Title: Iteris: Agentic Research Loops for Computational Mathematics

Original: arXiv:2606.02484v1 Announce Type: new Abstract: Recent advances in large language models and agentic AI systems have enabled significant progress in mathematical discovery, from solving competition problems to tackling research-level conjectures. However, open problems in computational mathematics have received comparatively less attention: research in this area often requires not only proofs but also numerical experimentation, adversarial constructions, and algorithm design. In this paper, we introduce an agentic research system, Iteris, designed for open problems in computational mathematics. We apply Iteris to two open problems from a recent Simons Workshop collection (arXiv:2602.05394). In these case studies, Iteris generated numerical evidence, constructions, and proof drafts that led, after expert review and correction, to verified results. The first result is a phase diagram for the asymptotic comparison between conjugate gradient and randomized coordinate descent on power-law spectra; the second is a counterexample showing that QR factorization with column pivoting can fail to select well-conditioned submatrices even under low coherence. These case studies suggest that agentic AI systems can participate meaningfully in research workflows for open problems in computational mathematics, while human validation remains essential.

Rewritten:

Title: Iteris: Agentic Research Loops for Computational Mathematics

Abstract: While large language models and agentic AI frameworks have recently driven substantial breakthroughs in mathematical discovery—ranging from competition-level solutions to complex research conjectures—open challenges within computational mathematics have seen relatively limited focus. Investigating these issues typically demands a multifaceted approach that extends beyond theoretical proofs to include algorithmic design, adversarial construction, and numerical experimentation. To address this gap, we present Iteris, an agentic research platform specifically tailored for tackling open problems in computational mathematics. We demonstrate the system’s capabilities through two case studies involving unresolved questions from a recent Simons Workshop collection (arXiv:2602.05394). In both instances, Iteris produced numerical data, structural constructions, and preliminary proof drafts. Following expert scrutiny and necessary corrections, these outputs resulted in confirmed findings. The first finding establishes a phase diagram comparing the asymptotic performance of conjugate gradient methods against randomized coordinate descent on power-law spectra. The second delivers a counterexample demonstrating that QR factorization with column pivoting may fail to identify well-conditioned submatrices, even when coherence levels are low. Collectively, these examples indicate that while human oversight and validation remain critical, agentic AI systems can play a substantive role in the research workflows associated with open problems in computational mathematics.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC