OpenAI & Anthropic Refute ETH with 2^o(n) 3-SAT Algorithm

Executive Overview

On October 27, 2026, OpenAI and Anthropic jointly announced that their AI models have discovered a deterministic algorithm that solves the 3-SAT problem in time 2^o(n), thereby refuting the Exponential Time Hypothesis (ETH) that has stood since 1999. The result, published on GitHub with full Lean 4 formalization and independent verification via the nanoda kernel, is purely theoretical and does not imply a break of current cryptography, but it opens a debate on the possibility of P=NP and calls for a review of scientific papers more than a decade old.

Deep-Dive Features & Architecture

The new algorithm solves the 3-SAT problem — a fundamental logical problem in complexity theory — in time 2^o(n) instead of the exponential bound 2^Ω(n) assumed by the ETH since 1999. More importantly, the algorithm solves the harder #3-SAT problem (counting satisfying assignments) in time 2^O(n log log n / √log n). The core idea relies on tensor networks on complete graphs K_K with λ bits per edge, where it is proven that e(9K^16) ≤ 0.9·e(K) + O(1/K), then nodes are ordered on a grid and rows or columns are merged according to the cheaper, and aggregated into K balanced classes. This construction gradually reduces complexity until it reaches sub-exponential.

The official GitHub documentation includes the file MainResults.lean which explicitly declares that ETH is false (ETHIsFalse) and contains the first paper's theorem (MainTheorem). The file contains no proofs but imports only Mathlib, and is kept short so the reader can verify it manually. The actual proofs are in proof/PaperIndex.lean which links each numbered statement in the paper to its Lean counterpart. Full verification is performed via the comparator-check.sh script which compares statements with Challenge.lean and rechecks proofs with the independent nanoda kernel.

According to Ryan Williams of MIT, parts of OpenAI's proof may be routine, but there are two or three parts that "look space-like" and call for the collaboration of the entire community. He confirmed that the result does not imply a break of current cryptography, and that the OpenAI team verified this and was reasonably confident.

Benchmark & Competitive Performance

At the results level, OpenAI presents solutions to three mathematical problems including the refutation of ETH, while Anthropic presented its own refutation hours later via Claude Opus 5.5. Anthropic's algorithm re-derives a weaker result: O(2^δn) for every δ>0, not the sharper bounds mentioned in the original paper. This means the two results agree in conclusion (ETH is false) but differ in strength. In terms of formal verification, Anthropic provides full Lean 4 documentation with a dual verification kernel (Lean kernel + nanoda), while OpenAI relied on publishing on GitHub and then forming a working group led by Williams, Russell Impagliazzo, and Ramamohan Paturi. This methodological contrast reflects a transitional phase in how labs handle AI-generated mathematical discoveries.

Industry Impact & Enterprise Adoption

For the Arab developer, the result is a double-edged sword: on one hand, it demonstrates the growing capability of AI models to contribute to fundamental mathematics; on the other hand, it underscores the need for rigorous formal verification and community review. Enterprises relying on cryptographic security should note that the result is theoretical and does not affect current encryption standards. However, the precedent of AI-assisted mathematical discovery may accelerate research in complexity theory and lead to new algorithmic breakthroughs. The open-source release under Apache 2.0 allows researchers worldwide, including in the Arab region, to access and verify the findings, fostering global collaboration.

Conclusion

The refutation of ETH by OpenAI and Anthropic marks a significant milestone in AI-driven mathematical discovery. While the result is purely theoretical and does not threaten current cryptography, it challenges long-held assumptions and invites a reevaluation of complexity theory. The use of Lean 4 and independent verification sets a new standard for rigor in AI-generated proofs. As the community digests these findings, the collaboration between AI labs and human mathematicians will likely intensify, paving the way for further breakthroughs.

Media Source: TechCrunch AI | البيان الرسمي للشركة: المصدر الأصلي | Fact Verification & Analysis: AI Tools Oasis