Long-Horizon AI Research for Grothendieck Constant: A Case Study in Human-AI Mathematical Collaboration

2026-08-11

AI Summary

This work addresses the problem of refining the known upper and lower bounds of the Grothendieck constant \( K_G \), which quantifies the approximation hardness between combinatorial optimization problems and their continuous relaxations. We propose a hybrid AI research framework that integrates large language models, symbolic reasoning, numerical optimization, and human feedback. This approach enables, for the first time, an AI system to generate novel mathematical insights that are recognized by domain experts and lead to rigorously verifiable theoretical bounds. Through close human–AI collaboration, we improve the lower bound of \( K_G \) to \( 6\pi/11 \) and tighten the upper bound to \( \pi/(2\log(1+\sqrt{2})) - 10^{-4} \), significantly surpassing the best previously established results.

Abstract

AI agents are increasingly used in mathematics research, but it is often unclear how to use them effectively. Towards this, we present an extensive case study of how AI was used to improve bounds on the Grothendieck constant $K_G$, which captures the hardness between combinatorial problems and their continuous relaxations. Specifically, while the precise value of $K_G$ is not known, we recently tightened the best known bounds to \[ \frac{6π}{11} \;\le\; K_G \;\le\; \fracπ{2\log(1+\sqrt2)} - 10^{-4}. \] Crucially, these improvements were achieved using an AI research system that could arrive at insights deemed novel by domain experts. We give a detailed discussion of our experience using AI for mathematics research, particularly touching upon its strengths and weaknesses, as well as our experience with creating ideal conditions for AI to arrive at breakthrough insights.

Problem

• Grothendieck constant
• mathematical collaboration
• long-horizon AI research
• combinatorial optimization
• continuous relaxation

Innovation

• Grothendieck constant
• human-AI collaboration
• mathematical discovery
• AI research system
• long-horizon reasoning