Discrete and computational geometry

Gilbert–Pollak Conjecture

Collaboration beta

For a finite set of points in the plane, compare the shortest network that may add Steiner junctions with the ordinary minimum spanning tree. The conjecture says the first length is always at least √3/2 of the second. This packet reports substantial local reductions and certified regions, but no complete proof.

S (V) \geq \frac{\sqrt{3}}{2} M (V)

Listed inOptimization Problems in Mathematics, problem 43

Known results and sources

An equilateral triple attains the proposed √3/2 ratio; this equality example does not settle the universal conjecture.

Research problem

Exact mathematical statement

For every finite terminal set $V\subset\mathbb R^2$ , let $S(V)$ be the Euclidean Steiner minimum-tree length and let $M(V)$ be the Euclidean minimum-spanning-tree length. The open conjecture asks whether

S(V)\ge \frac{\sqrt3}{2}M(V)

holds for every such $V$ . The governing source explicitly says that no theorem in the source is a full proof of this inequality.

Problem infographic

Problem at a glance

Three panels compare a two-edge spanning tree and a three-arm Steiner tree for the same equilateral terminals, then place seven arbitrary terminals beside the still-open universal ratio question. — The equilateral triple attains S(V)/M(V)=√3/2; the open conjecture asks whether every finite planar terminal set satisfies this lower bound.

Current mathematical picture

Where work on Gilbert–Pollak Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureOrientation-free Q-shallow compression

At Z exactly one of Q and R is shallow, and the proof using at most four descendants of Q requires O_Q; the current work withdraws the orientation-free formulation. Orientation-specific shallow-block compression remains live in O_Q, and a reflected graph-legal mechanism remains live in O_R.

Route status · Narrowed route

Main reductionLocal nonpositive-debt reduction

Using imported deepest-cherry, splitting-induction, and full-component interfaces, the source reduces the target to finding a legal nonpositive-debt split in each feasible normalized local configuration.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeCover the O_R, f≥d cells where the reflected margin is nonpositive or near zero.Task status · Ready to work on

Work mapped so far

Gilbert–Pollak Conjecture in numbers

1.2kretained lines of mathematical investigation1,224 in the current working snapshot

Argument development: 1,023 · 84%
Explored or eliminated routes: 38 · 3%
Computational analysis: 71 · 6%
Open obligations: 37 · 3%
Definitions and setup: 55 · 4%

7selected mapped statements2routes investigated5open questions5contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byEdgar N. Gilbert, Henry O. PollakProposed1968Status checkedAug 27, 2026

Source recordedAug 27, 2026

Recommended next task

Cover the O_R, f≥d cells where the reflected margin is nonpositive or near zero.

Suggested move: Build a finite proof-producing cover from the reflected function, GP-T01–GP-T03, nine full-block functions, and boundary-subtree feasibility inequalities.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeOrientation-free Q-shallow compression

Route status · Narrowed route

Narrowed routeUniformly positive reflected margin

The second audit reports a sharp symmetric configuration where the reflected margin equals a negative value. Combine reflected, full-block, and four-point functions in a proof-producing cell cover rather than relying on one tail inequality.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks

Cover the O_R, f≥d cells where the reflected margin is nonpositive or near zero.Suggested move: Build a finite proof-producing cover from the reflected function, GP-T01–GP-T03, nine full-block functions, and boundary-subtree feasibility inequalities.

Ready to work on

Complete the bounded regular hard-wedge residual domains for f≤d and for O_Q with f≥d.Suggested move: Emit rational cells and exact outward bounds from a deterministic interval or branch-and-bound cover, never optimizer state alone.

Ready to work on

Exact universal ratio question

For every finite terminal set V in the Euclidean plane, the conjecture asks whether S(V) is at least √3/2 times M(V).

