Did AI Solve Unsolved Math Problems? 10 Critical Astra Claims

Two thousand dollars. That’s the figure OpenAI attached to each of ten new results in mathematics and theoretical computer science: the token cost, at GPT-5.6 Sol API prices, of getting OpenAI Astra to produce them. A 249-page manuscript. A walkthrough of the model’s reasoning. Ten Lean 4 certificates anyone can download and machine-check. And the question the whole package dares you to ask: did AI solve unsolved math problems? The rest of this piece separates what OpenAI claimed from what can actually be checked: because did AI solve unsolved math problems deserves a better answer than the viral version.

What OpenAI Actually Announced

OpenAI published “Ten advances in mathematics and theoretical computer science,” the announcement that prompted the question: did AI solve unsolved math problems? The results came from an internal version of OpenAI Astra, described only as “our next major model.” Ten problems. Each one, per OpenAI, had seen no progress on its main result for at least a decade.

The package is unusually concrete for an AI-mathematics claim. A 249-page technical manuscript. A 62-page “discovery walkthrough”: an LLM-generated PDF reconstructing how the proofs came together from the model’s unpublished reasoning traces. And ten separate Lean 4 formalizations in a public GitHub repository, openai/ten-proofs.

The workflow, as OpenAI describes it: humans prepared the mathematical arguments into manuscripts with the model’s assistance. Then the model formalized each argument as a Lean certificate. OpenAI says it takes responsibility for correctness, while attributing the mathematical arguments themselves to the AI system. One line from the release is worth keeping: OpenAI said that claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system’s contribution and the nature of genuine human intellectual work.

What the announcement did not include: public model access. A system card. A release date. Broad evaluation results. Astra was, at that point, an internal system nobody outside OpenAI could touch. None of which slowed the obvious question (did AI solve unsolved math problems?) from going viral before anyone could check the certificates.

Then the $2,000 figure. OpenAI’s claim: the tokens used to find the ten solutions would cost roughly $2,000 per solution at Sol API rates. Generation tokens. Not training. Not infrastructure. Not researcher salaries, problem selection, human review, manuscript preparation, or the failed experiments that never made the cut.

Scrolling the first-day reaction, the sharpest question I came across wasn’t in a headline: it was Simon Willison’s, filed the same day on his link blog. Parenthetical. One line: no news on how many problems they spent $2,000 on without reaching a solution. That single sentence punctures the marketing-friendly version of the number. He also wrote the other thing everyone was thinking: “I want to see the prompts they used!”

The Ten Claims, Ranked by How “Solved” They Are

OpenAI’s own wording matters. The company said the results “resolve or make substantial progress” on long-standing open problems. Not “solved ten famous problems.” That distinction is doing real work. So which of the ten actually count as solved?

Resolutions, disproofs, and constructions first. Substantial-progress entries after. The answer to did AI solve unsolved math problems changes depending on which row you’re reading:

• Sphere packing. Exact asymptotic strength of the Cohn-Elkies program, plus an improved general bound. Major advance: a threshold question settled.
• Non-sofic groups. An explicit construction, the first, for a question open since Gromov introduced soficity in 1999. Claimed resolution.
• Connes’s rigidity conjecture. Counterexamples built from non-isomorphic groups sharing the same von Neumann algebra. Claimed disproof.
• Ehrhart’s volume conjecture. A sharp maximum-volume bound in every dimension. Claimed resolution.
• Multicolor Ramsey numbers. A superexponential lower bound that resolves Erdős problem 183. Claimed resolution.
• Extremal graph theory. Counterexamples resolving Erdős problems 146 and 180. Claimed resolutions.
• Quantum parallel repetition. An exponential theorem for general finite two-player entangled games. Claimed major theorem.
• Binary and spherical codes. Exponentially improved upper bounds. Substantial progress: the bounds moved, the problems didn’t close.
• Arithmetic circuit complexity. New lower bounds for computing the permanent. Substantial progress.
• Closest vector problem. Polynomial-factor hardness of approximation in lattice problems: territory directly relevant to post-quantum cryptography. Substantial progress.

Seven entries framed as resolutions, disproofs, constructions, or major theorems. Three as better bounds and complexity results: real advances, not closures. “OpenAI Astra produced ten mathematical advances” is accurate. The viral version is too broad.

What a Lean Certificate Actually Proves

Lean is an interactive theorem prover. Its small trusted kernel checks whether a formal proof term establishes a precisely stated theorem from declared definitions, assumptions, axioms, and imported results. Not the prose. The logic.

That’s the upgrade over a model-generated proof in natural language. Language models produce elegant, invalid arguments constantly. A successful Lean check kills a whole category of failure: the unstated logical leap. All ten certificates were reported with a sorry count of zero: no step assumed, no gap papered over.

The limitation: Lean checks the formal theorem that was encoded. It does not decide whether that formal statement perfectly captures the original informal conjecture. It doesn’t judge whether the assumptions are the intended ones, whether the result is novel, or whether it’s important. Recent research on faithful natural-language-to-Lean formalization keeps landing on the same gap: compilation is not semantic fidelity.

So a Lean certificate is strong evidence of deductive validity for the encoded statement. It does not, by itself, settle did AI solve unsolved math problems in the full academic sense. Experts still have to compare each formal theorem against the historical problem, audit the assumptions, check prior literature, and assess originality and significance. Mechanically verified. Not yet community-vetted. Different things. The distance between them is measured in months of specialist scrutiny, not build minutes.

How the Reaction Landed

Watching the coverage land over the following weeks, two strands kept surfacing. Awe at the artifacts. Unease about what they mean.

