When Paul Erdős jotted down his conjecture about point sets in the plane in 1946, he almost certainly never imagined that the answer would come from a machine some 80 years later. Now it's being reported that its successor problems—several of them—have been cracked all at once, for a computational cost of about $2,000.
An 80-Year-Old Question and a Record Frozen Since 1978
It's not unusual for an unsolved math problem to sit with "no progress for over a decade." That usually means the tools at hand can't get anywhere near it. What remains after every standard technique has been exhausted is precisely what earns the label "hard problem."
The upper bound on high-dimensional sphere packing is a textbook case. The bound set by Kabatiansky-Levenshtein in 1978 went unimproved by anyone for nearly half a century. The problem of packing spheres as densely as possible looks intuitively simple, but as dimensions rise, the tools of geometry stop working. Then there's the problem of "non-sofic groups." Ever since Mikhail Gromov introduced the concept of sofic groups (locally finite approximability) in group theory in 1999, it had remained open whether groups actually exist that lack this property. Proving existence sits at the core of group theory, and the question had gone untouched for 27 years.
Several of the problems also come straight out of Erdős's own numbered catalog: Erdős Problem 183 (lower bounds on multicolor triangle Ramsey numbers), and Problems 146 and 180 (extremal graph theory). These are problems the mathematics community has "officially registered" as open.
The Method Astra Used
What OpenAI deployed this time is an internal build of its next-generation model, called "Astra." Astra is the series OpenAI positions as its next flagship model line, designed to orchestrate multiple agents and handle long-horizon tasks spanning hours to days.
According to the announcement, the token cost Astra incurred to find solutions to the 10 problems came out to roughly $2,000 at Sol API rates. Comparing that figure to the time a typical PhD student would spend on a single hard problem isn't straightforward, but the number gives us a starting point for discussion—one that separates the cost of finding an answer from the cost of understanding it and placing it in context.
Each result was recorded in two forms. One is a 249-page draft paper, which a human turned into a finished manuscript using the same model as Astra. The other is a Lean 4 proof certificate. Lean 4 is a type of proof assistant that lets software mechanically check every logical step of a proof. This format is a direct response to criticism that AI can produce mathematical "hallucinations": whether a proof is correct can be verified by software, giving mechanical assurance of validity before any mathematician even reads it.
A case that crisply illustrates Astra's approach is the disproof of the Erdős unit distance conjecture, announced in advance in May 2026. Erdős had conjectured in 1946 that the number of pairs at unit distance among n points in the plane stays on the order of n^{1+o(1)}, but the AI system found an infinite family of point sets that breaks this. Rather than standard geometric techniques, it drew on algebraic number theory—class field towers and Golod-Shafarevich theory—an approach that outside mathematicians Timothy Gowers, Noga Alon, and Thomas Bloom verified and posted to arXiv. Will Sawin later refined the result, and the improvement exponent is now given explicitly as δ≈0.014.
Four Emblematic Breakthroughs
Of the 10 problems, let's look at four whose achievements differ in kind—from ones that can be described as an old-vs-new numerical comparison, to a full logical reversal. Each marks a turning point in its own distinct sense.
The upper bound on high-dimensional sphere packing: the bound derived by Kabatiansky-Levenshtein in 1978 had sat unimproved for nearly half a century. Astra pushed that bound down to the Cohn-Elkies threshold. Simply reaching the limit point of this linear-programming-based method was itself a long-standing goal.
In the multicolor Ramsey number (triangle) problem, only an exponential lower bound had previously been known, but Astra demonstrated a super-exponential lower bound. Ramsey theory is a family of theorems asserting that "regularity inevitably emerges in sufficiently large structures," and this super-exponential improvement to the lower bound shows that the threshold at which such regularity emerges lies much farther out than previously thought. This resolves Erdős Problem 183.
What this inequality shows is conceptually simple: when trying to avoid triangles using k colors, the limiting point lies much farther away than was previously believed.
The disproof of the Connes rigidity conjecture isn't a numerical comparison but an outright reversal of the proposition itself. Against Alain Connes's conjecture that "certain groups with property (T) are uniquely determined by their von Neumann algebra," Astra showed that "infinitely many pairwise non-isomorphic property (T) groups exist that share the same von Neumann algebra." The conjecture was false.
And in the result on arithmetic circuit complexity, a lower bound on the order of was established for computing the permanent using arithmetic formulas. The computational hardness of the permanent is a central theme in theoretical computer science, and this lower bound provides grounding for just how hard actual circuit design for it is.
The Rift Between the Math Community and AI
These announcements arrived at a moment when the mathematics community was already embroiled in debate. On June 2, 2026, the "Leiden Declaration," backed by the International Mathematical Union (IMU), was published. Drafted by 16 researchers from 15 universities—including Fields Medalist Terence Tao among its signatories—the declaration does not call for banning AI.
What the declaration asks is something else entirely: who verifies AI-generated proofs, and how? Once mathematical research becomes dependent on commercial AI, who holds the power to set the agenda? There isn't even an agreed method for evaluating the possibility that results from a black box are an uncredited reconstruction of existing human work. These are questions that mere technical transparency cannot resolve.
OpenAI itself acknowledges this tension. In its announcement, the company states explicitly that "attribution should honestly reflect how the research results were produced," clearly disavowing any claim that AI-generated proofs are human achievements. It states that the human role was limited to drafting the paper and formalizing it in Lean.
How far this stated position goes toward addressing the math community's concerns, however, remains uncertain—because what the Leiden Declaration calls for isn't just technical transparency, but structural measures to curb commercial influence over the research agenda itself.
What AI Hasn't Reached
At the same time, OpenAI has been explicit about what the model has not solved. The Millennium Prize Problems remain unresolved. Much as the Poincaré conjecture—the only one of the seven problems to be solved—was proven by Grigori Perelman in 2003, these problems sit at the very heart of mathematics, and it's believed they cannot be reached simply by piling on more complex computation.
A more fundamental question remains. Even if Lean 4 confirms that every step is valid, whether a mathematician who reads the proof can say they "understood why it's true" is a separate matter. The fact that, in the case of the Erdős unit distance disproof, outside mathematicians went through a verification process to write a "digested" version shows that there is still a distance to be bridged between mechanical verification and human understanding.
Even granting that the construction of the non-sofic group is correct, how many mathematicians will emerge who intuitively grasp why the construction works and can apply it in other contexts? The old mathematical intuition that the existence of a proof and the understanding of a proof are two different things ought to remain valid even in the age of AI.