Google DeepMind's theorem-proving system AlphaProof Nexus has now been published in Science. The first author is listed as Tsoukalas, and the whole research team comes from DeepMind. The work was first made public as a preprint at the end of May; this is the peer-reviewed journal version.
The scorecard is unchanged. The system attempted 353 unsolved Erdős problems and solved 9 of them, two of which had been open for 56 years. It proved 44 of 492 open conjectures in the OEIS, the database of integer sequences. It also answered a roughly 15-year-old question in algebraic geometry about Hilbert functions and improved a known bound in min-max optimization.
One Gemini, one Lean compiler
The system is built on Gemini 3.1 Pro. The model drafts proofs in the formal language Lean, the Lean compiler checks every step as it changes, and the error messages are fed straight back into the next attempt. Humans only look at the result at the very end.
The paper separates out four versions:
- Version A: the plainest generate-and-verify loop, with the large model writing and the compiler judging;
- Version B: A plus DeepMind's earlier prover AlphaProof, brought in specifically to fill in missing pieces of a proof;
- Version C: adds an evolutionary mechanism in which multiple sub-agents share a pool of draft proofs ranked by Elo score;
- Version D: everything combined, used to go after the Erdős problems.
The most intriguing result is that version A independently proved all 9 Erdős problems as well; it just spent more money on the hardest few. The inference cost cited by The Decoder is a few hundred dollars per problem. In the paper, the authors put it this way:
"power of compiler feedback in grounding LLM reasoning"
Reservations from Spanish scholars
On the day the paper went online, Spain's Science Media Centre gathered comments from several local academics, and opinion split into two camps.
Pablo Haya Coll, a researcher at UAM, believes the approach can systematically produce mathematical discoveries of real substance. Josep Curto of UOC offered the verdict "a dazzling display of technical capability marred by significant methodological inconsistencies". He listed three points. Versions A and D solved the same problems, which suggests that framework complexity does not necessarily determine discovery power. Costs were counted only for the successful runs, and the accounting for versions B and D also leaves out AlphaProof's own inference spend. And Lean only guarantees that a proof holds for the statement that was written down; whether that statement faithfully translates the original problem still has to be checked by mathematicians by hand, and the paper itself admits that formalization errors occurred.
Javier Aramayona, director of ICMAT, worries about something else: a growing body of results that are "known to be true but not fully understood by anyone" could weaken the understanding step in mathematical research, as well as the training of young researchers. Teodoro Calonge of the University of Valladolid put it more bluntly, saying the field still has considerable room for improvement.
The authors also spell out the limits themselves. AI's overall success rate on Erdős problems is roughly 1% to 2%, and the problems solved tend to be the relatively easy ones, concentrated in combinatorics, convex optimization and number theory, where the Mathlib library is well developed and problems break down readily into subgoals. Problems that require inventing large amounts of new theory are largely out of the system's reach.
Putting this year's events together
In May, this site reported on an OpenAI model cracking an 80-year-old Erdős conjecture. In September, a dispute over credit broke out between mathematicians and OpenAI. On October 6, OpenAI released 722 AI-generated mathematical manuscripts on GitHub in one go, through an open-source repository under the Apache 2.0 license rather than through a journal. According to The Decoder, a group of mathematicians led by Terence Tao then called for a boycott of the practice.
The difference between the two companies' paths is laid out clearly here. OpenAI chose volume and openness, leaving it to the community to filter. DeepMind chose far fewer results, each of which passed both the Lean compiler and journal peer review. The paper is also restrained about Lean's role, describing formal verification as something that "can serve as a filter for determining which proofs merit human review", that is, a way to decide which proofs deserve a human's attention.
A rough back-of-the-envelope check: at a few hundred dollars per problem, the inference spend for the 9 Erdős problems is far below one postdoc's monthly salary. But on the basis Curto points to, adding the failed runs and the AlphaProof calls would raise the bill by an amount the paper does not currently state. DeepMind has put the proofs and some explanatory material in the alphaproof-nexus-results repository on GitHub, so outside mathematicians can check them one by one.
Sources: Science journal paper, The Decoder, CocoLoop, expert commentary from Spain's Science Media Centre; the paper itself was checked for the 9 solved out of 353 Erdős problems, 44 proved out of 492 OEIS conjectures, and the division of labor among the four agent versions.