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arxiv: 2601.22365 · v2 · pith:KN5Q6RIGnew · submitted 2026-01-29 · 💻 cs.DM · cs.LG

Towards Solving the Gilbert-Pollak Conjecture via Large Language Models

Yisi Ke , Tianyu Huang , Yankai Shu , Di He , Jingchu Gai , Liwei Wang This is my paper

Pith reviewed 2026-05-22 11:45 UTC · model grok-4.3

classification 💻 cs.DM cs.LG
keywords Steiner ratioGilbert-Pollak conjecturelarge language modelsgeometric lemmasverification functionslower boundEuclidean Steiner treeminimum spanning tree
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The pith

Large language models generate geometric lemmas to certify a new lower bound of 0.8559 on the Steiner ratio.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The Gilbert-Pollak conjecture states that the Steiner minimum tree on any finite set of points in the Euclidean plane has length at least roughly 0.866 times the length of the minimum spanning tree. Prior mathematical work had only reached a certified lower bound of 0.824 with no major advances in thirty years. This paper instead asks large language models to produce rule-constrained geometric lemmas that are implemented directly as executable verification functions. Progressive refinement of those lemmas through repeated reflection steps allows the system to assemble a collection of functions that together prove the higher certified bound of 0.8559. The approach illustrates how language models can contribute to open research problems by systematically generating and verifying pieces of a geometric argument rather than attempting to solve the full conjecture in one step.

Core claim

The paper shows that an LLM-driven process of generating rule-constrained geometric lemmas, encoding them as verification functions, and iteratively refining the lemmas via reflection produces a collection of functions whose combined lower bounds rigorously establish that the Steiner ratio is at least 0.8559.

What carries the argument

Verification functions built from LLM-generated rule-constrained geometric lemmas that together compute certified lower bounds on the Steiner ratio.

If this is right

  • The proven lower bound on the Steiner ratio rises from 0.824 to 0.8559.
  • The gap between the proven bound and the conjectured value of approximately 0.866 narrows further.
  • The same lemma-generation and reflection loop can be continued to seek still tighter certified bounds.
  • Comparable LLM systems could be applied to other long-standing ratio conjectures in geometric optimization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the refinement process can be scaled without introducing errors, repeated application might eventually reach or surpass the conjectured value of 0.866.
  • Cross-checking the generated lemmas against known extremal point sets that achieve ratios near 0.866 could help detect hidden flaws before they propagate.
  • The method of turning LLM output into executable verification code may generalize to other problems where lower bounds are obtained by exhaustive case analysis over geometric configurations.

Load-bearing premise

The geometric lemmas produced by the large language model are free of subtle mathematical errors and the resulting verification functions truly certify rigorous lower bounds rather than heuristic estimates.

What would settle it

A finite point configuration whose Steiner minimum tree to minimum spanning tree length ratio is strictly less than 0.8559, or an explicit flaw in one of the verification functions that invalidates its claimed bound.

Figures

Figures reproduced from arXiv: 2601.22365 by Di He, Jingchu Gai, Liwei Wang, Tianyu Huang, Yankai Shu, Yisi Ke.

Figure 1
Figure 1. Figure 1: Progress on Steiner Ratio Lower Bound in the Euclidean plane, the Steiner Minimal Tree (SMT) is the shortest network that connects all points, allowing the introduction of additional auxiliary nodes, known as Steiner points. The Steiner ratio is the infimum of the ratio between the length of the SMT and that of the Minimum Spanning Tree (MST). Gilbert & Pollak (1968) conjectured that this ra￾tio is lower-b… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of our proposed LLM Math Research System for the Gilbert-Pollak Conjecture [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of minimum spanning tree (left) and Steiner [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustra￾tion of Pruning Pro￾cess. We try to prune V ∗ = {B}. Black dashed lines are S −. Black solid lines are S +. Green line is t ∗ . In this pruning process, the decrease in the Steiner tree length is given by ∆LS = LS− − LS+ , while the in￾crease in the spanning tree length is given by ∆Lm = Lt ∗ . Here, LX de￾notes the total edge length of graph X. The requirement for the induc￾tion to hold is that ∆… view at source ↗
Figure 5
Figure 5. Figure 5: Branch-and￾Bound on 2D Plane To make verification tractable, we employ a Branch-and￾Bound strategy that recur￾sively partitions the continu￾ous parameter space W into smaller hyperrectangles, as il￾lustrated in [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The Steiner Spiral Chain structure Due to the geometric constraints established in Lemma 11, every internal angle ∠Ai−1AiAi+1 along this chain is exactly 120◦ . We abstract this configuration as the Steiner Spiral Chain, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (AB)-(CD) type Steiner tree. We adopt the Valid 4-Point Steiner Tree Theorem from Du et al. (1987) to determine the existence of the (AB)- (CD) topology. The theo￾rem’s details are provided in Appendix A.2 as Theorem 14. The LLM agent’s task is to translate the geometric conditions of the theorem into al￾gebraic constraints on the parameter space w, simplifying them to satisfy the orthogonally convex prope… view at source ↗
Figure 8
Figure 8. Figure 8: Interaction Between Reward Model & LLM Agent Verification of the Proof Correctness. We ensure the re￾liability of our results through a dual-verification strategy. First, the LLM acts solely as a reasoning engine, offloading all validation tasks to Mathematica to perform exact com￾putation. This significantly reduces hallucinations in the generated lemmas. Second, we performed independent man￾ual verificat… view at source ↗
Figure 9
Figure 9. Figure 9: Illustration of the ratio change throughout the iteration [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Local substructures of a Steiner Minimal Tree. A.1.2. CORE LEMMAS In this section, We will provide a more detailed description of how induction works. To complete the induction step, we utilize a reduction argument introduced by Du & Hwang (1983). 13 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Sample of Subtree: D is regular Recall the subtree of a Steiner Minimal Tree as shown in [PITH_FULL_IMAGE:figures/full_fig_p035_11.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents an LLM-based system that generates rule-constrained geometric lemmas as executable code; these are iteratively refined via reflection to produce verification functions claimed to yield theoretically certified lower bounds on the Steiner ratio. The central result is a new lower bound of 0.8559, improving on the 0.824 bound established in the 1980s.

Significance. If the certification procedure is sound, the result would constitute the first substantial improvement on the Steiner ratio lower bound in over thirty years and would provide concrete evidence that LLM-driven lemma generation can address long-standing open problems in geometric optimization.

major comments (2)
  1. [Abstract] Abstract: the assertion that the 0.8559 bound is 'theoretically certified' is not supported by any description of the verification procedure, machine-checked correspondence between lemmas and code, or safeguards against subtle geometric errors (e.g., degenerate point configurations or angle-bound violations). This gap directly undermines the central claim.
  2. [Methods / Verification functions] The construction of verification functions from LLM-generated lemmas (described in the methods section) lacks an independent soundness argument; reflection alone does not guarantee detection of logically or geometrically invalid statements, leaving open the possibility that the reported bound is an artifact of an unsound implementation rather than a rigorous lower bound.
minor comments (2)
  1. [Section 3] Provide explicit pseudocode or a small worked example showing how a single generated lemma is translated into an executable verification function and how its output is aggregated into the final 0.8559 bound.
  2. [Experimental setup] Clarify the precise stopping criterion for the reflection loop and report the total number of LLM calls broken down by generation versus refinement phases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and insightful comments on our manuscript. We appreciate the opportunity to clarify the verification aspects of our LLM-based approach for bounding the Steiner ratio. Below, we provide point-by-point responses to the major comments and outline the revisions we will make to address these concerns.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the 0.8559 bound is 'theoretically certified' is not supported by any description of the verification procedure, machine-checked correspondence between lemmas and code, or safeguards against subtle geometric errors (e.g., degenerate point configurations or angle-bound violations). This gap directly undermines the central claim.

    Authors: We agree that the abstract would benefit from a more explicit reference to the verification procedure to support the claim of a theoretically certified bound. In the revised manuscript, we will modify the abstract to include a concise description of how the verification functions are derived from the lemmas and the built-in checks for geometric validity. Additionally, we will add a new paragraph in the introduction or methods section that outlines the safeguards against degenerate configurations and angle violations, ensuring the certification is more transparently justified. revision: yes

  2. Referee: [Methods / Verification functions] The construction of verification functions from LLM-generated lemmas (described in the methods section) lacks an independent soundness argument; reflection alone does not guarantee detection of logically or geometrically invalid statements, leaving open the possibility that the reported bound is an artifact of an unsound implementation rather than a rigorous lower bound.

    Authors: The referee raises a valid point regarding the need for an independent soundness argument. While our current description emphasizes the rule-constrained generation and the reflection process for refinement, we recognize that this may not fully address potential subtle errors. We will revise the methods section to include a dedicated soundness argument, detailing how the executable code enforces the geometric lemmas, the iterative reflection detects inconsistencies, and additional validation steps (such as testing on known configurations) provide further assurance against invalid statements. This will strengthen the rigor of the certification procedure. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the 0.8559 lower bound by having LLMs generate rule-constrained geometric lemmas as executable code, then constructing verification functions from those lemmas and refining them via reflection. This process produces new functions that are applied to finite point sets to certify the bound; it does not reduce the target ratio to a quantity already defined in terms of itself, fit parameters to data and relabel the fit as a prediction, or rely on a load-bearing self-citation whose content is itself unverified. The cited Gilbert-Pollak conjecture is an external statement, and no ansatz or uniqueness theorem is imported from prior work by the same authors. The derivation chain is therefore self-contained against the external benchmark of the Steiner ratio definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim depends on the unverified correctness of LLM-generated geometric lemmas and the assumption that the assembled verification functions produce rigorous certificates.

axioms (1)
  • domain assumption LLM-generated geometric lemmas are correct and can be safely compiled into executable verification functions that yield rigorous lower bounds.
    Invoked when the system uses the lemmas to construct certified bounds; no independent check is described in the abstract.
invented entities (1)
  • verification functions no independent evidence
    purpose: Combine generated lemmas to produce theoretically certified lower bounds on the Steiner ratio.
    New construct introduced by the paper to turn lemmas into bounds.

pith-pipeline@v0.9.0 · 5776 in / 1327 out tokens · 53417 ms · 2026-05-22T11:45:09.066831+00:00 · methodology

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

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    Theorem 6 (Vertex-Maximization Property)is the foundation of our entire verification strategy. It fundamentally addresses the challenge of computational tractability by reducing the verification problem over a continuous parameter domain to a check on its discrete vertices1 2.Pathways for Constructing New Verification Functions:The efficacy of Theorem 6 r...

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    The validity of the implication C=⇒Trapped(E) is independent of the topology of D or the value of f

    Trapped Regular Point Lemmas (Implicit E):These lemmas rely exclusively on the local geometry of the chain starting at A. The validity of the implication C=⇒Trapped(E) is independent of the topology of D or the value of f. Therefore, these lemmas areuniversally transplantedto all four cases

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    • Cases 2 & 4 ( f≥d ):As per our reduction strategy, the verification functions for the unbounded domain f∈(d,+∞)are mapped to the boundaryf=d

    4-Point Steiner Tree Lemmas:These lemmas (e.g., verifying the existence of (AD)-(QR) trees) were derived and symbolically verified under the strict assumptionf=d. • Cases 2 & 4 ( f≥d ):As per our reduction strategy, the verification functions for the unbounded domain f∈(d,+∞)are mapped to the boundaryf=d. Thus, these lemmas arevalid and applied. • Cases 1...

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    • ⃗QR× ⃗RA≤0: Verified to hold whenc+e≥1, which is explicitly enforced by inequalityI 1

    Convexity of QuadrilateralRADQ: • ⃗RA× ⃗AD≤0, ⃗AD× ⃗DQ≤0, and ⃗DQ× ⃗QR≤0: Verified unconditionally. • ⃗QR× ⃗RA≤0: Verified to hold whenc+e≥1, which is explicitly enforced by inequalityI 1

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    •∠U QD≤120 ◦: The condition I3 implies that ⃗QU· ⃗QD≥0 (i.e., the angle is ≤90 ◦), which sufficiently implies ≤120 ◦

    Simpson Point Angles: 25 Towards Solving the Gilbert-Pollak Conjecture via Large Language Models •∠U DQ≤120 ◦: The condition I2 implies that ⃗DU· ⃗DQ≥0 (i.e., the angle is ≤90 ◦), which sufficiently implies ≤120 ◦. •∠U QD≤120 ◦: The condition I3 implies that ⃗QU· ⃗QD≥0 (i.e., the angle is ≤90 ◦), which sufficiently implies ≤120 ◦. •∠V RA≤120 ◦: This is th...

