OpenAI announced three research claims attributed to its internal models on October 26, 2026, covering computational complexity, harmonic analysis and quantum field theory. Anthropic separately described AI-discovered algorithms for 3-SAT and the harder counting problem #3-SAT. OpenAI’s claim about the Exponential Time Hypothesis (ETH) does not solve P versus NP, a separate problem that remains open.
OpenAI’s three claims and their timing
OpenAI said its models refuted ETH, proved the local smoothing conjecture in every dimension, and established existence results for certain four-dimensional quantum field theories. The company’s announcement dates the work separately from its October 26 announcement: it says the ETH result was found on October 16 and formalized in Lean on October 17. OpenAI dates its local-smoothing output to October 18 and its Lean formalization to October 19.
The October 26 announcement was separate from OpenAI’s October 6 catalogue of 722 mathematical manuscripts. NeoTeo covered that earlier, reported GitHub plan in a previous article.
ETH is not the same as P versus NP
The Exponential Time Hypothesis is a conjecture about how the time needed to solve Boolean satisfiability problems grows as their formulas get larger. 3-SAT is a version of that problem in which each clause has three literals. OpenAI said its result refuted ETH; that claim concerns computational complexity, not a resolution of P versus NP.
An algorithmic bound describes how running time scales with problem size. It does not, by itself, tell you how fast an algorithm runs on formulas encountered in practice. That distinction is central to interpreting the separate result Anthropic describes.
Anthropic’s bounds for 3-SAT and #3-SAT
Anthropic’s formal-math repository attributes a deterministic 2^{o(n)}-time algorithm for 3-SAT to an internal Claude research model and says the work was formalized in Lean 4. Here, n is the number of variables, and o(n) denotes an exponent that grows more slowly than n.
The repository also gives a bound of 2^{O(n log log n / √log n)} for #3-SAT, which counts satisfying assignments rather than deciding whether a satisfying assignment exists. Anthropic describes both bounds as asymptotic: they concern how complexity behaves as n grows, and do not establish that the algorithm solves formulas faster in practical use.
What readers can inspect in OpenAI’s results
OpenAI described different release arrangements for the three fields. It said the mathematics and computer-science outputs were public. For computer science, the company also arranged external examination of the ETH work and said raw data would be public for independent scrutiny. OpenAI said the physics collaborators chose to withhold the AI-produced artifacts while preparing their own account.
OpenAI said participation in the ETH review would be voluntary and unpaid, involve no nondisclosure agreements, and leave the collaborators free to reach their own conclusions. The company also said that taking part should not be treated as endorsement or certification of its claim.
The local-smoothing and physics claims
OpenAI said its local-smoothing result applies in every dimension and implies results for the Kakeya and Stein restriction conjectures in every dimension. These are claims about harmonic analysis, a branch of mathematics that studies functions and their frequency components.
For physics, OpenAI described existence results for certain interacting four-dimensional quantum field theories, including quantum chromodynamics (QCD) and a completion of the Standard Model. It included pure Yang–Mills existence as a subproblem. The Yang–Mills Millennium Prize problem also involves the mass gap, which OpenAI did not claim to have solved.
