Did AI prove a major quantum-computing conjecture? PSI has a Lean-checked manuscript—but expert verification is still open
Physical Superintelligence says a system using a non-public Anthropic model produced a proof of the Aaronson-Ambainis conjecture. Public Lean files support a machine-checkable theorem, while statement matching, exposition and the quantum consequence remain unreviewed.
The short version
- What happened
- PSI says its internal research harness used instances of a non-public Anthropic model and produced a complete proof on the third day of a run, then improved the quantitative bound twice.
- Who it affects
- Mathematicians gain a public machine-checkable artefact while inheriting statement-matching and independent-replay work.
- What remains unproven
- Self-published and not peer reviewed.
Keep in mind
Self-published and not peer reviewed.
The non-public model and full discovery process cannot be reproduced.
Lean does not itself establish equivalence to the intended conjecture.
Not every prose claim or consequence is formalised.
One improved endpoint assumes an unformalised supporting theorem, and one successful run cannot estimate general AI research capability.
Multi-source analysis · 3 linked sources. Primary analysed source date: 6 November 2026. Earliest linked supporting source: 1 January 2014.
Read the full report · 7 min The evidence, methods, implications and limitations
The answer: there is a serious public proof package, but 'proved' still needs two kinds of review
Physical Superintelligence, or PSI, has released a manuscript and Lean 4 formalisation claiming to resolve the Aaronson-Ambainis conjecture, an open problem about the limits of quantum speedups. PSI says an internal research system using instances of a non-public Anthropic model produced a complete proof on the third day of a run. The company kept the system working and reports that it improved the quantitative bound twice before the final 6 November release.
This is more than an unsupported model answer. The public repository contains a formal statement, proof files, a paper-to-Lean map, an axiom audit and verification records. Its main printed theorem says that a bounded low-degree polynomial on the Boolean cube with non-trivial variance must have at least one influential variable. The simpler bound in the repository is at least epsilon divided by degree, raised to the seventh power.
Yet two questions remain distinct. One is whether Lean accepts the formal theorem from the definitions and dependencies supplied. The other is whether that theorem exactly captures the 2014 conjecture and supports the claimed implication for quantum query complexity. PSI reports a positive result on the first. The repository explicitly leaves statement matching for readers, and the manuscript has not completed peer review. The accurate status is a substantial machine-checkable proof claim, not an automatically settled result because an AI generated it.[1][2][3]
What the conjecture asks, in plain language
A quantum algorithm can inspect an input in superposition, so theorists compare how many queries it needs with an ordinary randomised algorithm. Famous exponential advantages often rely on highly structured inputs. Aaronson and Ambainis asked whether a comparable advantage can persist for typical, unstructured yes-or-no problems under a uniform distribution.
The bridge is a polynomial. After a bounded number of quantum queries, the probability that an algorithm answers yes can be written as a low-degree polynomial of the input bits. If the polynomial varies meaningfully, the conjecture says at least one bit must have noticeable influence: flipping it changes the value by an amount bounded below by a polynomial function of the variance and degree. An ordinary algorithm can inspect influential bits until little variation remains.
A polynomial lower bound prevents the cost of that classical approximation from exploding exponentially in the quantum query count for most inputs. It does not say quantum computers have no advantages, make Shor's factoring algorithm ordinary or compare complete machine runtimes. Hardware, error correction, data loading and practical problem structure sit outside the theorem. PSI's public explanation is useful, but each consequence still depends on the original reduction and its assumptions.[1][2][3]
The distinction that matters
The three checks behind an AI-generated formal proof
A checked theorem is necessary evidence, but every layer answers a different question.
Formal build
Replay Lean declarations and dependencies
Does the kernel accept the theorem?
Statement match
Compare definitions with the 2014 conjecture
Is it the intended claim?
Consequence
Audit the query-complexity reduction
Does the interpretation follow?
What PSI says the AI system produced
PSI attributes the mathematical statements, first manuscript draft and Lean formalisation to AI models inside a research harness developed by David Turturean. The company does not disclose the Anthropic model because it was non-public. Outsiders therefore cannot reproduce the generation process, prompts, sampling, tool calls, failed attempts or compute budget, even though they can inspect the resulting artefacts.
The proof follows a plan associated with Shachar Lovett and Jiapeng Zhang and builds on David Reimer's Butterfly Lemma and other earlier work. PSI says the system replaced an unresolved inequality in that plan with a new pairing argument. The AI did not work without human mathematical history: it searched, combined and extended an existing record. Turturean reviewed the paper and accepts responsibility for correctness, which preserves a named point of accountability.
