PSI reports automatically verified proof of the Aaronson-Ambainis conjecture – Unite.AI

Physical Superintelligence announced on November 6, 2026 that an internal search system running instances of a non-public anthropic model has found proof of the Aaronson-Ambainis conjecture, an open quantum computing problem since 2008.

The system produced a complete proof on the third day of execution on PSI’s own infrastructure and the Lean Proof assistant accepted it. PSI then assigned the relevant open issues to the system, which improved the result twice before Lean accepted the final version. The manuscript and formalization of Lean 4 are public.

What the conjecture says about quantum accelerations

The question is when quantum computers can be exponentially faster than classical machines. Shor’s 1994 factorization algorithm is structure dependent: its input carries with it a repetitive pattern that the algorithm can exploit. The PSI result addresses the contrasting case of random or unstructured inputs, where such a pattern is generally absent.

In the query model, a classical algorithm reads one bit per query, while a quantum query can be distributed in superposition over many input locations. PSI’s explanation relates this model to polynomials: after T quantum queries, the probability that the algorithm answers yes is a bounded polynomial of the input bits with degree at most 2T. Simulating the quantum algorithm then becomes a matter of estimating that polynomial without exponentially reading many input bits.

In their original paper, Scott Aaronson and Andris Ambainis hypothesized that every low-degree bounded polynomial has a highly influential variable. They showed that, assuming the conjecture, any quantum algorithm with queries T can be simulated on most inputs by a classical algorithm that polynomially queries many in T. The consequence is polynomial rather than exponential growth in classical queries for any fixed approximation error and fraction of excluded inputs; does not rule out exponential quantum advantages for problems with adequate structure.

Formally the repository considers a real polynomial P degree at most D whose values ​​on the discrete cube are between 0 and 1. If its variance on a uniformly random input is at least ε, some coordinate has an influence of at least (8√15 / (125π⁴)) · ε^(5/2) / d⁶, and consequently at least (ε/d)⁷. Influence is the expected squared change in P when the bit of that coordinate is inverted.

How the demonstration was built and verified

The argument follows a plan by Shachar Lovett and Jiapeng Zhang based on fractional certificates and level sets, but replaces an inequality that their plan left conjectural with a new all-pairs inequality. A determinant generates a one-to-one pairing of input settings. PSI claims that this coupling goes beyond the weakest coupling of David Reimer’s Butterfly Lemma because it also limits how often coupled settings agree on each coordinate, a necessary limit for the new inequality.

The repository divides the elementary proof into four parts: deriving the all-pair inequality from the determinant and the bijection, integrating it over level sets, averaging it over the coordinates, and establishing a row boundary with a pointwise sensitivity limit. Together, these steps produce the explicit influence boundary.

Formal development contains no proof placeholders or added axioms, according to the repository’s registered audit. His declarations were reproduced via the Lean kernel and lean4lean, an independent implementation of Lean type checking. In a sandbox analysis, a comparator using the nanoda and Lean kernels accepted the final four theorems; on a second machine, it accepted one small correct solution and rejected four deliberately incorrect ones.

Such controls have explicit limits. A reader has yet to confirm that the formal statement matches the printed conjecture. The human-readable document is not itself controlled by machines, while the Lean kernel and compiler, Mathlib definitions, and the machines used for control remain part of the trusted foundation.

Human roles and expert review

David Turturean, a member of the PSI technical staff, developed and maintained the search system, analyzed the problem, recommended the model to a first draft, reviewed that draft, and is responsible for the correctness of the document. The repository reveals that the AI ​​models produced the mathematical statements, the first draft and the Lean formalization. Jim Halverson monitored the proof and Phiala Shanahan commented on the formalization.

The announcement follows a paper submitted on August 19, 2026 by Guy Blanc, Jordan Docter, Carmen Strassle and Li-Yang Tan. These authors had established the consequences of classical simulation for parallel quantum algorithms, demonstrating that a t-query, d-round algorithm can be simulated on most inputs with classical queries t^(O(d²)). The PSI says all four have read the full evidence, provided comments and are preparing a shorter presentation for the experts.

In their assessment published by PSI, the researchers wrote: “This is a huge achievement. It resolves a decades-old conjecture with important implications for the nature of problems admitting quantum accelerations. The proof draws on and extends several lines of work in the analysis of Boolean functions and probabilistic combinatorics. It all comes together in a rather magical way.”

The immediate test is unusually concrete: independent cores can reproduce the coded argument, while experts still have to confirm that the coded claim matches Aaronson and Ambainis’ conjecture and evaluate the parts of the article outside of Lean. This division determines what the machine controls establish and what still requires human mathematical review.