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

Abstract: Recent progress in LLMs 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 LLM 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 LLM, this experiment illustrates both the promise and the current limitations of LLMs as mathematical collaborators.

• The paper demonstrates that GPT-5-Pro contributes to a rigorous proof of the first-order Taylor expansion for the gradient of a perturbed convex function.
• It utilizes advanced techniques like Alexandrov's theorem, local quadratic bounds, and subdifferential analysis to ensure precision in convex analysis.
• The study highlights the iterative refinement between human experts and the model, emphasizing both the potential and limitations of AI in mathematical research.

Accelerating Mathematical Research with LLMs: A Case Study of GPT-5-Pro on a Convex Analysis Problem

Overview and Motivation

This paper presents a detailed case study of collaborative mathematical research involving a human expert and GPT-5-Pro, OpenAI's latest LLM, focused on a nontrivial problem in convex analysis. The central mathematical objective is to establish a first-order Taylor expansion for the gradient of the biconjugation operator (i.e., the Fenchel biconjugate) applied to a strictly convex function perturbed by a smooth test function. The study not only provides a rigorous proof of the expansion but also documents the iterative interaction between the researcher and the LLM, highlighting both the model's contributions and its limitations.

Mathematical Context and Main Result

The problem is motivated by optimal transport theory, specifically the analysis of perturbations of convex potentials that generate optimal maps. The classical result by Gangbo [gangbo1994elementary] provides a first-order expansion for the Fenchel conjugate of a perturbed convex function. The present work seeks to extend this to the biconjugate and its gradient, aiming to prove that for a strictly convex, twice differentiable function $\phi$ and a compactly supported $C^2$ function $h$, the following holds for almost every $x \in \mathbb{R}^d$ and sufficiently small $|t|$: $$\nabla (\phi + t h)^{**}(x) = \nabla \phi(x) + t \nabla h(x) + o(t).$$ The proof establishes that the remainder term $o(t)$ is in fact zero for $|t|$ small enough (depending on $x$), and provides explicit bounds for the admissible $t$.

Technical Proof Structure

The proof leverages the following key ingredients:

• Alexandrov's Theorem: At almost every $x$, a convex function $\phi$ is twice differentiable with a positive-definite Hessian.
• Local Quadratic Lower Bound: At such $x$, $\phi$ admits a quadratic lower bound in a neighborhood, which is crucial for controlling the behavior of the perturbed function.
• Affine Envelope Characterization: The biconjugate $f^{**}$ at a point $x$ equals the supremum of all affine minorants of $f$ that touch $f$ at $x$.
• Carathéodory's Theorem: Any convex combination in finite dimensions can be represented with at most $d+1$ points, facilitating the analysis of convex envelopes.
• Taylor Expansion for $h$: The $C^2$ regularity of $h$ allows for uniform quadratic bounds on its deviation from linearity.

The proof proceeds by constructing an explicit affine minorant at $x$ and showing, via careful estimates, that for sufficiently small $|t|$ this minorant remains below $f = \phi + t h$ globally. The argument is split into two regions: inside a small ball around $x$, where quadratic bounds dominate, and outside, where a fixed gap (from the quadratic lower bound on $\phi$) and boundedness of $h$ ensure the minorant remains valid. The gradient equality follows from a subdifferential argument that does not require global convexity of $f$.

Interaction with GPT-5-Pro: Collaborative Reasoning Process

The documented interaction reveals several important aspects of LLM-assisted mathematical research:

• Model Contributions: GPT-5-Pro suggested relevant research directions, identified key conjectures, and provided partial proofs and technical lemmas. Notably, the model was able to articulate the necessity of a local (rather than global) argument and recognized the importance of the affine envelope characterization.
• Model Limitations: The model's reasoning required frequent correction, especially regarding the propagation of quadratic bounds and the distinction between pointwise and neighborhood properties. Several technical errors were identified and addressed by the human expert, particularly in the handling of bounds outside the local ball and the use of Carathéodory's theorem.
• Iterative Refinement: The proof was ultimately completed through a back-and-forth process, with the human expert guiding the model toward a correct and rigorous argument. The model's ability to adapt and refine its approach in response to feedback was instrumental in reaching the final result.

Implications for Mathematical Research and LLM Evaluation

Practical Implications

• Accelerated Discovery: The case study demonstrates that state-of-the-art LLMs can significantly accelerate mathematical research by suggesting plausible directions, generating technical lemmas, and providing partial proofs.
• Expert Oversight Required: Despite their capabilities, LLMs still require careful supervision by human experts to ensure rigor and correctness, especially in subtle technical arguments.
• Pointwise vs. Uniform Results: The proof highlights the importance of distinguishing between pointwise and uniform statements in analysis, a nuance that LLMs may not consistently handle without explicit guidance.

Theoretical Implications

• Local Convex Envelope Lemma: The result provides a precise characterization of when the biconjugate of a perturbed convex function coincides with the function itself at a point, with explicit dependence of the admissible perturbation size on local properties.
• Subdifferential Argument: The approach clarifies the relationship between differentiability and subdifferential inclusions for convex envelopes, extending classical convex analysis results.

Evaluation of LLMs

• Qualitative Assessment: The study argues that meaningful evaluation of mathematical competence in LLMs increasingly requires qualitative, expert-driven case studies rather than automated benchmarks.
• Scalability Challenges: Systematic and scalable frameworks for evaluating LLMs as research collaborators remain an open challenge, given the need for extended interactions and subjective judgment.

Future Directions

• Systematic Evaluation Protocols: Developing protocols for expert-driven, reproducible evaluation of LLMs in mathematical research is a critical next step.
• Enhanced Reasoning Capabilities: Improving LLMs' ability to handle subtle distinctions in analysis, such as pointwise versus uniform properties, will be essential for their broader adoption in mathematical research.
• Integration with Formal Verification: Combining LLMs with formal proof assistants could further enhance rigor and reliability in collaborative mathematical discovery.

Conclusion

This case study provides a rigorous proof of a first-order expansion for the gradient of the biconjugate of a perturbed strictly convex function, achieved through collaborative interaction with GPT-5-Pro. The documented process illustrates both the promise and current limitations of LLMs as mathematical research assistants. While LLMs can accelerate progress and suggest valuable directions, expert oversight remains indispensable. The study underscores the need for systematic, qualitative evaluation frameworks and points toward future developments in AI-assisted mathematical research.