The repository lists four final Lean theorems. Three cover the core printed and explicit bounds in two normalisations. A fourth improved-constant result assumes another theorem as a hypothesis because that supporting result is not formalised there. PSI says the core theorem does not depend on that hypothesis. Reviewers should keep those endpoints separate rather than treating the strongest constant and the main conjecture as one undifferentiated check.[1][2]
What Lean verifies—and what it cannot decide automatically
Lean is a proof assistant: a small kernel checks that each formal step follows from declared definitions, earlier theorems and accepted foundations. The repository says its final endpoints have no placeholders or added axioms and use standard foundations exposed by Mathlib. It also records checks through more than one kernel implementation. These features sharply reduce the risk of a hidden algebraic leap inside the formalised path.
Formal verification does not eliminate specification risk. A checked proof of the wrong proposition remains a proof of the wrong proposition. Reviewers must compare PrintedConjecture with the original statement, including the domain, normalisation, influence definition, variance condition and exponent. They must inspect dependencies and versions. The repository isolates the statement and makes this feasible, but it cannot perform the human act of deciding that the formal object matches the intended claim.
The manuscript also contains context, examples and consequences that do not all have corresponding Lean declarations. In particular, the link from the polynomial theorem to simulation of quantum query algorithms uses earlier mathematical results and interpretation. Expert checking must cover the formal theorem, the statement match and the external consequence. A green build is strong evidence for the first layer, not a certificate for every sentence around it.[2]
Why this matters for researchers and what would change the assessment
For mathematicians, the release offers an unusually inspectable AI-generated claim. PSI supplies definitions, code, a manuscript and a map between them instead of asking peers to trust a persuasive chat transcript. Researchers can replay the build, challenge a lemma and write a shorter exposition. That publication pattern is more useful than claiming autonomous scientific ability from a benchmark score.
Universities also gain new work: experts must audit provenance, check statement equivalence, verify citations, separate formal from informal claims and decide who is accountable. Institutions need policies for model disclosure, prompt and dependency preservation, credit, licensing and corrections. The public should not read the result as a new quantum computer or an encryption breach. Its immediate impact is intellectual, so it is consequential Latest research—not Breaking.
Confidence would rise with independent rebuilds, a published statement comparison, expert review of the quantum reduction and a shorter human explanation. It would weaken if the formal theorem uses a different condition, the files fail to rebuild, or an undeclared assumption appears. The broad AI capability claim needs many disclosed attempts, including failures. One successful internal run cannot estimate how often AI systems solve open problems.[1][2][3]
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Read next
Continue exploring mathematics: compare the finding and its evidence limits.
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Read the next reportExplore the full source and evidence record
What this means for people
- Mathematicians gain a public machine-checkable artefact while inheriting statement-matching and independent-replay work.
- Research institutions need disclosure and correction rules that keep a named human accountable.
- The public should read this as theoretical evidence, not as a new quantum computer or proof that AI routinely solves open science.
Global context
The proof draws on an international research record while the release comes from a US company using a non-public commercial model. Formal artefacts can be replayed globally, but frontier-model and compute access is uneven. Independent validation may influence publication and credit norms beyond quantum computing.
What the evidence does not yet show
- Self-published and not peer reviewed.
- The non-public model and full discovery process cannot be reproduced.
- Lean does not itself establish equivalence to the intended conjecture.
- Not every prose claim or consequence is formalised.
- One improved endpoint assumes an unformalised supporting theorem, and one successful run cannot estimate general AI research capability.
What to watch next
- Independent Lean rebuilds
- A statement-equivalence audit
- Review of the quantum consequence
- Corrections or an independent exposition
- Disclosure of comparable attempts and failures
Living evidence record
Impact record IAI-01EZA6V
Evidence stage
Studied
Confidence
Corroborated
Reporting basis
Multi-source analysis
Independent or research support
Present
Record status
Monitoring
Last checked
8 November 2026
Source trail
3 direct sources across 2 source types.
People impact
Documented in this record.
Uncertainty
Limits and next checks are explicit.
Stages describe the evidence available—not whether a technology is good or bad. See the public method.
Evidence trail
Sources used for this report
Links checked 8 November 2026
Research question, topics and regions
Whether PSI's public manuscript and Lean 4 development establish the Aaronson-Ambainis influence conjecture, and what remains outside the formal checker
Topics: AI for science · Formal verification · Quantum computing · Mathematics · Research integrity
Regions: United States · Global
This report is labelled multi-source analysis. We summarise and analyse source material in our own words; company statements remain attributed claims until independently supported. Translated summaries preserve the meaning of the original source and link back to it. Read our editorial standards.
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