AI-DRIVEN MATHEMATICAL DISCOVERY
AI Mathematician as a Partner in Advancing Mathematical Discovery
- A Case Study in Homogenization Theory
Artificial intelligence (AI) has demonstrated impressive progress in mathematical reasoning, yet its integration into the practice of mathematical research remains limited. In this study, we investigate how the AI Mathematician (AIM) system can operate as a research partner rather than a mere problem solver. Focusing on a challenging problem in homogenization theory, we analyze the autonomous reasoning trajectories of AIM and incorporate targeted human interventions to structure the discovery process. Through iterative decomposition of the problem into tractable subgoals, selection of appropriate analytical methods, and validation of intermediate results, we reveal how human intuition and machine computation can complement one another. This collaborative paradigm enhances the reliability, transparency, and interpretability of the resulting proofs, while retaining human oversight for formal rigor and correctness. The approach leads to a complete and verifiable proof, and more broadly, demonstrates how systematic human-AI co-reasoning can advance the frontier of mathematical discovery.
AI may still be a flawed individual researcher today.
Yet, it can already serve as a valuable research partner—if used wisely.
Key Enterprise Impact Metrics
Our analysis reveals the substantial efficiency gains and accelerated discovery facilitated by AI-human collaboration in complex mathematical research.
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Homogenization Subproblems Addressed
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1 Introduction
In recent years, artificial intelligence (AI) has made remarkable progress in mathematical reason- ing, achieving milestones once thought to be exclusive to human intelligence. In mathematical competitions, large language models (LLMs) have demonstrated outstanding performance. For example, several LLMs have achieved scores exceeding 90 on the AIME benchmarks [1-4], which are constructed from real American Invitational Mathematics Examination (AIME) problems [5, 6], and some have even reached perfect 100-point scores [1, 3, 4]. Furthermore, Gemini with Deep Think has officially attained a gold-medal standard at the 66th International Mathematical Olympiad (IMO 2025) [7], marking a symbolic moment in the competitive mathematical performance of AI.
Beyond competition-style problem solving, progress has also emerged in AI-assisted mathematical discovery. Romera-Paredes et al. [10] and Novikov et al. [11] demonstrate that LLMs can facilitate genuine mathematical discovery through guided program search. Similarly, GPT-5-Thinking [1] has been credited with helping renowned researchers resolve a challenging quantum computing problem [12]. Taken together, these developments suggest that AI is beginning to move beyond solving predefined problems toward a more engaged role in mathematical exploration.
2 Preliminaries
2.1 The Homogenization Problem
The mathematical research problem we investigate in this work is an instance of a Stokes-Lamé transmission system with a vanishing fluid inclusion, analyzed in the homogenization regime ε → 0. This problem will be referred to as the Homogenization Problem. It involves complex domains, Lamé pairs, elastostatic systems, and conormal derivatives, all defined with intricate mathematical rigor.
2.2 AIM: An AI Mathematician System
AIM is a multi-agent framework built upon large language models (LLMs) for conducting mathematical research [16]. Its design addresses two fundamental challenges: the intrinsic complexity of mathematical theory and the rigor of reasoning processes. AIM incorporates an exploration and memory mechanism that decomposes complex problems into multi-step explorations, generates intermediate conjectures, and iteratively reuses verified lemmas to refine reasoning. AIM employs Pessimistic Rational Verification (PRV), where multiple independent verifiers evaluate intermediate proofs. AIM consists of three core agents—the explorer, verifier, and optimizer—along with a memory module, operating iteratively to generate, verify, and refine proofs.
AIM has been evaluated on four mathematical research problems, including the homogenization problem examined here. While AIM made notable progress, it did not fully resolve the homogenization problem autonomously, highlighting the need for human-AI collaboration.
3 Overview
Based on an AI-human collaborative paradigm, we successfully completed the proof of the homogenization problem. The main conclusion for the homogenization problem is summarized as follows: We derived the homogenization equation in limit case and estimated the error between the original solution and the homogenized solution as: ||Uɛ – Ulim||H1(Ω) ≤ Cεα for some α ∈ (0, 1), specifically proven with α = 1/2.
