AI Proves Aaronson-Ambainis Conjecture Explained: Anthropic Model, Lean Check and What It Really Means

Image: Wikimedia Commons — Scott Aaronson, who posed the conjecture with Andris Ambainis in 2008. Public domain.

TL;DR: AI proves Aaronson-Ambainis conjecture explained in one line: on 6 November 2026, Physical Superintelligence (PSI) said a research system running an unreleased Anthropic model found a proof of the 2008 Aaronson–Ambainis conjecture — and the Lean proof assistant checked every step. It is a real, machine-checked result, but it is not yet peer reviewed, and neither Aaronson nor Ambainis has publicly assessed it yet.

📌 KEY HIGHLIGHTS

• PSI announced on 6 November 2026 that its internal research system, running instances of a non-public Anthropic model, proved the Aaronson–Ambainis conjecture, open since 2008.
• The system found a complete proof on the third day of its run, and Lean 4 — a proof assistant that verifies every logical step by computer — accepted it. It then improved the result twice, tightening the bound exponent and ending at 7.
• The result says exponential quantum speedups on yes-or-no questions need structured inputs; on random inputs, an ordinary computer can keep up with only polynomially more reads.
• Caveats that matter: the paper is not peer reviewed, the model used is not public, and human mathematicians still have to confirm the formal statement matches the original conjecture.

What happened?

On Friday 6 November 2026, Physical Superintelligence — PSI, a public benefit corporation that describes itself as an AI-native laboratory for physics — announced that it had settled one of the central open problems in quantum computing. Its research system, developed in-house and run on PSI's own infrastructure, used instances of a non-public Anthropic model to find a proof of the Aaronson–Ambainis conjecture, a question about the limits of quantum speedups that has been open since 2008.

The claim is striking, but what makes it different from an ordinary "AI solves maths problem" headline is the verification. The proof was formalised in Lean 4, a proof assistant in which a computer checks every step of a mathematical argument, and Lean's kernel accepted it. PSI has published both the manuscript and the full Lean formalisation, so anyone can replay the check. That is the strongest form of evidence a mathematical claim can carry short of peer review — and it is also where the honest limits begin, which this article keeps in view throughout.

AI proves Aaronson-Ambainis conjecture: how the proof was found

According to PSI, the run worked like a research team rather than a single prompt. Multiple instances of the Anthropic model searched the existing literature, proposed ideas, combined approaches and tested arguments against each other. The system had a complete proof during the third day of the run. PSI then pointed it at related open problems, and along the way the system improved its own result twice — each improved version was checked in Lean before PSI accepted it.

The proof follows a plan laid out by mathematicians Shachar Lovett and Jiapeng Zhang, who had left one key inequality unproved. PSI says its system replaced that missing piece with a new inequality, proved by pairing up input settings one to one using a determinant. The final theorem states that if v measures how much the polynomial varies and d is its degree, some bit has influence at least c·v^(5/2)/d^6, where c is about 0.0025 — which is at least (v/d)^7.

Credit is assigned carefully in PSI's own account, and this article follows it. David Turturean, a member of PSI's technical staff, led the work: he developed the research system, chose the problem, ran the system on it, prompted the model to package the results into a first draft, reviewed that draft and is named as responsible for the paper's correctness. Advisor Jim Halverson audited the proof, and advisor Phiala Shanahan commented on the Lean formalisation. The repository is blunt that the mathematical statements, the first draft of the paper and the Lean formalisation "were produced by AI models" — the human role was scoping, running, reviewing and taking responsibility.

Stage	What happened
Day 3 of the run	Complete proof found; Lean 4 accepts it
First improvement	Bound exponent tightened; Lean accepts the new version
Second improvement	Exponent tightened to 7; Lean accepts the final version
Independent re-check	Declarations replayed through Lean's kernel and lean4lean, an independent implementation of Lean's type checker; no placeholders, no added axioms
Still to come	Peer review, and public assessment by Aaronson and Ambainis; Stanford team writing a shorter exposition for experts

What the conjecture actually says

The question behind the conjecture is simple to state and hard to answer: when can a quantum computer be exponentially faster than an ordinary one? The famous quantum speedups all exploit structure — Shor's 1994 factoring algorithm works because the problem hides a repeating pattern the quantum computer can detect. But what about random inputs, which almost never have such structure? Could a quantum computer still win by an exponential margin there?

Aaronson and Ambainis found a way to settle this with pure mathematics. A quantum algorithm that looks at its input T times produces a yes-or-no answer whose probability is a polynomial in the input bits, of degree at most 2T. The "influence" of a bit is the average change in that polynomial's value when the bit is flipped. They conjectured that every bounded, low-degree polynomial that is not constant must have at least one bit with substantial influence — and showed that, if true, an ordinary computer could imitate the quantum computer's answer on most random inputs without needing exponentially more looks at the input.

Now that PSI says the conjecture is proven, the consequence is concrete: for yes-or-no questions, an ordinary computer can predict the quantum computer's probability of saying "yes", up to a small error, for all but a small fraction of inputs, using a number of looks that grows at most like a fixed power of T — not exponentially. Quantum speedups, on this measure, stay confined to problems with exploitable structure.