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    PART2: ORTHOGONALCONVEXITYCHECK We verify that each defining inequality inSis monotonic with respect to the coordinate axes within the feasible region

    Intersection Angle: • The condition that the angle between diagonalsRDandAQis≤120 ◦ corresponds directly to inequalityI 6. PART2: ORTHOGONALCONVEXITYCHECK We verify that each defining inequality inSis monotonic with respect to the coordinate axes within the feasible region. Linear Inequalities ( I1, I2, I3, I4): These are linear combinations of variables....

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    • ⃗QR× ⃗RA≤0: Verified (Requiresc≥1)

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    •∠U RQ≤120 ◦: Verified (Requirese≥1)

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    Intersection Angle(via transformed inequality): • Angle between diagonalsAQandDR≤120 ◦: Verified (Requiresc≥1)

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    Since these are intersections of convex sets, the resulting domain is convex

    Orthogonal Convexity: The validity region is defined by the intersection of linear inequalities. Since these are intersections of convex sets, the resulting domain is convex. Step 6 Lemma 28(Step 6).If e <(2/ √ 3−1)·c , then there exists a regular point E such that |AE| ≤ max{|AY|,|AZ|,|AR|}, and|DE| ≤max{|DY|,|DZ|,|DR|, p 3/4·(c+e) 2 + 3/2·(c+e)d+d 2}. P...

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    Convexity of QuadrilateralBDQR(via cross products): • ⃗BD× ⃗DQ≤0 , ⃗DQ× ⃗QR≤0 , ⃗QR× ⃗RB≤0 , and ⃗RB× ⃗BD≤0 : All verified to hold unconditionally within the geometric domain

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    •∠U QR≤120 ◦: Verified (Requiresd≥b)

    Simpson Point Angles(via transformed inequality): •∠U RQ≤120 ◦ and∠V BD≤120 ◦: Verified (Unconditional). •∠U QR≤120 ◦: Verified (Requiresd≥b). •∠V DB≤120 ◦: Verified (Requiresd≥b)

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    Intersection Angle(via transformed inequality): • Angle between diagonalsBQandDR≤120 ◦: Verified (Unconditional)

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    Step 8 Lemma 30(Step 8).The region defined by the system of inequalities S below is an orthogonally convex set

    Orthogonal Convexity: The validity region is defined by the linear inequality d≥b , which is automatically convex. Step 8 Lemma 30(Step 8).The region defined by the system of inequalities S below is an orthogonally convex set. Further- more, for anywin this region, the(AD)-(QR)type Steiner tree is valid. S:    I1 :d(c+e−1) +e(s+c)≥0 I2 :e≥ ...

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    • At vertexR, the condition ⃗QR× ⃗RA≤0is algebraically equivalent to inequalityI 1

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    •∠U RQ≤120 ◦: Solving the algebraic inequality corresponding to this angle condition yields exactly inequalityI 2

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    This corresponds to the condition ⃗AQ· ⃗DR≥0

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    We initializeQwith the fundamental orthant splits to ensure the initial partition covers the entire unbounded spaceW

    Initialization of Region Partitioning.We maintain the queue Q of hyperrectangles that partition the search space. We initializeQwith the fundamental orthant splits to ensure the initial partition covers the entire unbounded spaceW

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    This ensures that the maximum value is attained at the finite lower bound

    Monotonicity Check for Unbounded Regions.(Line 7) To handle regions extending to +∞, we verify that the verification function f is monotonically non-increasing along the unbounded dimension. This ensures that the maximum value is attained at the finite lower bound. The implementation details of the monotonicity check are provided in Appendix Section B.4

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    By verifying f(w)≤0 at all vertices, we mathematically guarantee that the inequality holds for every point w within the region

    Feasibility Verification for Bounded Regions.(Line 8) For the finite vertices, we invoke the Vertex Maximum Property. By verifying f(w)≤0 at all vertices, we mathematically guarantee that the inequality holds for every point w within the region

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    logical length

    Refinement via Subdivision.(Lines 10-13) Unverified regions are subdivided along the dimension with the longest edge. For unbounded dimensions [l,+∞) , we define a “logical length” of1/l and apply ageometric doubling strategy, partitioning the interval into a bounded segment [l,2l] and a remaining unbounded tail [2l,+∞) . This recursive process continues ...