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Reflected-margin failure cells

A complete proof still needs a uniform cover of the O_R cells with f≥d where the reflected margin is nonpositive or near zero.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Complete the bounded Steiner residual box and the remaining f≤d and non-hard-wedge topology-aware regions.Suggested move: Partition by additive arm order, apply the retained imbalance and common-arm tails, and add exact certificate families for the uncovered topology cases.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 27, 2026

Current statusOpen conjecture

The universal Euclidean-plane Gilbert-Pollak conjecture remains open. In the convention rho = inf L_SMT/L_MST over finite planar terminal sets, it asserts rho = sqrt(3)/2. Du and Hwang published a claimed proof in 1990–1992, but later peer-reviewed analyses identified gaps in essential continuity and characteristic-area steps, so those papers do not establish the conjecture. A 2026 preprint reports a certificate-backed improvement of the universal lower bound to 0.8559; that source-reported value remains below sqrt(3)/2 and is not a resolution. The false unrestricted regular-simplex generalization in dimensions d >= 3 is a separate statement.

[10][11][13]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026-01-29
PreprintKe, Huang, Shu, He, Gai, and Wang reported a new certified lower bound rho >= 0.8559 and released certificate and pipeline material. This remains below sqrt(3)/2 and therefore does not resolve the conjecture.[14][15]
2025
Peer reviewedA peer-reviewed specialist paper stated the planar conjecture explicitly and described it as still open, while separately explaining that the unrestricted higher-dimensional regular-simplex generalization is false.[13]
2010–2012
Peer reviewedInnami, Kim, Mashiko, and Shiohama challenged a continuity step essential to the Du-Hwang proof, and Ivanov and Tuzhilin published a clarification recording that the conjecture remained open.[10][11]
1991
Peer reviewedRubinstein and Thomas proved the conjectured inequality for six planar terminals; together with the earlier results, the conjecture is established for terminal sets of cardinality at most six.[5]

Mathematical neighborhood

Related results and reusable starting points

Current focusGilbert–Pollak Conjecture

Solved special casePlanar terminal sets of cardinality at most six

The planar inequality is established for all terminal sets with at most six points. These fixed-cardinality theorems do not imply the arbitrary-cardinality conjecture.

[1][3]

Solved special caseCocircular planar terminal sets

The conjectured inequality holds for arbitrary finite sets of planar terminals lying on a common circle.

[7]

Weaker or relaxed formSource-reported universal lower bound rho >= 0.8559

The 2026 preprint reports rho >= 0.8559. This is a stronger lower bound than the classical 0.82416874... result but remains strictly weaker than rho >= sqrt(3)/2.

[14][15]

Stronger or generalized formUnrestricted higher-dimensional regular-simplex generalization

The unrestricted conjecture that a regular d-simplex minimizes the Steiner ratio in Euclidean dimension d is false for every d >= 3. This does not bear negatively on the still-open d = 2 statement.

[8]

Related problemRegular simplex among configurations with d+1 terminals

A 2025 paper proposes a different open bounded-terminal conjecture: among configurations of d+1 terminals, the regular d-simplex minimizes the ratio. It explicitly differs from the false unrestricted higher-dimensional generalization.

[13]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

certificate · source linked; not reproduced by ProofAtlasGilbert-Pollak 0.8559 certificate and verification pipeline
The preprint authors provide public certificate and pipeline directories supporting their source-reported 0.8559 lower bound. ProofAtlas has not independently replayed or validated them.
[14][15]
software · source linked; not reproduced by ProofAtlasGeoSteiner 5.3
GeoSteiner provides exact algorithms and downloadable software for finite Euclidean Steiner-tree instances in the plane. It is useful for instance exploration but does not certify the universal Steiner-ratio conjecture.
[12]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA statement-aligned formal definition of finite Euclidean terminal configurations, Euclidean minimum spanning trees, Steiner minimum trees with auxiliary vertices, their lengths, and the planar Steiner-ratio infimum
Formalization targetMachine-checked existence and structural results for Euclidean Steiner minimum trees, including full-component decomposition and the 120-degree local geometry
Formalization targetMachine-checked versions of the established fixed-cardinality cases and classical universal lower bounds
Formalization targetA formally verified reduction from the 2026 verification-function framework to the universal planar lower bound
Formalization targetA small independently specified checker, proof-object format, and complete coverage proof for the released 0.8559 computational certificate

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements5 open questions2 narrowed routes