Image: Wikimedia Commons / Marozols, CC BY-SA 3.0 — Andris Ambainis, who posed the conjecture with Scott Aaronson.

Why it matters for readers — and what it does not mean

First, the boundary it draws is about queries — how many times an algorithm reads its input — not about total computing time. The theorem does not say quantum computers are useless, and it does not settle which real applications will deliver useful speedups. It says the advantage, where it exists, has to come from structure in the problem. That is a clearer research direction for the whole field, including India's quantum research community working under the National Quantum Mission: chase structured problems, and do not expect a free exponential win on unstructured data.

Second, it is a milestone for AI as a research tool, not just a chatbot. This is the second kind of AI-maths result in a week of them — OpenAI's GPT-6 Astra was recently shown fixing another model's maths — but the PSI result goes further: an AI system did not just assist a proof, it produced the mathematical statements, the draft and the formalisation, with humans scoping and verifying. The Lean check is what makes the claim auditable: a wrong proof cannot pass the kernel, whatever produced it.

Third, the limits are part of the story, not a footnote. The model that found the proof is not public and Anthropic has not said which model it was, so nobody outside PSI can re-run the discovery. The prose paper is not machine-checked — only the Lean formalisation is — and a reader must still confirm that the formal statement matches the conjecture as Aaronson and Ambainis meant it. PSI has put that statement in a short file that imports nothing but Lean's maths library, precisely so experts can compare it. The Stanford team that settled a special case of the conjecture in August 2026 — Guy Blanc, Jordan Docter, Carmen Strassle and Li-Yang Tan — has read the proof, sent comments and is writing a shorter exposition, which is the next credibility test to watch.

Should you care?

✅ Yes — if you follow AI or quantum computing. This is the most significant AI-proved maths result of the year so far, and the Lean verification makes it checkable rather than a press-release claim. But treat it as "claimed and machine-checked, awaiting peer review" — not as settled science — until Aaronson, Ambainis or independent mathematicians publicly confirm the formalisation matches the original conjecture. If you work in quantum algorithms, the practical takeaway is immediate: structure is where the advantage lives.

What to watch next

Three things will decide how this result is remembered. The first is the Stanford exposition: when Blanc and colleagues publish their shorter version for experts, read it as the first independent reading of the proof. The second is any public comment from Scott Aaronson or Andris Ambainis, whose assessment carries the most weight. The third is whether PSI's research system repeats the trick — a one-off proof is a headline, a repeatable system is a new way of doing mathematics. Anthropic, meanwhile, has not commented on which model ran inside PSI's system, so do not assume it was any public Claude model. Related reading: Quantinuum's Nexus roadmap for fault-tolerant quantum computing shows the hardware side of the same race, and Claude helping find a bacterial vault structure is the other recent example of an Anthropic model doing real scientific work.

FAQs

What is the Aaronson-Ambainis conjecture?
It is a 2008 conjecture by Scott Aaronson and Andris Ambainis about when quantum computers can be exponentially faster than ordinary ones. In plain terms, it says that every bounded, low-degree polynomial has at least one highly influential variable — which would imply that on random inputs, a classical computer can imitate a quantum computer's yes-or-no answer without exponentially more reads of the input.

How did AI prove the Aaronson-Ambainis conjecture?
Physical Superintelligence says its internally built research system, running instances of an unreleased Anthropic model, searched the literature, proposed and combined ideas, and found a complete proof on the third day of a run. The proof was then formalised in Lean 4, which checked every step, and improved twice before the final version was accepted.

Is the proof verified or peer reviewed?
It is machine-checked but not yet peer reviewed. Lean 4's kernel — and an independent implementation of Lean's type checker, lean4lean — accepted the formalisation, which contains no placeholders or added axioms. But the paper has not been through journal peer review, neither Aaronson nor Ambainis has publicly assessed it, and human mathematicians still have to confirm the formal statement matches the original conjecture.

What does the proof mean for quantum computers?
It means exponential quantum speedups on yes-or-no questions require structured inputs. On random, unstructured inputs, an ordinary computer can approximate the quantum answer with only polynomially more input reads. It does not rule out practical quantum advantages — it limits data reads, not total computing time — but it sharpens where researchers should look for an advantage.

Takeaway: An AI system has, PSI says, proved an 18-year-old conjecture about the limits of quantum speedups — and a computer checked every step. The result is real enough to take seriously and unreviewed enough to keep the caveats attached. Watch the Stanford exposition and the conjecture's authors for the verdict that matters.

Want the primary evidence first-hand? Read PSI's announcement and the public Lean formalisation linked in the Sources below — the machine-checked proof is the part anyone can verify for themselves.

Sources

• PSI — A proof of the Aaronson-Ambainis conjecture (primary source, 6 November 2026)
• Unite.AI — PSI Reports Machine-Checked Proof of Aaronson-Ambainis Conjecture
• The Quantum Insider — AI-Assisted Proof Settles Longstanding Question About Quantum Speedups
• Dagstuhl / CCC 2022 — Influence in Completely Bounded Block-Multilinear Forms and Classical Simulation of Quantum Algorithms (background on the conjecture)