“OpenAI revealed it had produced solutions to 10 long-standing mathematics problems, some of which had confounded academics for decades, unsettling and dividing mathematicians about AI's impact on their field.”

OpenAI published a post on 1 August 2026 titled "Ten advances in mathematics and theoretical computer science," stating that an internal, unreleased version of a model family it calls Astra produced new results on ten problems. The listed results include a disproof of Connes's rigidity conjecture, new arithmetic circuit and formula lower bounds for computing the permanent including a bound of order n⁴/log n, an exponential parallel repetition theorem for general two-player quantum games, and polynomial-factor hardness of approximation for the closest vector problem, plus Ehrhart's volume conjecture and multicolor Ramsey numbers. The accompanying repository states that it contains Lean 4 formalizations of the results, including improved asymptotic upper bounds on sphere-packing density reaching the Cohn-Elkies threshold, stronger upper bounds for binary and spherical codes, and a construction of a non-sofic group. Press accounts report a 249-page manuscript, reasoning walkthroughs, Apache 2.0 licensing, and a claimed token cost of roughly $2,000 at OpenAI's own API rates. At least one of the ten concerns a question open for decades: Mikhail Gromov introduced the concept of soficity in 1999, and for 27 years no mathematician proved or disproved whether every countable group must be sofic. Press reporting also describes the first improvement to the general upper bound on high-dimensional sphere-packing density since 1978. The community reaction is documented and divided. An independent group theorist published a same-week paper stating that the key technical ingredient crucially builds on work of Kun and Kun-Thom, contrary to the claim in the OpenAI announcement of 1 August that there had been no progress on the main result for at least a decade. He separately wrote that the proof crucially relies on work of Kun and Kun-Thom for the technical Proposition 2.3, applied to a construction involving elementary matrices over the binary Leavitt algebra and Thompson's group V. Another group theorist wrote that the non-sofic existence result answers a genuinely impressive question, but the "no progress for at least a decade" framing is dishonest because considerable theory had recently been building up. Scientific American reported that two of the most prominent results incorporate preexisting ideas from recent literature without proper citation, contradicting OpenAI's initial press release language, and that OpenAI has since updated the language, with a named mathematician saying OpenAI is running roughshod over the work of others who came before, and arguing his own research was effectively plagiarized. Institutional friction predates the announcement: in June the International Mathematical Union endorsed the Leiden Declaration, which warns that AI companies are using published research without consent, bypassing peer review, and threatening the integrity of proof and attribution.