mathematics

OpenAI claims its unreleased internal model resolved the Navier–Stokes existence and smoothness problem, one of seven $1 million Millennium Prize Problems. NYU professor Tristan Buckmaster disputes the timeline, alleging OpenAI moved after hearing rumors of his own near-year-long collaboration with Anthropic's Levent Alpöge.

Axiom Math's AI system AxiomProver has formally verified the proof of the '246 theorem,' a landmark result from the Polymath8b collaboration on prime gaps. The company says the achievement builds a reusable library for future formalization work and points toward AI verification of software code.

Anthropic says an as-yet-unreleased model significantly increased the lower bound of solutions for which the 150-year-old Riemann hypothesis holds true, coordinating 60 sub-agents across 650 tested ideas. The result was verified by in-house mathematicians and formalized in the Lean proof assistant.

OpenAI claims an internal version of its next major model, Astra, produced solutions to ten mathematical and theoretical computer science problems that had seen no progress in at least a decade. The company says each solution cost less than $2,000 in GPT-5.6 Sol token pricing, and published Lean 4 formalizations along with a paper describing the results.

OpenAI published a counterexample disproving the Unit Distance Conjecture, a geometric graph theory problem open since 1946, in what many mathematicians call the most significant AI math result yet. Reactions range from Terence Tao's cautious optimism to Timothy Gowers describing 'mixed feelings' about having the rug pulled out from under him.

OpenAI claims its new reasoning model has produced an original mathematical proof disproving a geometry conjecture first posed by Paul Erdős in 1946. The company says this is the first time AI has autonomously solved a prominent open problem central to a field of mathematics, with verification from mathematicians including Thomas Bloom and Noga Alon.

Zyphra has released ZAYA1-8B, a mixture-of-experts language model with 760M active parameters and 8.4B total parameters. The model scores 89.1% on AIME 2026, competitive with models exceeding 100B parameters, while maintaining efficiency for on-device deployment.

OpenAI's GPT-5.4 Pro model has reportedly solved Erdős open problem #1196 in approximately 80 minutes, with another 30 minutes to format the solution as a LaTeX paper. Mathematician Terence Tao notes the solution reveals a previously undescribed connection between integer anatomy and Markov process theory.