OpenAI and Anthropic have released AI-generated mathematical proofs that challenge the Exponential Time Hypothesis (ETH), a longstanding assumption about computational complexity. OpenAI announced its findings on October 26, 2026, alongside proposed solutions to major problems in mathematics and physics, while Anthropic published a separate refutation hours later.
The ETH, introduced in 1999, assumes that certain computational problems require exponentially increasing processing time as their complexity grows. Both companies claim their AI models have developed algorithms capable of solving these problems in subexponential time, potentially overturning an assumption widely used in theoretical computer science.
AI-Generated Proofs Challenge Computational Complexity Theory
According to OpenAI’s official announcement, its AI system produced a natural-language refutation of ETH on October 16, which was subsequently formalized using the Lean proof assistant on October 17. The company also reported AI-generated results concerning the Kakeya conjecture, local smoothing problems and quantum field theory.
The ETH is closely related to the P versus NP problem, which asks whether problems whose solutions can be verified efficiently can also be solved efficiently. A successful refutation of ETH would challenge established assumptions about computational difficulty, although it would not establish that P equals NP.
Anthropic independently published its own ETH refutation, following comments from mathematician Levent Alpöge questioning whether OpenAI’s approach relied on existing research. Both companies’ findings remain subject to broader scientific examination.
Researchers Organize Independent Review of OpenAI’s Findings
MIT computer scientist Ryan Williams is coordinating an independent review alongside University of California San Diego researchers Russell Impagliazzo and Ramamohan Paturi, who helped introduce the hypothesis. The researchers have requested public access to OpenAI’s supporting materials and funding for a conference examining the findings and their implications.
Williams said parts of OpenAI’s proof contain unfamiliar techniques that warrant examination by the wider research community. He also warned that previously published work relying on ETH assumptions may require reassessment if the findings withstand independent scrutiny.
Potential cybersecurity implications are also being examined, particularly whether faster algorithms could weaken cryptographic systems that depend on computational difficulty. Williams said OpenAI had assessed the issue and expressed reasonable confidence that its findings did not immediately compromise existing encryption systems.
OpenAI Changes Approach to Publishing AI Research
OpenAI has adopted different review processes for its three research areas, releasing mathematical proofs publicly while establishing expert working groups for computer science and physics. The company also agreed to support discussions covering verification standards, scientific attribution and access to AI-generated research.
Williams had previously criticized OpenAI’s large-scale publication of mathematical proofs, arguing that the volume placed considerable demands on researchers responsible for reviewing them. Despite those concerns, he agreed to participate in evaluating the latest findings and coordinating responses from the scientific community.
Featured image credits: Wikimedia Commons
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