ProofAtlas

Dynamical systems, convex billiards, and rigidity

Birkhoff–Poritsky billiard conjecture

Collaboration beta

Does a complete smooth family of invariant curves close to the boundary of a strictly convex billiard force the boundary itself to be an ellipse?

full smooth grazing foliation⇒Γis an ellipse
Known results and sources
A smooth convex billiard with reflected paths and nested boundary caustics faces a second convex shape marked by an open question.
The local conjecture asks whether a full smooth foliation near a strictly convex billiard boundary forces the boundary to be an ellipse.

Research problem

Exact mathematical statement

Let Γ⊂ℝ2Γ\subset\mathbb R^2 be a smooth strictly convex billiard table. The local Birkhoff–Poritsky conjecture asks whether a full sufficiently smooth grazing collar foliated by rotational invariant curves, or convex caustics in the source's formulation, forces ΓΓ to be an ellipse.

full smooth grazing foliation⇒Γis an ellipse\text{full smooth grazing foliation}\;\Longrightarrow\;Γ\text{ is an ellipse}

A stronger global form asks the analogous question for a suitably globally integrable billiard annulus. The source keeps these formulations, smooth-first-integral hypotheses, and endpoint regularity distinct; it does not prove either unrestricted form.

Problem infographic

Problem at a glance

A landscape plate separates specular reflection, a foliated grazing collar, the ellipse question, and a distinct global annulus inset.
Local collar rigidity and global annular integrability are distinct open formulations; the image does not present either as proved.

Current mathematical picture

Where work on Birkhoff–Poritsky billiard conjecture stands

Open conjecture

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

Useful failureUnweighted fixed-boundary averaging

The source records that moving concentration spikes remain possible and explicitly classifies linear elimination, scalar distortion, and unweighted averages as exhausted. Gauge-invariant pairings, derivative control, or exact rational reconstruction may still address the global obstruction; Ward divisibility remains the distinct local frontier.

Route status · Narrowed route
Main reductionCurrent reduction

The local source-reported program seeks full Ward divisibility and grouped exponential-growth control before applying an exact adjacent-numerator determinant; the distinct global program seeks control of a gauge-invariant phase-drift or moving-spike obstruction.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve full Ward divisibility of the stationary forcing after all lower primitive resonance equations are imposed.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

Birkhoff–Poritsky billiard conjecture in numbers

4.2kretained lines of mathematical investigation4,172 in the current working snapshot
Argument development
3,663 · 88%
Explored or eliminated routes
86 · 2%
Computational analysis
25 · 1%
Open obligations
129 · 3%
Definitions and setup
269 · 6%
6selected mapped statements1routes 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 byGeorge D. Birkhoff, Harry PoritskyStatus 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

12 selected steps

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

12 selected steps

Scroll horizontally to explore the route

Working route overview for Birkhoff–Poritsky billiard conjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Does a full foliated grazing collar force an ellipse? — Depends on missing premiseDoes a full foliated grazingcollar force an ellipse?Current reduction — Depends on missing premiseCurrent reductionGauge-invariant global obstruction — Depends on missing premiseGauge-invariant globalobstructionClosing target — Depends on missing premiseClosing targetFinite identities do not supply uniform control — Depends on missing premiseFinite identities do notsupply uniform controlMacroscopic-numerator primitivity — Depends on missing premiseMacroscopic-numeratorprimitivityUnweighted fixed-boundary averaging — stoppedUnweighted fixed-boundaryaveragingProve full Ward divisibility of the stationary forcing after all lower primitive resonance equations are imposed. — OpenProve full Ward divisibilityof the stationary forcingafter…Establish grouped exponential-growth and uniform remainder control along the rational sequence for one analytic billiard table. — OpenEstablish groupedexponential-growth anduniform…Control the global gauge-invariant phase-drift or moving-spike obstruction under strict chord concavity. — OpenControl the globalgauge-invariant phase-driftor…Local grazing-collar rigidity — OpenLocal grazing-collarrigidityGlobal annular rigidity — OpenGlobal annular rigidity
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 routeUnweighted fixed-boundary averaging

The source records that moving concentration spikes remain possible and explicitly classifies linear elimination, scalar distortion, and unweighted averages as exhausted. Gauge-invariant pairings, derivative control, or exact rational reconstruction may still address the global obstruction; Ward divisibility remains the distinct local frontier.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Prove full Ward divisibility of the stationary forcing after all lower primitive resonance equations are imposed.Suggested move: Audit forcing and first-slope vanishing at each lower resonance before claiming the quadratic zero needed for divisibility.
Ready to work on
02
Establish grouped exponential-growth and uniform remainder control along the rational sequence for one analytic billiard table.Suggested move: Formulate a uniform radius and Sobolev estimate that survives the passage from finite primitive pairs to the analytic limit.
Ready to work on
03
Local grazing-collar rigidity

Prove that a full sufficiently smooth foliated grazing collar forces the boundary to be an ellipse.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Global annular rigidity

Prove the separate global annular-integrability form without assuming equivalence to the local collar form.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Control the global gauge-invariant phase-drift or moving-spike obstruction under strict chord concavity.Suggested move: Test gauge-invariant pairings or direct bounds on the reconstructed nonresonant and resonant modes, with endpoint hypotheses explicit.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

Open for unrestricted smooth strictly convex planar billiards. Peer-reviewed theorems cover local perturbations of ellipses and a centrally symmetric foliated region; a preprint treats sufficiently nearly centrally symmetric domains. These special cases do not establish the arbitrary local or global rigidity statement.

