis ·
Illustration: The physics proofs aim to provide a mathematically rigorous foundation for the Standard Model.From: OpenAI Models Claim Disproof of Exponential Time Hypothesis Alongside Math and Physics Breakthroughs ·
Illustration: OpenAI reported that the model required an average of just three hours of compute time per proof.From: Mathematicians Dissect AI-Generated Proof of the Quasi-Riemann Hypothesis ·
The released manuscripts include step-by-step reasoning traces formatted for automated theorem provers.From: OpenAI Publishes 722 AI-Generated Math Manuscripts Following Mathematician Backlash ·
Illustration: Mathematicians are currently analyzing the 722 manuscripts to determine if the AI system discovered novel mathematical techniques.From: OpenAI Publishes 722 AI-Generated Math Manuscripts Following Mathematician Backlash ·
The broader mathematical community is waiting to evaluate the underlying proofs for the 100-plus claimed solutions.From: OpenAI Forms Independent Math Advisory Group After Claiming AI Solved 100+ Open Problems ·
The AI's proof demonstrates that under specific forced conditions, a fluid vortex can reach infinite velocity in finite time.From: Mathematicians Criticize OpenAI's Navier-Stokes Proof as Incomprehensible Amid Plagiarism Row ·
OpenAI deployed 10,000 autonomous agents over 88 hours to generate the proof, sparking a debate over data privacy and academic credit.From: Mathematicians Criticize OpenAI's Navier-Stokes Proof as Incomprehensible Amid Plagiarism Row ·
Independent mathematicians allege OpenAI rushed to replicate their unpublished approach after rumors of their progress circulated.From: OpenAI Claims AI Solved Navier-Stokes Millennium Prize Problem in 88 Hours Amid Credit Dispute ·

Timeline

October 2026

  1. OpenAI and Anthropic AI Models Independently Refute the Exponential Time Hypothesis
  2. OpenAI Models Claim Disproof of Exponential Time Hypothesis Alongside Math and Physics Breakthroughs
  3. Mathematicians Dissect AI-Generated Proof of the Quasi-Riemann Hypothesis
  4. OpenAI Publishes 722 AI-Generated Math Manuscripts Following Mathematician Backlash

September 2026

  1. OpenAI Forms Independent Math Advisory Group After Claiming AI Solved 100+ Open Problems
  2. Mathematicians Criticize OpenAI's Navier-Stokes Proof as Incomprehensible Amid Plagiarism Row
  3. OpenAI Claims AI Solved Navier-Stokes Millennium Prize Problem in 88 Hours Amid Credit Dispute

August 2026

  1. OpenAI's Astra Solves 10 Unsolved Math Problems, Publishing Formal Proofs for $2,000 in Compute