The proof was achieved through a staged process, dividing the original problem into six subproblems:
- Two-Scale Expansion: Manual derivation due to AIM's errors in complex symbolic reasoning.
- Cell Problem and Homogenization Equation: Manual derivation as AIM lacked sufficient understanding of geometric structures for correct results.
- Existence and Uniqueness: AIM successfully applied the correct theorem.
- Ellipticity of Operator: AIM provided a proof with a high degree of completion.
- Error Estimation and Control: AIM presented the correct proof approach, which required human adjustments for completion.
- Regularity of Cell Problem: AIM provided a complete proof after multiple human-AI interactions and theoretical guidance.
4 Modes of Human-AI Interaction
In pursuing the complete proof of the homogenization problem, we found that effective human-AI collaboration plays a crucial role. Based on extensive experimentation, we summarize five representative modes of interaction that proved particularly effective:
- Direct Prompting: Guides the agent toward promising proof directions and optimizes its reasoning path through targeted yet concise instructions.
- Theory-Coordinated Application: Agent is provided with a coherent body of mathematical theory, enabling it to derive related results within the theoretical framework.
- Interactive Iterative Refinement: Follows a “Feedback - Revision - Re-reasoning" cycle, where human experts and AIM collaboratively refine proofs.
- Applicability Boundary and Exclusion Domain: Assigns challenging tasks (e.g., decomposing proof strategies) to human experts, reserving AIM for reliable domains.
- Auxiliary Optimization Strategies: Enhance correctness and robustness by iteratively providing additional contextual information and optimizing tool selection.
Enterprise Process Flow: Human-AI Interaction Modes
Direct Prompting
→
Theory-Coordinated Application
→
Interactive Iterative Refinement
→
Applicability Boundary and Exclusion Domain
→
Auxiliary Optimization Strategies
| Steps | Hardness | Current Status |
|---|---|---|
| Two-Scale Expansion | Easy | Correct expansion and derivation were completed manually. |
| Cell Problem and Homogenization Equation | Medium | The cell problem and homogenization equation were manually constructed and derived. |
| Existence and Uniqueness | Hard | AIM applied the correct theorem to get this conclusion. |
| Ellipticity of Operator | Medium | AIM provided a proof with a high level of completeness. |
| Error Estimation and Control | Hard | AIM presented the correct proof approach, which after human adjustments, led to a complete proof process. |
| Regularity of Cell Problem | Hard | By AIM, complete proof process was provided. |
Case Study: Human-AI Collaboration in Error Estimation
The Error Estimation and Control subproblem was identified as the most complex, requiring rigorous analytical and detailed derivation. Initially, AIM produced a seemingly convincing proof, but human examination revealed it relied on an unjustified property of cell problem equations.
A human expert conjectured the property should hold and prompted AIM to prove it. AIM initially failed, prompting deeper analysis which revealed the intrinsic difficulty. The expert then suggested mathematical tools like Difference Quotient, Galerkin Method, and Schauder Theory. Notably, the expert was uncertain of their applicability.
Eventually, AIM successfully proved the property using Schauder Theory. This experience vividly illustrates that while AIM may still be a flawed individual researcher, it can already serve as a valuable research partner—if used wisely, demonstrating how human intuition and machine computation can complement one another.
Key Finding: This collaborative process enhances the reliability, transparency, and interpretability of complex proofs, retaining human oversight for formal rigor.
α = 1/2
Proven Error Estimation Exponent for ||Uɛ – Ulim||H1(Ω) ≤ Cεα
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Your AI Mathematical Discovery Roadmap
A structured approach to integrating AI into your mathematical research pipeline for optimized outcomes.
Phase 01: Initial Assessment & Pilot
Evaluate current research workflows, identify suitable problems for AI-human collaboration, and conduct a small-scale pilot project leveraging AI-assisted tools.
Phase 02: Framework Integration & Training
Integrate AIM-like multi-agent frameworks, train researchers on effective human-AI interaction modes (direct prompting, iterative refinement), and refine theoretical knowledge packages.
Phase 03: Scaled Deployment & Continuous Improvement
Expand AI assistance to broader research areas, establish feedback loops for model optimization, and continuously adapt AI strategies based on performance and new mathematical discoveries.
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