Physical Superintelligence (PSI), a public benefit corporation, says a research system it built found a proof of the Aaronson–Ambainis conjecture, a problem in quantum complexity theory that its announcement says has been “Open since 2008”. PSI says “Instances of a non-public Anthropic model were used in the run.”
PSI says the system “had a complete proof of the conjecture during the third day of the run”, and that Lean, a proof assistant, accepted it. That is 18 years from 2008 to 2026, by our arithmetic. The claim has not yet been published in a peer-reviewed journal, and we found no public assessment by Scott Aaronson or Andris Ambainis.
Business Pill · AI Agents Explained
A short explainer of how AI agents work. It teaches the general idea only and does not discuss PSI, Anthropic or this proof.
The key insight: As we read PSI’s announcement and paper, the claim is that an AI research system running a non-public Anthropic model produced a Lean-checked proof of a conjecture open since 2008 during the third day of a run, 18 years later by our arithmetic; the paper is not peer reviewed, and we found no public assessment by Aaronson or Ambainis.
The conjecture, stated by Aaronson and Ambainis in their paper on the need for structure in quantum speedups, concerns bounded polynomials of low degree. PSI’s paper, by David Turturean, states the result: “We prove their conjecture: every function f : {โ1, 1}n โ [โ1, 1] of degree at most d โฅ 1 has a variable of influence at least c Var(f )5/2/d6, where c > 0 is an absolute constant.”
The abstract spells out what that means for quantum computers: “Thus, as Aaronson and Ambainis anticipated, superpolynomial quantum speedups in the query model require structure, such as a promise on the inputs.” PSI’s announcement puts it this way: “So on yes-or-no questions, a quantum computer cannot save exponentially many looks unless the string has a rare pattern.”
The exponent in the influence bound across three versions of the proof, as stated in Appendix A of PSI’s paper: 22 in the first complete proof, written during the third day of the run, then 11, then 7, each checked in Lean, the paper says. Lower is stronger. 22 divided by 7 is about 3.1, our arithmetic.
How the Proof Was Found
The paper’s AI disclosure says: “The mathematical statements in this paper were produced by AI models.” It says the run used “instances of a non-public Anthropic model, serving within the harness to search for existing ideas, to propose new ideas, to combine and test these ideas, and to supervise the work and the communication between other instances.”
Turturean, PSI’s announcement says, “developed the research system, scoped the problem and ran the system on it”, and “is responsible for the paperโs correctness”. The paper says he “prompted an instance of a non-public Anthropic model to package the results into a first draft of this paper”. Advisor Jim Halverson “audited the proof”, PSI says.
From Exponent 22 to 7
The paper’s timeline says the first complete proof “gave the bound Var(f )8/(223ฯ12d12) in place of the bound of Theorem 1.1, that is, the exponent 22 in place of 7 in Corollary 1.2.” After the model instances were asked to push on open problems, the paper says, “the integration over all levels gave the exponent 11, checked in Lean, and then the averaging over coordinates” gave the exponent 7, also checked in Lean.
The final exponent is about 3.1 times smaller than the first, by our arithmetic (22 divided by 7). PSI’s repository publishes the Lean 4 formalization and the paper. The repository was created on 6 November, according to GitHub’s records.
What Other Researchers Said
PSI says Guy Blanc, Jordan Docter, Carmen Strassle and Li-Yang Tan, who in August settled a special case of the conjecture’s prediction, “have since been reading our proof of the full conjecture, have sent us comments and are writing a shorter exposition of it for experts.” PSI quotes them: “This is a huge result. It resolves a decades-old conjecture with important implications for the nature of problems that admit quantum speedups.”
PSI’s announcement places all four at Stanford University; its thread on X places Blanc at MIT. Turturean wrote on X that the system took “upwards of 3 days” and that “After sitting with the proof for some time, indeed it still feels a bit alien.”
What Anthropic Has Said
We found no statement from Anthropic about the run or the model when we checked its X account at 15:58 UTC on 6 November. Neither PSI’s announcement nor the paper names the model; both call it non-public.
As we read the paper’s AI disclosure, the human role it describes is scoping the problem, building and running the harness, and reviewing the draft, while it says the mathematical statements were produced by AI models.
As we read Appendix A, the first proof was weaker than the final one: the paper says the exponent went from 22 to 11 to 7, each version checked in Lean, about a 3.1-fold cut by our arithmetic.
As we read PSI’s materials, the model is identified only as non-public and from Anthropic; we found no Anthropic statement about it.
David Turturean, PSI, on X, 6 November 2026
“After sitting with the proof for some time, indeed it still feels a bit alien.”
Three Implications
FOR QUANTUM COMPUTING PSI’s announcement says that on yes-or-no questions a quantum computer cannot save exponentially many looks unless the input has a rare pattern; this is PSI’s statement of what the proof implies.
FOR AI RESEARCH CLAIMS The paper says its statements were produced by AI models and that a human is responsible for its correctness; PSI says four researchers who worked on a special case are reading the proof.
FOR READERS Which Anthropic model was used, and on what terms, is not stated; we did not check the Lean files ourselves.
The framework says: “The surface, not the model, governs autonomy.”
As we read it, PSI’s account puts weight on the harness: the paper says model instances were used to search, propose, combine, test and supervise inside a research system PSI built. The framework does not discuss PSI, this conjecture or mathematics.
What Is Not Established
The proof has not been through journal peer review, and we found no public assessment by Scott Aaronson or Andris Ambainis. Which Anthropic model was used, and on what terms PSI had access, are not stated. We did not check the Lean formalization ourselves, and we did not contact PSI or Anthropic.
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PSI says a research system running a non-public Anthropic model found a proof of the Aaronson–Ambainis conjecture on the third day of a run, 18 years after it was posed, by our arithmetic. PSI says Lean accepted the proof, and the paper says the system later improved the bound’s exponent from 22 to 7.
A note on sourcing. The claims about the proof are PSI’s, from its announcement, its paper and its posts on X; the paper has not been through journal peer review. We did not check the Lean formalization ourselves. Years and ratios are our arithmetic. We did not contact PSI or Anthropic.
Gennaro Cuofano is a CRO and tech executive who has worked in AI since late 2015, starting with bringing NLP, natural-language generation, voice and chatbot products to the marketing industry. His background is in law and finance: he holds a Master's degree in Law and an International MBA with an emphasis on corporate finance (LUISS Business School and the University of San Diego, 2012), and worked as a financial analyst at a real-estate investment firm in San Diego and as an assistant controller. His work focuses on business model strategy, business engineering and, more broadly, structural analysis: how companies make money, read from their own filings. He created FourWeekMBA and leads research there and at The Business Engineer, his newsletter on AI and business strategy, with over 95,000 subscribers and more than 1,000 published analyses. His writing has also appeared on Entrepreneur (for example, How to Digitize a Family Business in Europe) and HackerNoon. Find him on LinkedIn and Substack. See how we source.
