AI Proves a 2008 Quantum Conjecture in Three Days

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A research system built by Physical Superintelligence (PSI) produced a complete proof of the Aaronson-Ambainis conjecture, open since 2008, on the third day of a run using a non-public Anthropic model. Lean accepted the proof, and the system then improved it twice. Here is what was claimed, what was checked, and what it does not mean.

What PSI actually announced

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The post is specific about the setup. The run used instances of a non-public Anthropic model on PSI's own infrastructure, not a public API model. PSI says the system reached a complete proof of the conjecture during the third day, that Lean — a proof assistant that machine-checks every step — accepted it, and that the system was then pointed at related open problems, where it improved the result twice, with Lean accepting the final version.

The proof is written up in a paper by David Turturean of PSI and posted to GitHub on November 6, 2026. PSI's contributions note is unusually candid about the division of labour: Turturean developed the research system, scoped the problem and ran it, then prompted the model to package the results into a first draft of the paper, reviewed that draft and is responsible for the paper's correctness. A Lean-checked proof and a posted manuscript are strong evidence that something real exists, but they are not the same as peer review. As of November 8, 2026, the work has not been published in a journal.

Why the conjecture sat open for eighteen years

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Quantum computers are not uniformly faster than classical ones. Almost every dramatic speedup we know — Shor's factoring algorithm being the famous case — depends on the input having hidden structure, a promise that a rare pattern exists. Strip the structure away and the advantage mostly evaporates. The open question was whether that evaporation is a law or an accident of the algorithms we happen to have found.

Aaronson and Ambainis turned the question into pure mathematics. In the query model, where cost is measured by how many times an algorithm reads its input, the acceptance probability of a T-query quantum algorithm is a polynomial of degree at most 2T in the input bits, with values in [0,1] — a result from Beals, Buhrman, Cleve, Mosca and de Wolf. If a classical machine can approximate that polynomial while reading only a few bits, then it can imitate the quantum machine cheaply. Aaronson and Ambainis conjectured exactly that approximation is possible, and showed it would imply that quantum computers cannot achieve exponential query speedups on random inputs.

For eighteen years nobody could prove the polynomial statement. It became one of the recognised central open problems in quantum complexity theory, cited in survey work on quantum speedups and revisited in later papers. In August 2026, a Stanford group — Guy Blanc, Jordan Docter, Carmen Strassle and Li-Yang Tan — settled the conjecture's prediction in a special case, covering quantum algorithms that read their input in a fixed number of parallel rounds.

What the proof does — and does not — show

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If the theorem holds as stated, the consequence is narrow and important: on total boolean functions with random inputs, a quantum computer needs at most polynomially fewer queries than a classical one, for any fixed error and any fixed fraction of inputs. Exponential quantum advantage requires structure. That is a statement about the query model, not about runtime, memory, or every possible computational setting.

Three limits are worth stating plainly. First, it does not touch structured problems, so Shor's algorithm and the cryptographic urgency around it are unaffected. Second, query complexity is a lower-bound tool: a polynomial query overhead does not by itself say anything about whether a specific practical algorithm is fast. Third, the theorem bounds worst-case-ish behaviour on random inputs; it is not a claim that quantum computers are useless, only that they are not exponentially better for free.

How the proof was checked

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Lean acceptance is the strongest part of the story. A proof assistant does not care who or what wrote a proof; it checks each inference against a kernel, so a Lean-accepted theorem is, modulo the correctness of Lean and the statement formalised, actually proved. The usual failure mode in AI-for-math claims is a proof that reads well and is wrong, or one that proves a subtly different statement. Formalisation removes the first problem and exposes the second, which is why the Lean detail carries more weight than the prose in the announcement.

The remaining checks are human. PSI says the Stanford researchers who proved the special case have been reading the full proof, have sent comments, and are writing a shorter exposition for experts. Their quoted reaction calls it "a huge result" that "resolves a decades-old conjecture" and notes the proof draws on and extends work in boolean function analysis and probabilistic combinatorics. That is a strong signal from people positioned to judge it — but a quote in the discoverer's own announcement is not a referee report.

Why the result matters beyond the theorem

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Two things are being claimed at once, and they have different evidence standards. The mathematical claim is checkable and, if the Lean development is sound, checkable by anyone with the toolchain. The methodological claim — that a research system running a non-public frontier model can close a recognised open problem in days rather than years — rests on PSI's account of its own run, with the model unnamed and the system undescribed.

That second claim is the one with industrial consequences. If it generalises, the bottleneck in formal mathematics shifts from finding proofs to choosing problems, formalising statements, and verifying that the thing proved is the thing you wanted. It also sharpens the accountability question: a proof assistant tells you the theorem is true, not whether the result was already known, whether the statement is the interesting one, or whether the system's search was steered.

What to watch next

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• The Lean development itself. Whether the formalisation is published, which axioms it assumes, and whether independent parties can build it from source.
• The Stanford exposition. A shorter expert write-up from Blanc, Docter, Strassle and Tan is the nearest thing to independent review.
• Peer review. A journal or conference acceptance would settle the mathematical side; nothing else will.
• Details of the run. Model identity, compute, number of attempts, and how candidate proofs were selected. Without those, the speed claim cannot be evaluated.

Bottom line: if the Lean-checked proof is what PSI says it is, this is the most significant AI-assisted mathematics result to date — a genuine open problem in quantum complexity theory closed, with machine verification. The theorem's scope is narrower than the headlines suggest, and the claim about how it was found is still a vendor account. Both can be true at once, and both deserve to be stated.

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