# Accelerating mathematical research with language models: A case study of an interaction with GPT-5-Pro on a convex analysis problem

arXiv: https://arxiv.org/abs/2510.26647 · HTML: https://mlsanity.com/paper/2510.26647/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2025.

## Built on by (most cited first, 5 of 5)

- [Towards Autonomous Mathematics Research](https://mlsanity.com/paper/2602.10177/) (2026)
- [Solving a Research Problem in Mathematical Statistics with AI Assistance](https://mlsanity.com/paper/2511.18828/) (2025)
- [The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learning](https://mlsanity.com/paper/2603.15914/) (2026)
- [Shaping the Future of Mathematics in the Age of AI](https://mlsanity.com/paper/2603.24914/) (2026)
- [The Benjamini--Hochberg Procedure Can Fail to Control the FDR for Correlated Two-Sided Gaussian Tests](https://mlsanity.com/paper/2607.12208/) (2026)

## Abstract (arXiv)

Recent progress in large language models has made them increasingly capable research assistants in mathematics. Yet, as their reasoning abilities improve, evaluating their mathematical competence becomes increasingly challenging. The problems used for assessment must be neither too easy nor too difficult, their performance can no longer be summarized by a single numerical score, and meaningful evaluation requires expert oversight. In this work, we study an interaction between the author and a large language model in proving a lemma from convex optimization. Specifically, we establish a Taylor expansion for the gradient of the biconjugation operator--that is, the operator obtained by applying the Fenchel transform twice--around a strictly convex function, with assistance from GPT-5-pro, OpenAI's latest model. Beyond the mathematical result itself, whose novelty we do not claim with certainty, our main contribution lies in documenting the collaborative reasoning process. GPT-5-pro accelerated our progress by suggesting, relevant research directions and by proving some intermediate results. However, its reasoning still required careful supervision, particularly to correct subtle mistakes. While limited to a single mathematical problem and a single language model, this experiment illustrates both the promise and the current limitations of large language models as mathematical collaborators.