Statements by mathematical role7 selected mapped statements

theorem candidate1 of 71
reduction2 of 72
lemma4 of 74

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 2 routes included

retained route statementCan adding Steiner junctions ever save more than the equilateral-triangle ratio?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementLocal nonpositive-debt reductionintermediate
retained route statementLow-root strip closureintermediate
retained route statementRegular hard-wedge compactificationintermediate
retained route statementSteiner hard-wedge compactificationintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureOrientation-free Q-shallow compressionreported failure
Useful failureUniformly positive reflected marginreported failure
Research targetCover the O_R, f≥d cells where the reflected margin is nonpositive or near zero.open
Research targetComplete the bounded regular hard-wedge residual domains for f≤d and for O_Q with f≥d.open
Research targetComplete the bounded Steiner residual box and the remaining f≤d and non-hard-wedge topology-aware regions.open
Research targetExact universal ratio questionopen
Research targetReflected-margin failure cellsopen
Narrowed routeOrientation-free Q-shallow compressionAt Z exactly one of Q and R is shallow, and the proof using at most four descendants of Q requires O_Q; the current work withdraws the orientation-free formulation. Orientation-specific shallow-block compression remains live in O_Q, and a reflected graph-legal mechanism remains live in O_R.
Narrowed routeUniformly positive reflected marginThe second audit reports a sharp symmetric configuration where the reflected margin equals a negative value. Combine reflected, full-block, and four-point functions in a proof-producing cell cover rather than relying on one tail inequality.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeCover the O_R, f≥d cells where the reflected margin is nonpositive or near zero.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointCover the O_R, f≥d cells where the reflected margin is nonpositive or near zero.

Gilbert–Pollak Conjecture · ready to start

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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

For a finite set of points in the plane, compare the shortest network that may add Steiner junctions with the ordinary minimum spanning tree. The conjecture says the first length is always at least √3/2 of the second. this work reports substantial local reductions and certified regions, but no complete proof.

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references16 cited works · next context review by Nov 27, 2026

The mathematical context was checked on Aug 27, 2026. Status can be refreshed sooner after a material result or claim.

