ProofAtlas

Diophantine approximation · combinatorics · circle dynamics · exact finite classification

Lonely Runner Conjecture

Collaboration beta

Given finitely many distinct constant speeds around a unit circle, must there be a time when every moving runner is far enough from a stationary reference? The general answer remains unknown.

‖vit‖≥1n+1(1≤i≤n)
Known results and sources
A dark elliptical unit track with distinct luminous phase points, precise division marks, and one highlighted safe arc around a stationary reference.
Distinct constant speeds generate phases on a unit circle; the conjecture asks whether one time makes every phase simultaneously far enough from zero.

Research problem

Exact mathematical statement

For distinct positive integers v₁,…,vₙ, prove that there is a real t such that

‖vit‖≥1n+1(1≤i≤n),\lVert v_i t\rVert \ge \frac{1}{n+1} \qquad (1\le i\le n),

where ‖x‖ is distance to the nearest integer. For the source's tactical thirteen-speed case, the threshold is 1/14. Excluding that case would be a finite-case advance only; the source explicitly says it would not settle every n.

Problem infographic

Problem at a glance

Three-panel Lonely Runner explainer with the exact v=(1,2), t=1/3 example, the general nearest-integer distance inequality, and thirteen moving speeds with the 1/14 bad arc and open status.
The circle model makes the exact threshold visible. The packet studies the next tactical thirteen-speed case, while peer-reviewed and preprint finite cases do not settle arbitrary n.

Current mathematical picture

Where work on Lonely Runner Conjecture stands

Open conjecture

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

Useful failureRaw triple-overlap threshold

The source says the family (1,N,N²) destroys that implication. The centered safe-box separator, quotient-torus classification, exact pencil lifts, and sparse/dense elimination remain source-proposed branches.

Route status · Narrowed route
Main reductionCurrent reduction

For thirteen speeds, let B={x∈T: ‖x‖<1/14}. A counterexample is exactly a primitive speed vector for which the thirteen bad-strip indicators satisfy S(t)≥1 for every t. The route studies the integer relation lattice and quotient-torus covers forced by that condition.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeIndependently audit the complete rank-seven safe-box and Fourier-transfer chain, including every non-centered approximation constant.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Lonely Runner Conjecture in numbers

2.8kretained lines of mathematical investigation2,838 in the current working snapshot
Argument development
2,285 · 81%
Explored or eliminated routes
155 · 5%
Computational analysis
131 · 5%
Open obligations
155 · 5%
Definitions and setup
112 · 4%
8selected mapped statements1routes investigated4open questions4contribution-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 byJ. M. Wills, T. W. CusickStatus checkedAug 14, 2026
Source recordedAug 14, 2026

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

13 selected steps

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

13 selected steps

Scroll horizontally to explore the route

Working route overview for Lonely Runner ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Every finite distinct-speed system should have a simultaneous lonely time. — Depends on missing premiseEvery finite distinct-speedsystem should have asimultaneous…Current reduction — Depends on missing premiseCurrent reductionDense complement rows — Depends on missing premiseDense complement rowsExact lonely-time target — Depends on missing premiseExact lonely-time targetModulo-seven alternatives — Depends on missing premiseModulo-seven alternativesSparse rank-seven proof candidate — Depends on missing premiseSparse rank-seven proofcandidateThirteen-speed cover criterion — Depends on missing premiseThirteen-speed covercriterionClosing target — Depends on missing premiseClosing targetRaw triple-overlap threshold — stoppedRaw triple-overlap thresholdIndependently audit the complete rank-seven safe-box and Fourier-transfer chain, including every non-centered approximation constant. — OpenIndependently audit thecomplete rank-seven safe-boxand…Formalize the six-pencil and seven-pencil incidence reduction before relying on the source-reported reduced finite enumeration. — OpenFormalize the six-pencil andseven-pencil incidencereduction…Close one exact quotient-lift or sparse/dense-elimination branch for the thirteen-speed tactical case. — OpenClose one exactquotient-lift orsparse/dense-elimination…Uniform all-n bridge — OpenUniform all-n bridge
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

1 recorded
Narrowed routeRaw triple-overlap threshold

The source says the family (1,N,N²) destroys that implication. The centered safe-box separator, quotient-torus classification, exact pencil lifts, and sparse/dense elimination remain source-proposed branches.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Independently audit the complete rank-seven safe-box and Fourier-transfer chain, including every non-centered approximation constant.Suggested move: Check the rational-dependent disintegration, product independence, orbit-versus-comparison-torus agreement, and normalized rank-extension support claim line by line.
Ready to work on
02
Formalize the six-pencil and seven-pencil incidence reduction before relying on the source-reported reduced finite enumeration.Suggested move: Produce an ordinary projective or matroid proof, or a machine certificate beginning before the reduction to affine support-two cases.
Ready to work on
03
Uniform all-n bridge

Find a uniform mechanism producing sufficient strict local savings at every later critical rank for arbitrary n.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Close one exact quotient-lift or sparse/dense-elimination branch for the thirteen-speed tactical case.Suggested move: Work in Smith/Hermite quotient coordinates modulo 7, 14, and 49, preserve saturation and labels, and emit an exact safe point or short-relation certificate.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The conjecture for arbitrary runner counts remains open. Published computer-assisted work covers nine and ten total runners, and a 2026 preprint reports coverage through thirteen total runners; these finite cases do not cover the fourteen-runner target in the source material.

