Human-ai Interactive Theorem Proving Enables Scientific Discovery and Preserves Mathematical Rigor
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The pursuit of mathematical proof often demands immense creativity and painstaking verification, and researchers are now exploring how artificial intelligence can accelerate this process. Chenyi Li, Zhijian Lai from Beijing International Center for Mathematical Research, Peking University, Dong An, and Jiang Hu, with Zaiwen Wen also from Beijing International Center for Mathematical Research, Peking University, demonstrate a new workflow that combines human expertise with the power of large language models. Their approach allows mathematicians to maintain control over the core logic of a problem, while the AI assists in searching for potential proofs, suggesting new properties, and constructing solutions that meet specific criteria. This collaborative framework, tested successfully on complex problems linking manifold optimization and Grover’s search algorithm, promises to significantly speed up mathematical discovery and algorithm design, all while ensuring the transparency and rigor essential to scientific advancement. Riemannian Descent, Convergence Rate, and Improvements Scientists are achieving faster convergence rates in a specific type of optimization algorithm, Riemannian gradient descent, by exploiting the unique structure of the mathematical functions involved.
This research focuses on functions defined on unitary matrices, where the inherent properties allow for a more efficient search for optimal solutions.
The team demonstrates that by carefully analysing the function’s behaviour, it is possible to establish a stronger relationship between the function’s value and the magnitude of its gradient, leading to a significant improvement in convergence speed. This advancement moves beyond traditional results, potentially reducing the computational effort required to solve complex optimization problems.
Language Models Guide Manifold Optimization Research Researchers have developed a collaborative workflow integrating large language models into mathematical research, accelerating discovery while upholding rigorous standards of correctness. This human-in-the-loop system empowers experts to define problems and acceptable assumptions, while the language model explores potential proofs, identifies contradictions, and proposes candidate theorems and properties. The system generates structures and parameters that satisfy explicit constraints, supported by numerical experiments and simple verification checks, providing a foundation for expert refinement and the formulation of precise, rigorous proofs. In a case study connecting manifold optimization and Grover’s quantum search algorithm, the pipeline successfully identified invariant subspaces and explored Grover-compatible retractions, ultimately establishing convergence guarantees for the retraction-based gradient method. This demonstrates the system’s capacity to address complex mathematical challenges and deliver verifiable results, offering a new paradigm for mathematical exploration.
Language Models Accelerate Theorem Proving Scientists are accelerating mathematical theorem proving by integrating large language models into a human-in-the-loop workflow. This approach centres on interactive theorem proving and discovery, where human experts retain control over problem formulation and admissible assumptions, while the language model searches for proofs or contradictions and proposes candidate properties. This collaboration allows experts to refine model outputs and organize results into precise statements and rigorous proofs.
The team validated this workflow in a case study connecting manifold optimization and Grover’s quantum search algorithm, successfully identifying invariant subspaces and exploring Grover-compatible retractions. Experiments demonstrate the pipeline’s ability to obtain convergence guarantees for the retraction-based gradient method, a crucial step in optimizing complex systems. This framework provides a practical template for integrating large language models into frontier mathematical research, accelerating the research cycle and enhancing the efficiency of theorem proving.
Language Models Aid Mathematical Theorem Proving Researchers have developed a novel workflow integrating large language models into the process of mathematical research and theorem proving.
The team created a system where human experts maintain control over problem definition and assumptions, while the language model assists in exploring proof spaces, proposing candidate theorems and properties, and constructing solutions that meet specific constraints. This approach facilitates faster exploration of complex mathematical problems while preserving the necessary rigor and transparency of reasoning, effectively transforming model outputs into verifiable definitions, lemmas, and proofs. The researchers validated this workflow in a case study connecting manifold optimization and Grover’s quantum search algorithm, successfully identifying invariant subspaces and exploring convergence guarantees for related methods. Specifically, the workflow supports three key stages of research: conceptualizing topics with AI assistance, goal-guided proving for fixed targets, and open-ended theorem discovery when the conclusion is unknown. 👉 More information 🗞 Advancing Research via Human-AI Interactive Theorem Proving 🧠 ArXiv: https://arxiv.org/abs/2512.09443 Tags: