Anthropic deposited a preprint on October 27, 2026, claiming that Claude found a deterministic algorithm for 3-SAT that also solves its counting variant, #3-SAT. The deposit followed OpenAI’s October 26 announcement of three AI-generated research claims in mathematics, computer science and physics. Both companies describe results with mathematical implications; OpenAI says further research is needed before theoretical advances translate into practical improvements.
Anthropic’s preprint claims an algorithm for 3-SAT and #3-SAT
Anthropic’s preprint describes a deterministic algorithm with running time 2^{o(n)} for 3-SAT on n variables. In complexity theory, that notation describes subexponential growth: the exponent grows more slowly than a fixed multiple of n. 3-SAT asks whether a Boolean formula whose clauses contain three literals can be satisfied. The harder counting problem, #3-SAT, asks how many satisfying assignments it has.
The preprint says Claude found the core algorithm on October 16, using 14 million output tokens and no human input, and that the result was formalized in Lean 4. October 16 is the reported discovery date; the preprint’s first deposit came on October 27.
OpenAI made a separate claim about the Exponential Time Hypothesis
The Exponential Time Hypothesis (ETH) is the conjecture that 3-SAT has no subexponential-time algorithm. OpenAI said its system produced an ETH-refuting result on October 16 and formalized it in Lean on October 17. The company released a natural-language write-up and Lean proof. Anthropic’s preprint describes its own algorithm claim; the shared reported discovery date does not establish that the two efforts used the same method.
Lean is a proof assistant, software that checks whether steps in a formal proof follow from its encoded rules and assumptions. OpenAI named Russell Impagliazzo, Ramamohan Paturi and Ryan Williams to lead an external collaboration reviewing its ETH result.
The ETH claims concern computational complexity, not an attack on a deployed cryptographic system. OpenAI said significant further research would be needed before theoretical advances produced practical improvements.
OpenAI also reported a local-smoothing result
OpenAI said its models produced a local-smoothing result in every dimension on October 18 and formalized it in Lean on October 19. Local smoothing is a problem in harmonic analysis, the study of functions and their frequency components. OpenAI said its result implies the Kakeya and Stein restriction conjectures in every dimension.
NeoTeo previously covered OpenAI’s reported Navier–Stokes result, a separate mathematical claim from the work announced in October.
OpenAI’s Yang–Mills claim concerns existence
OpenAI said its model produced a proof concerning Yang–Mills existence on September 9, 2026, and a proof for quantum chromodynamics (QCD) on September 18. The company said it completed a Lean formalization of the Yang–Mills result on October 16. These physics claims concern the existence component of the Yang–Mills problem. The problem also requires a positive mass gap, which remains a separate open component.
OpenAI said its physics collaboration chose to hold the AI-produced artifacts while preparing its own write-up. The collaboration is led by David Gross and includes Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov.
Formalization and review address different parts of the work
A Lean formalization gives researchers a computer-checkable chain of encoded proof steps. OpenAI released Lean and natural-language artifacts for its local-smoothing and ETH claims, while its physics collaboration held the AI-produced artifacts during its write-up. Anthropic’s preprint says its ETH-related result was formalized in Lean 4.
The public artifacts and the announced review arrangements differ by research area. OpenAI’s ETH collaboration was set to examine that result; its physics group was preparing a write-up. These processes accompany the companies’ claims, while OpenAI says further research is needed to turn theoretical progress into practical improvements.
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