The following claim is rejected or insufficient in the recorded route: Separate marginal invariance at block scale automatically yields invariance of the full joint block law under the carry map. Turn rational Følner-box or finite-orbit approximations into either fixed-frequency Fourier cancellation and Lebesgue measure or stabilized rational support and a finite orbit.
Route status · Narrowed routeErgodic theory and arithmetic dynamics
Furstenberg’s ×2, ×3 Conjecture
Collaboration betaCan a genuinely diffuse ergodic measure be invariant under both doubling and tripling without being uniform Lebesgue measure?
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Known results and sources 
Research problem
Exact mathematical statement
On , let
Every jointly ergodic Borel probability measure invariant under both and is either Lebesgue measure or the invariant probability on a finite rational joint orbit.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Furstenberg’s ×2, ×3 Conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
Known positive-entropy rigidity and the atomic reduction leave only jointly ergodic atomless zero-entropy invariant measures.
Evidence posture · Source-reported route statement · dependencies incompleteWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Furstenberg’s ×2, ×3 Conjecture in numbers
1.8kretained lines of mathematical investigation1,766 in the current working snapshot
- Argument development
- 1,582 · 90%
- Explored or eliminated routes
- 28 · 2%
- Computational analysis
- 4 · 0%
- Open obligations
- 49 · 3%
- Definitions and setup
- 103 · 6%
7selected mapped statements1routes investigated3open questions3contribution-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.
Recommended next task
Close the large-conductor remainder of the anisotropic HNF Fourier analysis.
Suggested move: Estimate the exact weighted additive-character sum in the long transverse cyclic coordinate and force cancellation, a second relation, or a bounded rational kernel.
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
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
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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
The following claim is rejected or insufficient in the recorded route: Separate marginal invariance at block scale automatically yields invariance of the full joint block law under the carry map. Turn rational Følner-box or finite-orbit approximations into either fixed-frequency Fourier cancellation and Lebesgue measure or stabilized rational support and a finite orbit.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Close the large-conductor remainder of the anisotropic HNF Fourier analysis.Suggested move: Estimate the exact weighted additive-character sum in the long transverse cyclic coordinate and force cancellation, a second relation, or a bounded rational kernel.
Ready to work on
Classify weak limits of complete finite rational joint-orbit measures under the source’s exponential shadowing assumptions.Suggested move: Prove fixed-frequency equidistribution or stabilization of a rational kernel while keeping empirical boxes, complete orbits, and graph joinings distinct.
Ready to work on
Prove a transverse-return or positive-entropy alternative in the logarithmically thick zero-entropy regime.Suggested move: Count deterministic modal carry fields on selected patches and show that avoiding transverse returns is incompatible with the observed entropy profile.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen conjecture
A current peer-reviewed source continues to describe the measure-classification conjecture as a central open problem and supplies equivalent formulations rather than a proof.
[2]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Mathematical neighborhood
Related results and reusable starting points
Current focusFurstenberg’s ×2, ×3 Conjecture
The measure conjecture is equivalent to selected complex-analytic and operator-algebraic formulations developed in the paper.
[2]Periodic approximation and periodic equidistribution conjectures form two related triples of formulations.
[2]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetNo statement-aligned formalization of the full invariant-measure dichotomy was located.
- Formalization targetA formal development would need entropy, ergodicity, invariant measures on the circle, and the positive-entropy rigidity input.
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 removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
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.
5 standing statements2 proposed statements3 open questions1 narrowed routes
- theorem candidate
1 of 7 1 - reduction
3 of 7 3 - lemma
3 of 7 3
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
- retained route statementAre Lebesgue measure and finite rational orbits the only ergodic ×2, ×3 measures?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementExact measure dichotomyintermediate
- retained route statementZero-entropy residual caseintermediate
- retained route statementSubcritical collision reductionintermediate
- retained route statementThin HNF modelintermediate
- 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 failureSource-reported limitationreported failure
- Research targetClose the large-conductor remainder of the anisotropic HNF Fourier analysis.open
- Research targetClassify weak limits of complete finite rational joint-orbit measures under the source’s exponential shadowing assumptions.open
- Research targetProve a transverse-return or positive-entropy alternative in the logarithmically thick zero-entropy regime.open
- Research targetLarge-conductor Fourier zonesuperseded
- Research targetGlobal approximation and thick-regime closuresuperseded
- Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: Separate marginal invariance at block scale automatically yields invariance of the full joint block law under the carry map. Turn rational Følner-box or finite-orbit approximations into either fixed-frequency Fourier cancellation and Lebesgue measure or stabilized rational support and a finite orbit.
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
1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.
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.
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Prepared starting pointClose the large-conductor remainder of the anisotropic HNF Fourier analysis.
Furstenberg’s ×2, ×3 Conjecture · ready to start
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Research contextPrepared context for any AI agent
Can a genuinely diffuse ergodic measure be invariant under both doubling and tripling without being uniform Lebesgue measure?
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- Current routes and known obstacles
- What a useful result should report
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Sources and references2 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.
- 1Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximationoriginal source · Hillel Furstenberg · Mathematische Systeme Theorie · 1967 · DOI 10.1007/BF02392240 · accessed Aug 14, 2026
- 2Formulations of Furstenberg’s ×2 ×3 conjecture in complex analysis and operator algebraspeer reviewed result · Peter Burton, Jane Panangaden · Annals of Functional Analysis / Springer · 2026 · ARXIV 2410.22701 · DOI 10.1007/s43034-026-00512-1 · accessed Aug 14, 2026
Important qualifications
- Selective primary-source status pass; not an exhaustive survey of entropy-rigidity and approximation results.
- An unreviewed proof claim was not treated as status-changing evidence because a current peer-reviewed source still identifies the conjecture as open.
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