Google Cogentic and Meta Muse Spark Each Prove Open Math Problems

Google's Cogentic system and Meta's Muse Spark models have each produced verified proofs for previously unsolved mathematics problems, following different approaches. Cogentic, built on Gemini and described in arXiv preprint 2609.40324, autonomously discovered proofs for five open problems in online learning, auction theory, and mechanism design, starting from problem statements alone with no expert hints. Most problems required around 100 Gemini API calls; the hardest required approximately 1,000. Domain experts independently verified all five proofs, and results are being published in companion papers. Cogentic's architecture centers on an orchestrator that allocates independent provers across distinct proof directions, subjects their output to adversarial verification by multiple specialized components, and promotes confirmed intermediate results into a persistent verified ledger that later rounds build on. Meta's approach paired Muse Spark 1.1 and 1.2 in Thinking Mode with working mathematicians via the standard meta.ai chat interface, producing six papers, five of which answer previously open questions. A separate group of mathematicians reviewed the work independently, and each paper marks which passages were drafted primarily by humans or AI. Specific results include a theoretical benchmark on the limits of exact data fitting, a counterexample disproving a conjecture about mathematical group classification, and a characterization of when a relaxation of a binary polynomial optimization problem is exact. A separate AI agent called Nilradical independently reported a different counterexample to the same group-classification conjecture on September 16, 2026.