Antonio and Pablo Acquaviva described five results in Banach space theory for which a model proposed key ideas and draft proofs
Antonio and Pablo Acquaviva described five results in Banach space theory for which a model proposed key ideas and draft proofs
On July 19, Antonio and Pablo Acquaviva posted a preprint presenting five results in Banach space theory. The authors report that the model identified the key steps and drafted the proofs, while they checked, corrected, and developed the final arguments.
Banach spaces provide a way to measure distances between objects and work with infinite sequences. Mathematicians use this framework to describe functions and transformations of functions. A single missing assumption can invalidate a proof in this field, so the authors examined every step, corrected errors, and checked the arguments against earlier theorems.
According to the Acquavivas, the model's initial answers already contained the main proof in four of the problems. The authors had to correct references, specific errors, and the presentation. For the fifth problem, the model proposed a plan for a long proof, and the authors completed the intermediate steps and assembled them into a coherent argument.
In the same paper, the authors describe a system for finding problems in the scientific literature. Scripts retrieved the source files of papers from arXiv and flagged phrases such as “question,” “problem,” and “conjecture.” An agent then read the relevant passage, searched for a proof, a counterexample, or an existing answer, and saved its findings in a package for a mathematician to review. Each package contains the original paper, the exact statement of the question, the reasoning behind the proposed answer, and the references found during the search.
The main run processed a queue of 1 433 papers in functional analysis. The mathematicians reviewed the resulting packages individually: 31 were marked as verified, and 10 were rejected. The five results in the first part of the preprint came from problems selected by the authors themselves, not from this queue.
The work therefore involved two distinct modes. In one, the model works on a problem selected by people. In the other, it finds questions in papers and prepares material for review.
In the work on a lower bound for convex optimization, the argument is written in a formal language, and Lean checks it line by line. In the Acquavivas' work, the mathematicians check the proofs themselves: their package assembles material for review but does not replace that review with a formal certificate.
In Google DeepMind's FunSearch, the model also proposes candidate solutions, but it expresses them as programs. An automated evaluator runs the code and selects the successful candidates. A proof cannot be tested in the same way. A mathematician must check every step and compare the theorem with both the original question and the published literature.
The model can explore possible approaches quickly, while the mathematician selects the question, checks the argument, and determines how it relates to known results. In this work, a model's answer becomes a research result only after that review.