How LLMs Raised the Steiner Ratio Lower Bound to 0.8559, Closing in on the Gilbert‑Pollak Conjecture

Problem definition

The Gilbert‑Pollak (Steiner ratio) conjecture states that for any finite set of points in the Euclidean plane the length of a minimum Steiner tree (SMT) is at least a fixed fraction ρ of the length of the minimum spanning tree (MST). The best proven lower bound before this work was ρ = 0.824 (established in 1985).

Human inductive approach

Mathematicians treat the conjecture as a max‑min problem. For a given tree shape w they seek a verification function F (a way of splitting the tree) such that the ratio bound holds for the remaining part after pruning. Formally the goal is max_w min_F. Historically ten different families of verification functions were explored, yielding the 0.824 bound.

LLM‑driven pipeline

The system introduces a Reward Model that scores candidate verification functions F. A divide‑and‑conquer search automatically generates many candidate F by filling structured parameters into two lemma templates:

Trapped Regular Point Lemma

4‑Point Steiner Tree Lemma

The LLM proposes the parameters as code snippets. A Mathematica engine translates each snippet into a formal lemma and checks it against a rule set, guaranteeing logical correctness.

Correctness verification

Every generated F is fed to Mathematica; only those that produce provably correct lemmas are retained, ensuring that the large‑scale search does not introduce invalid arguments.

Bottleneck detection and iterative guidance

After each reward‑model evaluation the improvement δ (e.g., 0.0001) is recorded. A subsequent run that fails to achieve the same improvement identifies a bottleneck region in the parameter space—areas not yet covered by any F. The next LLM iteration is directed to generate functions that specifically target these bottlenecks, preventing wasted effort on trivial or duplicate candidates.

Iterative loop

The process repeats the cycle Reward → Bottleneck detection → LLM‑generated lemma → Mathematica translation . Approximately ten such iterations were performed.

Results

The iterative search raised the Steiner‑ratio lower bound from 0.824 to 0.8559, leaving only ~0.01 to the conjectured optimum. Human experts verified the final lemmas. The same pipeline produced comparable results with GPT‑5, Gemini 3, and Claude 4.6, demonstrating model‑agnostic robustness.

Resources

Code and proofs are publicly available at https://github.com/teorth/optimizationproblems and https://github.com/keyisi2006/Steiner-Ratio. The paper (arXiv:2601.22365) was accepted to ICML 2026 and the new bound has been recorded in Terence Tao’s constant list (problem 43).