Iteris: Agentic Research Loops for Computational Mathematics

Mathematical research has long been a domain dominated by human intuition, painstaking calculations, and iterative trial-and-error. Despite advances in computational tools, automating the discovery process—especially for open problems—remains a formidable challenge. Traditional algorithms excel at solving well-defined tasks but falter when faced with complex, multi-faceted questions involving performance analysis, counterexample construction, and proof verification. Recognizing this gap, the authors introduce Iteris, a pioneering AI system designed to participate actively in mathematical research workflows.

At its core, Iteris employs an explore-plan-execute loop, orchestrating multiple specialized agents that collaboratively navigate the intricate landscape of open problems. The exploration agent probes the current research state by reading project files, retrieving relevant facts, and drafting potential research routes. It signals promising directions without making definitive decisions, thus maintaining flexibility. The planning agent then analyzes the global context—problem statements, prior results, route status—and selects a structured set of tasks for the next iteration, including numerical experiments, proof drafts, or counterexamples.

The execution agents, categorized into foundation, experiment, proof, and review types, perform targeted research actions based on the task specifications. They generate structured output files that update the project state, ensuring traceability and reproducibility. This modular design allows the system to handle diverse research modes while maintaining a clear separation of concerns. The entire process is underpinned by a file-based communication protocol, which facilitates long-term tracking and expert validation.

Applying Iteris to two challenging open problems from the Simons Workshop collection, the authors demonstrate its effectiveness. In the first case, the system derived a fixed-parameter phase diagram comparing the asymptotic performance of conjugate gradient (CG) and randomized coordinate descent (RCD) on power-law spectra. The analysis revealed that for p<1/3, CG outperforms RCD, whereas for p>1/3, RCD is superior, with performance differences exceeding 20%. These results were validated through extensive numerical simulations on large random matrices, confirming the theoretical predictions.

In the second case, Iteris constructed low-coherence counterexamples showing that QR factorization with column pivoting (QRCP) can fail to select well-conditioned submatrices, even under favorable conditions. The constructed matrices, with orthonormal rows, demonstrated failure probabilities above 80%, with error amplification factors reaching 15 times. This finding challenges existing assumptions about QRCP’s reliability and provides new insights into its limitations.

These case studies exemplify how an agentic AI system can actively contribute to mathematical research, generating valuable artifacts that, after expert review, lead to rigorous conclusions. The approach not only accelerates discovery but also enhances understanding of complex phenomena in spectral analysis and matrix conditioning. Looking ahead, the authors plan to incorporate reinforcement learning and meta-learning techniques to further improve the system’s autonomy, aiming for fully automated theorem discovery and validation.

Overall, Iteris represents a significant step toward autonomous mathematical research, bridging numerical experimentation, symbolic reasoning, and expert validation. Its flexible, modular architecture and demonstrated success on challenging problems suggest a promising future for AI-assisted science, with potential applications spanning algorithm design, scientific computing, and theoretical mathematics. This work paves the way for a new era where AI systems can participate meaningfully in the frontiers of mathematical knowledge, transforming how fundamental research is conducted and validated.