OpenAI says its AI models produced major proofs in mathematics, computer science and physics

OpenAI said on Oct. 26 that its internal AI models produced three new results addressing longstanding scientific questions: a claimed refutation of the Exponential Time Hypothesis in computer science, a proof of the local smoothing conjecture in every spatial dimension in mathematics, and a rigorous construction of interacting quantum field theories in four dimensions in physics, including an existence proof for pure Yang–Mills theory that covers only part of the Clay Mathematics Institute's Millennium Prize problem. OpenAI has published writeups and Lean formalizations for the mathematics and computer-science results, but the external physics group withheld the AI-produced materials while preparing its own account, prompting the Advisory Group on Mathematics and Artificial Intelligence to call for immediate release. Hours later, Anthropic posted a separate claimed refutation of the Exponential Time Hypothesis, credited to "Claude (Anthropic)," saying an internal Claude research model found the core algorithm without human input on Oct. 16, the same discovery date OpenAI reports.

Nearly seven weeks after announcing an AI-generated proposed solution to the Navier–Stokes problem https://openai.com/index/navier-stokes-solution/ , and three weeks after releasing hundreds of mathematical manuscripts https://openai.com/index/sharing-ai-progress-in-mathematics/ , OpenAI said on Oct. 26 that its internal AI models have produced three more results addressing longstanding scientific questions. OpenAI says its computer-science result refutes the Exponential Time Hypothesis https://openai.com/index/building-scientific-understanding-with-academic-communities/ , a decades-old conjecture about how quickly computers can solve certain logic problems that underpins hundreds of studies of computational limits. In mathematics, OpenAI says it proved the local smoothing conjecture in every spatial dimension, resolving a longstanding question about how waves concentrate and, through its implications, the Kakeya and Stein restriction conjectures. In physics, the company claims a rigorous mathematical construction of interacting quantum field theories in four dimensions, including quantum chromodynamics, which describes the strong force binding quarks inside protons and neutrons. That work includes an existence proof for pure Yang–Mills theory, which OpenAI calls part of the Clay Mathematics Institute’s Yang–Mills Millennium Prize problem. The Clay problem also requires proving a mass gap; the announcement covers existence alone. OpenAI co-founder and president Greg Brockman wrote https://x.com/gdb/status/2114837059462627533 that AI is “starting to help us improve our understanding of the universe.” “If the claim is correct, it will change an important part of theoretical computer science,” Russell Impagliazzo and Ramamohan Paturi of UC San Diego and Ryan Williams of MIT wrote in a letter to their research community https://docs.google.com/document/d/1adB0IhJgLzsmbfCgoLrtoJTjnlPf74UnyjMo35P5JHo/edit . ETH has connected decades of research on computational difficulty, they explained, and the techniques developed around it retain value even if their interpretation changes. “Our involvement should not be taken as certification of the claim,” they added. The independent Advisory Group on Mathematics and Artificial Intelligence https://agmai.org/ called on OpenAI to release the physics proofs immediately, arguing that “understanding should develop in the open, with everyone given the same access.” OpenAI has published writeups for its mathematics and computer-science results, along with formalizations in Lean, software for checking formal mathematical proofs, but says the external physics group chose to withhold the AI-produced materials while preparing its own account. Hours after OpenAI’s announcement, Anthropic posted a separate claimed refutation of the Exponential Time Hypothesis https://github.com/anthropics/formal-math/tree/main/eth , credited to “Claude Anthropic .” The paper says an internal Claude research model found the core algorithm without human input on Oct. 16, the same discovery date OpenAI reports; Anthropic researchers subsequently helped improve its presentation. The release includes a Lean formalization. Anthropic cautions that the theoretical result does not establish a practical speedup for solving real-world problems.