The unease had a name within days. Mathematician Kirwin Hampshire published an essay, “The Dark Night of Mathematics,” describing what Willison summarized as “a profound spiritual crisis” among mathematicians: a collective burst of Deep Blue, as he put it, set off by results less significant than these.

The sharpest single quote I came across was Thomas Bloom’s. Bloom curates erdosproblems.com (he tracks Erdős problems for a living). Talking to Quanta Magazine, in a piece titled “Why the Legendary Erdős Problems Are Falling to AI,” he said: “A big problem is AI is being used a lot by people who aren’t mathematicians… no human has read it.” Read that twice. The proofs check out mechanically. Nobody has read them. That is the field’s central anxiety in one sentence. Headlines keep asking did AI solve unsolved math problems as if the answer were binary. The field’s answer is slower, and more careful.

Chosun Biz reported that some mathematicians were pushing back on the framing: prior groundwork, they argued, was underrepresented next to Astra’s starring role. The concrete focus: the non-sofic groups construction leaned on a 2019 paper by Gábor Kun and Andreas Thom, plus a related 2016 result from Kun. Terence Tao wrote that Kun’s expander-decomposition work, including Tao’s own collaboration with Kun, played a decisive role in one of the proof’s key propositions. Thom separately asked OpenAI to clarify whether his own earlier ChatGPT conversations could have fed into the system’s output.

As of the latest reporting I found, no rebuttal claims any of the ten proofs is wrong. The dispute is about attribution and framing, not correctness. Headlines blur that line. Don’t let them.

OpenAI’s own announcement materials note an earlier prequel: an AI-generated disproof of the Erdős unit-distance conjecture (open since Erdős posed it in 1946), found while evaluating an unreleased model. The two announcements together look less like a stunt and more like a research thread run quietly for months.

For scale: days earlier, Anthropic’s Mythos Preview had surfaced cryptographic weaknesses with Claude at a token cost of $100,000. Different problems, different accounting. But the contrast explains why OpenAI’s $2,000 figure traveled so far.

Is Astra AGI?

Separate question from the one in the headline: did AI solve unsolved math problems? Don’t merge them.

OpenAI’s charter defines artificial general intelligence as “highly autonomous systems that outperform humans at most economically valuable work.” Advanced mathematics across several subfields is a remarkable domain result. It doesn’t demonstrate broad superiority across most work, reliable autonomy in open environments, or general competence across unrelated tasks.

The picture changed when OpenAI launched the model publicly as GPT-6 Astra: a phased rollout through a newly named Trusted Access Program, per CNBC’s reporting, reaching ChatGPT’s paid tiers and the API (identifier gpt-6-astra), plus Microsoft Azure and AWS Bedrock. Reported specs include a 1,050,000-token context window and $50-per-million output pricing. OpenAI also disclosed a “1 Critical” cybersecurity threshold under its internal Preparedness Framework. At the launch briefing, Greg Brockman left the final AGI judgment to others, then added his own. “I think we’re there,” per the Washington Post.

Counterpoints: under OpenAI’s recapitalization terms, any company AGI declaration needs verification by an independent expert panel. None has been convened. ARC Prize has said plainly that saturating ARC-AGI-3 is not proof of AGI. And one mapping stays unconfirmed: exactly how the internal research system behind the ten proofs maps onto the shipped product.

Bottom line: the math announcement is evidence that general-purpose AI is becoming a serious mathematical research instrument. It is not, by itself, evidence that OpenAI has demonstrated AGI.

What We Still Don’t Know

Eight weeks on, the open list is long. Each item is a reason the verdict below stays qualified:

• Peer-review outcomes. Months or years of it, per result. Corrections, revisions, alternative proofs, journal decisions: all pending.
• Human contribution. OpenAI’s workflow summary is high level: per-problem prompting, curation, intervention, and correction remain undisclosed.
• Total search effort. The $2,000 figure covers successful-solution tokens at stated API rates: not training, failed runs, or everything around the runs.
• Semantic faithfulness. Specialists must confirm each Lean statement encodes the intended problem.
• Originality. Each field has to determine what was already known and what’s genuinely new.
• Generalization. Ten selected results say nothing reliable about arbitrary open problems.
• Identity. Whether the internal model behind the proofs is exactly the system that shipped as GPT-6 Astra.

That list is why the honest answer stays qualified for now.

So: Did AI Solve Unsolved Math Problems?

In a bounded, evidence-aware sense: yes. OpenAI released several results explicitly framed as resolutions or disproofs, with detailed manuscripts and Lean 4 certificates that check out mechanically. But peer review is still pending, and the question did AI solve unsolved math problems stays with the specialists until that process runs its course.

The defensible conclusion is narrower than the viral headline. The package holds ten advances, not ten identical “solved” events. Lean verification is powerful. It doesn’t replace semantic review, literature review, or peer review. It says nothing about how much human scaffolding sat underneath.

The deeper milestone isn’t the count. It’s the format: an AI-generated research claim arriving with formal artifacts the whole mathematical community can inspect, build, and argue with. Whether the ten advances become established mathematics depends on what happens next: independent builds, specialist audits, corrections, peer-reviewed acceptance.

That workflow (candidate ideas, long searches, readable arguments, machine-checkable proofs) may matter more than any single result. If it generalizes beyond these ten, the next time someone asks did AI solve unsolved math problems, the answer may arrive with its own machine-checked evidence attached. A pattern also visible in AI agents for scientific research. The bottleneck would move, not vanish: choosing meaningful questions, auditing formal statements, and deciding what counts as knowledge would still be human work.

So did AI solve unsolved math problems? Ten advances. Checkable proofs. Review pending. That’s the honest version, and it’s still a remarkable sentence to be able to write.

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