Towards Solving the Gilbert-Pollak Conjecture via Large Language Models

2026-01-29

AI Summary

This work addresses the long-standing stagnation in the Gilbert–Pollak conjecture (also known as the Steiner ratio conjecture) by introducing a novel paradigm that integrates large language models with formal verification. The approach employs constraint-guided generation of geometric lemmas, automatically synthesizes executable verification code, and iteratively refines lemma structures through a reflection mechanism. For the first time, large language models are leveraged to produce certifiable mathematical lemmas aimed at resolving a prominent open problem in theoretical mathematics. Using only a few thousand model invocations, the method derives a new lower bound of 0.8559 for the Steiner ratio—substantially surpassing the previous best-known bound of 0.824, which had remained unimproved for over three decades—and thereby achieves a significant leap from competition-style problem solving to rigorous theoretical research.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Large language models for search
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

The Gilbert-Pollak Conjecture \citep{gilbert1968steiner}, also known as the Steiner Ratio Conjecture, states that for any finite point set in the Euclidean plane, the Steiner minimum tree has length at least $\sqrt{3}/2 \approx 0.866$ times that of the Euclidean minimum spanning tree (the Steiner ratio). A sequence of improvements through the 1980s culminated in a lower bound of $0.824$, with no substantial progress reported over the past three decades. Recent advances in LLMs have demonstrated strong performance on contest-level mathematical problems, yet their potential for addressing open, research-level questions remains largely unexplored. In this work, we present a novel AI system for obtaining tighter lower bounds on the Steiner ratio. Rather than directly prompting LLMs to solve the conjecture, we task them with generating rule-constrained geometric lemmas implemented as executable code. These lemmas are then used to construct a collection of specialized functions, which we call verification functions, that yield theoretically certified lower bounds of the Steiner ratio. Through progressive lemma refinement driven by reflection, the system establishes a new certified lower bound of 0.8559 for the Steiner ratio. The entire research effort involves only thousands of LLM calls, demonstrating the strong potential of LLM-based systems for advanced mathematical research.

Problem

• Gilbert-Pollak Conjecture
• Steiner Ratio
• Steiner Minimum Tree
• Euclidean Minimum Spanning Tree
• Lower Bound

Innovation

• Large Language Models
• Steiner Ratio
• Automated Theorem Proving
• Verification Functions
• Geometric Lemma Generation