用大模型生成可验证几何引理，将斯坦纳比下界提升至0.8559
Towards Solving the Gilbert-Pollak Conjecture via Large Language Models
- 让大模型生成可执行的几何引理，自动构造验证函数
- 通过迭代优化，将斯坦纳比下界提高到0.8559（原为0.824）
- 证明了大模型在高阶数学研究中的可行性，适合数学与AI交叉研究者
Gilbert-Pollak猜想（又称斯坦纳比猜想）指出：对于欧几里得平面上任意有限点集，斯坦纳最小树长度至少为欧几里得最小生成树长度的√3/2 ≈ 0.866倍（即斯坦纳比）。20世纪80年代一系列改进后，下界已达到0.824，此后三十年未见突破。尽管大语言模型在竞赛级数学问题上表现优异，但其对开放性研究问题的应用仍待探索。本文提出一种新型AI系统，通过指令大模型生成规则约束的几何引理，并将其转化为可执行代码，构建一系列可理论认证的验证函数，用于计算斯坦纳比的下界。通过反思驱动的引理迭代优化，系统获得新的严格下界0.8559。整个过程仅需数千次大模型调用，证明了大模型在高级数学研究中的巨大潜力。
原文摘要 · Abstract (English)
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.
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