Math progress seems to be accelerating faster than ever with AI tools.
OpenAI announced on August 1 that an internal version of Astra, the model it has been previewing to regulators and lawmakers in Washington over the past week, produced new results on ten open problems spanning sphere packing, group theory, complexity theory, and extremal combinatorics. The company says the total inference cost across all ten problems would run to roughly $2,000 at Sol API rates.
The list covers some genuinely hard territory. Astra produced new upper bounds on sphere-packing density approaching the Cohn–Elkies threshold, exponentially improved bounds on binary and spherical codes, and a construction establishing the existence of non-sofic groups, a question that has sat unresolved in group theory for decades. It also disproved Connes’s rigidity conjecture, delivered new lower bounds on arithmetic circuit complexity for computing the permanent, extended parallel repetition theorems to two-player quantum games, and established polynomial-factor hardness of approximation for the closest vector problem, which has direct relevance to post-quantum cryptography.
Rounding out the set: a resolution of Ehrhart’s volume conjecture across every dimension, a superexponential lower bound on multicolor triangle Ramsey numbers that closes out Erdős problem 183, and results on the compactness and degeneracy conjectures in extremal graph theory that resolve Erdős problems 146 and 180.
OpenAI’s process here is more deliberate than some of its earlier math claims. Human researchers took Astra’s arguments and wrote them up into full manuscripts, then the model itself formalized each proof into a Lean certificate, all published on GitHub. The company is also releasing the model’s own narration of its reasoning for each problem alongside the formal writeups. That level of verification matters given the company’s history here. Last October, Google DeepMind CEO Demis Hassabis publicly called out OpenAI after employees suggested GPT-5 had solved several Erdős problems, when the model had in fact surfaced existing solutions the maintainer of erdosproblems.com hadn’t found. This time, every claim comes with a machine-checked proof attached, which leaves considerably less room for dispute.
OpenAI is also being unusually direct about who did what. In its post, the company states plainly that claiming human authorship for a proof an AI system generated would misrepresent both the system’s contribution and the nature of actual human intellectual work. It frames its own role as preparing the manuscripts and formalizing the Lean proofs, and says it takes responsibility for their correctness, while crediting the mathematical arguments themselves to Astra. The company nods to the Leiden declaration on AI and Mathematics, signed by mathematicians concerned about how AI-generated results get credited and absorbed into the field, and says it wants the mathematical community to engage with the results rather than simply accept them.
The announcement lands in the middle of a fairly crowded few months for AI-assisted mathematics. Google DeepMind’s AlphaProof Nexus autonomously solved nine open Erdős problems in May, two of which had been open for 56 years, at a similarly small inference cost. Harmonic’s Aristotle, backed by Robinhood CEO Vlad Tenev, claimed a 30-year-old Erdős problem earlier this year, and Anthropic’s Fable model recently helped disprove the 85-year-old Jacobian conjecture. Even Donald Knuth, no easy convert to AI hype, was reportedly stunned when Claude cracked an open problem he’d been stuck on for weeks. The pattern across all of these releases is the same: labs increasingly treat Lean formalization as the price of entry for a credible math claim, not an optional extra.
Astra itself remains unreleased. Sam Altman has been demonstrating the model to senators and administration officials in Washington this week, positioning it as a system built around multiple agents collaborating on long-running tasks, with math as the headline proving ground. Whether it ships as GPT-6, a GPT-5.x update, or its own tier alongside Sol, Terra, and Luna hasn’t been decided publicly. What’s clear is that OpenAI wanted these ten proofs out before that decision gets made, and before the model goes through whatever federal review process is about to land on it.