[1][3][4]
External progress

What the literature has established

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

  1. Peer reviewedTrakulthongchai published computer-assisted proofs for nine and ten total runners.[3]
  2. PreprintSungkawichai and Trakulthongchai report computer-assisted proofs for eleven, twelve, and thirteen total runners.[4]
  3. Peer reviewedBarajas and Serra proved the then-first open case with seven total runners.[2]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusLonely Runner Conjecture
Solved special caseseven-runner case

The conjecture is proved for seven total runners.

[2]
Solved special casenine- and ten-runner cases

The conjecture has published computer-assisted proofs for nine and ten total runners.

[3]
Solved special caseeleven-, twelve-, and thirteen-runner cases

A 2026 preprint reports computer-assisted proofs for eleven through thirteen total runners.

[4]

Formal and computational footholds

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

  • computation · statement onlyComputer-assisted nine- and ten-runner proofs

    The journal article reports computer-assisted proofs; no executable artifact or certificate was inspected in this collection pass.

    [3]
  • computation · statement onlyComputer-assisted eleven-through-thirteen-runner proofs

    The preprint reports computer-assisted proofs; no executable artifact or certificate was inspected in this collection pass.

    [4]

Formalization opportunities

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

  • Formalization targetNo checked formal statement alignment for arbitrary runner counts was located.
  • Formalization targetNo independently reproduced certificate for the source-reported finite-case computations was assessed.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

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

3 standing statements5 proposed statements4 open questions1 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction4 of 84
  • lemma1 of 81
  • equivalence2 of 82
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 1 route included
  • retained route statementEvery finite distinct-speed system should have a simultaneous lonely time.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact lonely-time targetintermediate
  • retained route statementThirteen-speed cover criterionintermediate
  • retained route statementSparse rank-seven proof candidateintermediate
  • retained route statementDense complement rowsintermediate
  • retained route statementModulo-seven alternativesintermediate
  • 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
  • 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 failureRaw triple-overlap thresholdreported failure
  • Research targetIndependently audit the complete rank-seven safe-box and Fourier-transfer chain, including every non-centered approximation constant.open
  • Research targetFormalize the six-pencil and seven-pencil incidence reduction before relying on the source-reported reduced finite enumeration.open
  • Research targetClose one exact quotient-lift or sparse/dense-elimination branch for the thirteen-speed tactical case.open
  • Research targetUniform all-n bridgeopen
  • ComputationThe current work says verify_v9_claims.py checks exact rational inequalities, relation-extension bounds, hyperplane-section counts, exponent pairs, and parity-lift distances, while verify_v10_pencil_audit.py exhausts the reduced finite pencil cases. Intake did not execute submitted attachments.The recorded calculations are source-reported and remain advisory here. The current work itself says they do not certify the infinite disintegration, Fourier-transfer, or incidence-reduction arguments. · reported unreproduced
  • Narrowed routeRaw triple-overlap thresholdThe source says the family (1,N,N²) destroys that implication. The centered safe-box separator, quotient-torus classification, exact pencil lifts, and sparse/dense elimination remain source-proposed branches.
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 bridgeIndependently audit the complete rank-seven safe-box and Fourier-transfer chain, including every non-centered approximation constant.

1 approach has 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

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointIndependently audit the complete rank-seven safe-box and Fourier-transfer chain, including every non-centered approximation constant.

Lonely Runner Conjecture · ready to start

Mathematical updatesFollow this problem

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

Given finitely many distinct constant speeds around a unit circle, must there be a time when every moving runner is far enough from a stationary reference? The general answer remains unknown.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    The Lonely Runner Conjecture turns 60survey or monograph · Guillem Perarnau, Oriol Serra · arXiv · 2025-08-12 · ARXIV 2409.20160 · accessed Aug 14, 2026
  2. 2
    The Lonely Runner with Seven Runnerspeer reviewed result · J. Barajas, O. Serra · The Electronic Journal of Combinatorics · 2008-03-20 · DOI 10.37236/772 · accessed Aug 14, 2026
  3. 3
    Nine and Ten Lonely Runnerspeer reviewed result · Tanupat Trakulthongchai · The Electronic Journal of Combinatorics · 2026-06-05 · DOI 10.37236/14972 · accessed Aug 14, 2026
  4. 4
    Eleven, twelve, and thirteen lonely runnerspreprint · Touch Sungkawichai, Tanupat Trakulthongchai · arXiv · 2026-04-26 · ARXIV 2604.23906 · accessed Aug 14, 2026

Important qualifications

  • This was a bounded source check, not a systematic literature review.
  • The thirteen-runner result checked here is a 2026 preprint and is not represented as peer reviewed.
  • Computer assistance is source-reported; no software, certificate, or independent reproduction was inspected or executed.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image