1
Steiner Minimal Treesoriginal source · Edgar N. Gilbert, Henry O. Pollak · Society for Industrial and Applied Mathematics · 1968-01 · DOI 10.1137/0116001 · MR MR0223269 · ZBMATH Zbl 0159.22001 · accessed Aug 27, 2026
2
A New Bound for Euclidean Steiner Minimal Treespeer reviewed result · Fan R. K. Chung, Ronald L. Graham · Annals of the New York Academy of Sciences · 1985-05 · DOI 10.1111/j.1749-6632.1985.tb14564.x · accessed Aug 27, 2026
3
The Steiner Ratio Conjecture Is True for Five Pointspeer reviewed result · Ding-Zhu Du, Frank K. Hwang, E. Y. Yao · Journal of Combinatorial Theory, Series A · 1985-03 · DOI 10.1016/0097-3165(85)90073-1 · accessed Aug 27, 2026
4
The Steiner Ratio Conjecture of Gilbert and Pollak Is Truepeer reviewed result · Ding-Zhu Du, Frank K. Hwang · Proceedings of the National Academy of Sciences · 1990-12 · DOI 10.1073/pnas.87.23.9464 · accessed Aug 27, 2026
5
The Steiner Ratio Conjecture for Six Pointspeer reviewed result · J. Hyam Rubinstein, Doreen A. Thomas · Journal of Combinatorial Theory, Series A · 1991-09 · DOI 10.1016/0097-3165(91)90073-P · MR MR1119701 · ZBMATH Zbl 0739.05034 · accessed Aug 27, 2026
6
A Proof of the Gilbert-Pollak Conjecture on the Steiner Ratiopeer reviewed result · Ding-Zhu Du, Frank K. Hwang · Algorithmica · 1992 · DOI 10.1007/BF01758755 · accessed Aug 27, 2026
7
The Steiner Ratio Conjecture for Cocircular Pointspeer reviewed result · J. Hyam Rubinstein, Doreen A. Thomas · Discrete & Computational Geometry · 1992 · DOI 10.1007/BF02187826 · accessed Aug 27, 2026
8
Disproofs of Generalized Gilbert-Pollak Conjecture on the Steiner Ratio in Three or More Dimensionspeer reviewed result · Ding-Zhu Du, Warren D. Smith · Journal of Combinatorial Theory, Series A · 1996-04 · DOI 10.1006/jcta.1996.0040 · accessed Aug 27, 2026
9
Steiner Ratio for Hyperbolic Surfacespeer reviewed result · Nobuhiro Innami, Byung Hak Kim · Proceedings of the Japan Academy, Series A · 2006-06-12 · DOI 10.3792/pjaa.82.77 · accessed Aug 27, 2026
10
The Steiner Ratio Conjecture of Gilbert-Pollak May Still Be Openpeer reviewed result · Nobuhiro Innami, B. H. Kim, Yukihiro Mashiko, Katsuhiro Shiohama · Algorithmica · 2010-08 · DOI 10.1007/s00453-008-9254-3 · accessed Aug 27, 2026
11
The Steiner Ratio Gilbert-Pollak Conjecture Is Still Open: Clarification Statementpeer reviewed result · Alexander O. Ivanov, Alexey A. Tuzhilin · Algorithmica · 2012 · DOI 10.1007/s00453-011-9508-3 · accessed Aug 27, 2026
12
GeoSteiner: Software for Computing Steiner Treessoftware or dataset · Daniel Juhl, David M. Warme, Pawel Winter, Martin Zachariasen · GeoSteiner project (version 5.3) · 2026-04-04 · accessed Aug 27, 2026
13
On Steiner Trees of the Regular Simplexpeer reviewed result · Henry Fleischmann, Guillermo Gamboa Quintero, Karthik C. S., Josef Matějka, Jakub Petr · Journal of Computational Geometry · 2025-02-19 · ARXIV 2312.01252 · DOI 10.20382/jocg.v16i1a1 · accessed Aug 27, 2026
14
Towards Solving the Gilbert-Pollak Conjecture via Large Language Modelspreprint · Yisi Ke, Tianyu Huang, Yankai Shu, Di He, Jingchu Gai, Liwei Wang · arXiv · 2026-01-29 · ARXIV 2601.22365 · accessed Aug 27, 2026
15
Steiner-Ratio: Supplementary Material for Towards Solving the Gilbert-Pollak Conjecture via Large Language Modelssoftware or dataset · Yisi Ke, Tianyu Huang, Yankai Shu, Di He, Jingchu Gai, Liwei Wang · GitHub · 2026 · accessed Aug 27, 2026
16
A Collection of Optimization Problems in Mathematicsmaintained problem list · Damek Davis, Paata Ivanisvili, Terence Tao · GitHub · accessed 2026-08-27 · accessed Aug 27, 2026

Important qualifications

This was a scoped authoritative-source pass, not a systematic review of every Steiner-ratio result.
No packet content was used to establish identity, status, priority, milestones, relations, or readiness.
The record consistently uses rho = inf L_SMT/L_MST. Sources using the reciprocal convention state the conjectured constant as 2/sqrt(3).
The 2026 lower bound 0.8559 is source-reported in an arXiv preprint with public code and certificate material; ProofAtlas did not replay or independently validate the reduction, case coverage, or certificate.
The 2025 peer-reviewed open-status source predates the 2026 preprint, but the preprint itself claims only a stronger lower bound below sqrt(3)/2 and does not claim to resolve the conjecture.
The Du-Hwang proof papers were not found to have been formally retracted, but later peer-reviewed work identifies gaps in essential steps, so they are retained only as historical proof claims.
An Encyclopedia of Mathematics entry still describes the conjecture as proved; it was excluded as status authority because later peer-reviewed specialist literature explicitly says the conjecture remains open.
The generalized regular-simplex conjecture is false in every Euclidean dimension at least three; that result does not disprove or resolve the two-dimensional conjecture.
Scoped searches of Lean/mathlib, Isabelle/AFP, Coq/Rocq, Mizar, Metamath, and HOL repositories found no statement-aligned formalization of the planar conjecture or its strongest lower bounds. This does not establish nonexistence.
GeoSteiner computes exact solutions for finite planar instances; its availability is not a proof of the universal ratio inequality.
Results for hyperbolic surfaces, normed planes, manifolds, and fixed terminal counts are separate variants or special cases and must not be promoted to the universal Euclidean-plane statement.

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