Mathematical research with GPT-5: a Malliavin-Stein experiment (2509.03065v1)

Abstract: On August 20, 2025, GPT-5 was reported to have solved an open problem in convex optimization. Motivated by this episode, we conducted a controlled experiment in the Malliavin--Stein framework for central limit theorems. Our objective was to assess whether GPT-5 could go beyond known results by extending a \emph{qualitative} fourth-moment theorem to a \emph{quantitative} formulation with explicit convergence rates, both in the Gaussian and in the Poisson settings. To the best of our knowledge, the derivation of such quantitative rates had remained an open problem, in the sense that it had never been addressed in the existing literature. The present paper documents this experiment, presents the results obtained, and discusses their broader implications.

• The paper demonstrates that GPT-5 can assist in deriving a quantitative fourth-moment theorem in the Malliavin–Stein framework with explicit bounds based on fourth cumulants.
• It establishes sharp Gaussian bounds for sums of Wiener–Itô integrals and outlines necessary conditions in Poisson settings for convergence to a normal distribution.
• The study highlights the potential of AI to accelerate mathematical research while emphasizing the need for expert oversight in verifying complex proofs.

Mathematical Research with GPT-5: A Malliavin-Stein Experiment

Introduction and Motivation

This paper investigates the capabilities of GPT-5 as a mathematical research assistant, focusing on its performance in the Malliavin–Stein framework for central limit theorems. The experiment was motivated by a publicized episode in which GPT-5 reportedly solved an open problem in convex optimization, prompting the authors to assess whether GPT-5 could extend known qualitative results in probability theory to quantitative formulations with explicit convergence rates. Specifically, the paper targets the fourth-moment theorem for sums of multiple Wiener–Itô integrals, aiming to derive explicit bounds in both Gaussian and Poisson settings—problems that had not previously been addressed in the literature.

Mathematical Background

Malliavin–Stein Method

The Malliavin–Stein method combines Stein’s method for distributional approximation with Malliavin calculus, providing a framework for both qualitative and quantitative central limit theorems. In the context of Wiener chaos, the method yields explicit rates of convergence to Gaussianity, typically expressed in terms of cumulants or moments.

Fourth-Moment Theorems

The classical fourth-moment theorem of Nualart and Peccati states that for a sequence of normalized multiple Wiener–Itô integrals of fixed order, convergence of the fourth moment to three is equivalent to convergence in distribution to the standard normal. Quantitative refinements within the Malliavin–Stein framework relate the total variation distance to the fourth cumulant, enabling explicit convergence rates.

Main Results

Quantitative Two-Chaos Fourth-Moment Theorem (Gaussian Case)

The authors, with the assistance of GPT-5, establish a quantitative version of the fourth-moment theorem for sums of two multiple Wiener–Itô integrals of different parities. Let $X = I_p(f)$ and $Y = I_q(g)$, with $p$ odd, $q$ even, and $Z = X + Y$ normalized to unit variance. The main result is:

$$d_{\mathrm{TV}}(Z, N(0,1)) \leq \sqrt{6\,\kappa_4(Z)}$$

where $\kappa_4(Z) = \mathbb{E}[Z^4] - 3$ is the fourth cumulant and $d_{\mathrm{TV}}$ denotes total variation distance.

This bound is sharp in the sense that all terms are nonnegative and the mixed odd moments vanish due to parity. The proof leverages the Malliavin–Stein bound, orthogonality of Wiener chaoses, and explicit cumulant decompositions. The result provides a direct quantitative link between the fourth cumulant and the rate of convergence to Gaussianity for sums of multiple integrals of different orders.

Poisson Analogue and Limitations

In the Poisson setting, the situation is more subtle due to the non-vanishing of mixed odd moments such as $E[X^3Y]$ when $X$ and $Y$ are multiple Poisson–Itô integrals of different orders. The authors identify sufficient conditions under which a similar fourth-moment theorem holds:

If $X_n = I_p^\eta(f_n)$, $Y_n = I_q^\eta(g_n)$, $Z_n = X_n + Y_n$ with $E[Z_n^2] = 1$, and $E[X_n^3Y_n] \to 0$, $E[X_nY_n^3] \to 0$, then

$$E[Z_n^4] \to 3 \implies Z_n \Rightarrow \mathcal{N}(0,1)$$

The necessity of the vanishing mixed odd moments is demonstrated via a counterexample: there exist choices of $X$ and $Y$ such that $E[Z^2]=1$, $E[Z^4]=3$, but $Z$ is not Gaussian due to a nonzero third moment.

GPT-5 as a Mathematical Assistant

The experiment documents the interaction protocol with GPT-5, highlighting its strengths and limitations. GPT-5 was able to produce correct statements and proofs when guided, but made critical errors in reasoning that required expert intervention. In the Gaussian case, GPT-5 initially produced an incorrect formula for the covariance term, which was only rectified after explicit feedback. In the Poisson case, GPT-5 failed to recognize the positivity of covariance until directed to a specific result in the literature.

The role of GPT-5 was essentially that of an executor, capable of combining known ideas and producing incremental results, but not of generating fundamentally new insights or autonomously verifying correctness. The workflow resembled that of supervising a junior researcher, with the human expert responsible for steering the direction and validating the output.

Implications and Future Directions

Practical Implications

The results provide explicit quantitative bounds for central limit theorems in both Gaussian and Poisson settings, extending the applicability of the Malliavin–Stein method. These bounds are directly implementable in probabilistic analysis of functionals of stochastic processes, particularly in high-dimensional settings where explicit rates are crucial for applications in statistics, machine learning, and stochastic modeling.

Theoretical Implications

The experiment demonstrates that LLMs such as GPT-5 can assist in producing technically correct, incremental mathematical results when properly guided. However, the necessity of expert oversight for error correction and conceptual direction remains paramount. The findings suggest that while LLMs can accelerate certain aspects of mathematical research, they are not yet capable of autonomous discovery or deep verification.

Risks and Educational Impact

The authors caution that widespread use of AI in mathematical research may lead to a proliferation of incremental results, potentially diluting the scientific landscape and making it more difficult for original work to gain recognition. There is also concern regarding the impact on doctoral training, as reliance on AI-generated arguments may impede the development of independent problem-solving skills and mathematical intuition.

Future Developments

Further advances in LLM capabilities are anticipated, with potential for more sophisticated mathematical reasoning and verification. The extension of the Malliavin–Stein method to other settings, such as Gaussian subordinated fields and non-Gaussian frameworks, remains a natural direction for future research. The question of whether AI systems will eventually displace human mathematicians in creative research is unresolved and warrants ongoing scrutiny.

Conclusion

This paper provides a rigorous assessment of GPT-5’s utility in mathematical research, specifically in deriving quantitative central limit theorems via the Malliavin–Stein method. While GPT-5 can facilitate incremental progress under expert supervision, its current limitations underscore the continued necessity of human expertise in mathematical discovery and verification. The explicit quantitative results obtained herein contribute to the theory of normal approximations for functionals of stochastic processes and highlight both the promise and the challenges of integrating AI into mathematical research workflows.