According to Scott Aaronson’s blog post, OpenAI released a collection of 372 mathematical results yesterday, many of which were generated by its latest internal model and reviewed by an advisory group that included Timothy Gowers, Edward Witten and other distinguished mathematicians. Among the results was a proof of Subhash Khot’s Unique Games Conjecture (UGC), a statement that Aaronson’s wife, complexity theorist Dana Moshkovitz, has pursued for much of her career. The blog notes that the proof is accompanied by a Lean certificate, but adds that ‘we’re pretty sure that it’s a proof’ and that, as of the time of writing, no human has fully understood most of the newly released proofs; the effort to digest them has just begun.
Aaronson shared a series of text messages from Moshkovitz describing her reaction. She wrote that the paper ‘feels like something written by someone who’s on psychedelics’, that it is ‘horribly written’ and ‘impossible to read without AI help’, and that she relied on an AI assistant named Astra to extract reasonable completeness and soundness claims from the noise gadget construction. She observed that the proof introduces a ‘completely new bizarre code with a noise test’, a recursive construction that is neither the long code nor the short code, and that the citations appear often irrelevant and confusing. Despite the difficulty, Moshkovitz noted two mitigating feelings: vindication that the UGC is true, a belief she never doubted, and the sense that the entire theoretical‑computer‑science community now faces the same challenge of interpreting AI‑generated mathematics.
The blog also lists a sampling of the other breakthroughs included in the release. These comprise a proof that L equals BPL (derandomizing probabilistic logspace), an integer‑multiplication algorithm running in O(n log^{0.9999999999999} n) time, a positive solution to the Unitary Synthesis Problem, a proof that Parity is not in QAC⁰, a near‑fourth‑power separation between randomized and quantum query complexity for total Boolean functions, a superquadratic separation between sensitivity and block sensitivity, an area law for 2D gapped Hamiltonians, a matrix‑multiplication algorithm in O(n^{9/4}) time, improved lower bounds on the determinantal complexity of the permanent, randomized near‑linear‑time algorithms for approximate perfect‑matching counting and maximum matching in general graphs, and an uncomputability result for solving polynomial equations over the rational numbers. Aaronson notes that any single item could have been ‘result of the year’ in its subfield.
In addition, Aaronson mentions that the day before the OpenAI release, Virginia Williams and Josh Alman posted an arXiv preprint solving the 3SUM problem in O(n^{1.9992}) time and All‑Pairs Shortest Paths in O(n^{2.9995}) time, work that was produced with an Anthropic model. He contrasts the two emerging models for sharing AI‑generated mathematics: the OpenAI approach, which posts the raw AI output and sparks a race among humans to explain it, and the Anthropic approach, which selects particular mathematicians to write a digested version in exchange for compensation.