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    Boundary Dominance:The verification bottlenecks (regions where the feasible ρ is minimized) predominantly concentrate at the boundaryf=d

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    In this limit, the structure topologically degenerates to the case whereDis a regular point

    Topological Degeneracy:In cases where D is a Steiner point, the most critical constraints typically occur when the terminal edges (|DU| and |DV| ) approach zero length. In this limit, the structure topologically degenerates to the case whereDis a regular point. Consequently, Case 2 represents the most challenging and representative slice of the parameter ...

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    •A 2 =A 1 +a 2(cos(−60◦),sin(−60 ◦)) •A 3 =A 2 +a 3(cos(−120◦),sin(−120 ◦)) •

    Recursive Definition:For k≥2 , point Ak is derived from Ak−1 by moving distance ak at an inclination angle of−60×(k−1)degrees. •A 2 =A 1 +a 2(cos(−60◦),sin(−60 ◦)) •A 3 =A 2 +a 3(cos(−120◦),sin(−120 ◦)) •. . .and so on. 3.Point D:Dis connected toA 2 with lengthdat angle0 ◦. •D=A 2 + (d,0) PHASE 2: The Logical Implication Problem You are working with two t...

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    •Meaning:These are the constraints that the sequencea i always satisfies

    Properties (A) - Steiner Minimality: •Definition:For any0≤u < s≤v≤n, the Euclidean distance satisfies|A uAv| ≥a s. •Meaning:These are the constraints that the sequencea i always satisfies

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    • The Trap:IF H lies on the ray AvAv+1 ANDthe distance |AuH|< a s,THENthe entire tail of the chain (An) is trapped inside the polygon defined by verticesA u, Au+1,

    Properties (B) - The Polygon Trap: • Definition:For 0≤u < s≤v < n , let H be the foot of the perpendicular fromAu to the line containing AvAv+1. • The Trap:IF H lies on the ray AvAv+1 ANDthe distance |AuH|< a s,THENthe entire tail of the chain (An) is trapped inside the polygon defined by verticesA u, Au+1, . . . , Av, H. •Meaning:These are your goals. Pr...

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    Identify which ones are likely to hold (i.e., where the inequality is satisfied or close to satisfied)

    Analyze the Box:Check the provided Box values against the B conditions in the list. Identify which ones are likely to hold (i.e., where the inequality is satisfied or close to satisfied)

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    Select Constraints:Select a small subset (e.g., 1 B property and 0–3 A properties) that seem most relevant to the Box

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    worst case

    Formulate Logic:You want to prove that the A constraints prevent a5, a6 from violating the B goal. It is OK thatBgoals do not involvea 5, a6, in which case you may not needAproperties. IMPORTANT Tip:You should first check B properties with v≤4 and identify if there are any that already hold. In these cases, you do not need to derive upper bound fora 5. Ti...

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    Quadratic:Inequalities in the form Ax2 +Bx+C≥0 (or ≤0 ), subject to monotonicity rules explained in the prompt

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    Functional Boundaries:Inequalities in the form e > f(b, c, s, d) or e < f(b, c, s, d) , where monotonic- ity offmust be verified. Coverage Requirement: Your condition P must be general (not hardcoded numbers), but it must strictly cover the following ”Test Box”:{Bottleneck} Important:You MUST verify the APSC Property in theENTIREdomain, not in the Test Bo...

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    Detailed Derivation:Show how you calculated coordinates and how you derived the relaxed inequalities

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    APSC Proof:Provide a logical argument explaining why your condition P satisfies the APSC property, specifically addressing the Linearity/Monotonicity requirements for each part of your condition

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    4.C++ Implementation: Provide a C++ code block containing these exact functions: // Returns true if your condition $P$ is met

    Mathematica Output:Paste the code and the output confirming the False result for the implication checks. 4.C++ Implementation: Provide a C++ code block containing these exact functions: // Returns true if your condition $P$ is met. bool steiner_cond(double b, double c, double d, double s, double e) { // Your condition here } // Returns the total length of...

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