[2][3][4]
External progress

What the literature has established

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

  1. PreprintKaloshin, Koudjinan, and Zhang announced a perturbative extension from centrally symmetric to sufficiently nearly centrally symmetric integrable domains.[4]
  2. Peer reviewedBialy and Mironov proved the conjecture for centrally symmetric C2 convex billiards under a foliated phase-cylinder region, including a C1 caustic-foliation form through rotation number 1/4.[3]
  3. Peer reviewedKaloshin and Sorrentino established a complete local Birkhoff theorem in a neighborhood of an ellipse under their stated smoothness and rational-integrability hypotheses.[2]
  4. Peer reviewedAvila, De Simoi, and Kaloshin proved a local rigidity form for sufficiently small perturbations of ellipses of small eccentricity.[1]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusBirkhoff–Poritsky billiard conjecture
Solved special caselocal Birkhoff conjecture near ellipses

Local rigidity is known in a sufficiently small neighborhood of an ellipse under the paper's stated regularity and rational-integrability assumptions.

[2]
Solved special casecentrally symmetric Birkhoff–Poritsky conjecture

Centrally symmetric convex billiards satisfying the paper's foliation hypotheses are ellipses; central symmetry is essential to the theorem's recorded scope.

[3]
Solved special casenearly centrally symmetric integrable billiards

A preprint gives a perturbative theorem for domains sufficiently close to centrally symmetric ones, not arbitrary convex billiards.

[4]

Formalization opportunities

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

  • Formalization targetA formal statement needs smooth strictly convex planar curves, billiard reflection maps, phase cylinders, rotation numbers, invariant curves, and convex caustics.
  • Formalization targetIt must encode complete boundary-collar foliations and the exact differentiability assumptions without replacing them by finitely many rational caustics.
  • Formalization targetA formal theorem would need rigidity of the boundary up to Euclidean ellipse and a clean separation of local, global, and first-integral hypotheses.
  • Formalization targetNo scoped problem-level formalization was recorded in this bounded pass.

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
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.

Corrected the research recordCorrection note

Correction details
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.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

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 statements3 proposed statements5 open questions1 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction2 of 62
  • lemma2 of 62
  • negative result1 of 61
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
  • retained route statementDoes a full foliated grazing collar force an ellipse?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementMacroscopic-numerator primitivityintermediate
  • retained route statementFinite identities do not supply uniform controlintermediate
  • retained route statementGauge-invariant global obstructionintermediate
  • 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
  • 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 failureUnweighted fixed-boundary averagingreported failure
  • Research targetProve full Ward divisibility of the stationary forcing after all lower primitive resonance equations are imposed.open
  • Research targetEstablish grouped exponential-growth and uniform remainder control along the rational sequence for one analytic billiard table.open
  • Research targetControl the global gauge-invariant phase-drift or moving-spike obstruction under strict chord concavity.open
  • Research targetLocal grazing-collar rigidityopen
  • Research targetGlobal annular rigidityopen
  • Research targetTwo unresolved bridgessuperseded
  • Narrowed routeUnweighted fixed-boundary averagingThe source records that moving concentration spikes remain possible and explicitly classifies linear elimination, scalar distortion, and unweighted averages as exhausted. Gauge-invariant pairings, derivative control, or exact rational reconstruction may still address the global obstruction; Ward divisibility remains the distinct local frontier.
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 bridgeProve full Ward divisibility of the stationary forcing after all lower primitive resonance equations are imposed.

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

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Prepared starting pointProve full Ward divisibility of the stationary forcing after all lower primitive resonance equations are imposed.

Birkhoff–Poritsky billiard 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

Does a complete smooth family of invariant curves close to the boundary of a strictly convex billiard force the boundary itself to be an ellipse?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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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
    An integrable deformation of an ellipse of small eccentricity is an ellipsepeer reviewed result · Artur Avila, Jacopo De Simoi, Vadim Kaloshin · Annals of Mathematics · 2016 · DOI 10.4007/annals.2016.184.2.5 · accessed Aug 14, 2026
  2. 2
    On the local Birkhoff conjecture for convex billiardspeer reviewed result · Vadim Kaloshin, Alfonso Sorrentino · Annals of Mathematics · 2018 · DOI 10.4007/annals.2018.188.1.6 · accessed Aug 14, 2026
  3. 3
    The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tablespeer reviewed result · Misha Bialy, Andrey E. Mironov · Annals of Mathematics · 2022 · ARXIV 2008.03566 · DOI 10.4007/annals.2022.196.1.2 · accessed Aug 14, 2026
  4. 4
    Birkhoff Conjecture for nearly centrally symmetric domainspreprint · Vadim Kaloshin, Comlan Edmond Koudjinan, Ke Zhang · arXiv · 2023 · ARXIV 2306.12301 · accessed Aug 14, 2026

Important qualifications

  • The local Birkhoff–Poritsky formulation, global integrability formulation, rational integrability variants, and smooth-first-integral formulations have different hypotheses; this record does not silently identify them.
  • The full unrestricted convex-domain rigidity conjecture remains open. Near-ellipse, centrally symmetric, and nearly centrally symmetric theorems are retained only at their stated regularity and integrability scopes.
  • The historical proposer year and original Birkhoff or Poritsky source are left unresolved because this bounded pass did not inspect the original historical publication.
  • No current problem-level formalization or canonical computation was identified, but this pass did not perform an exhaustive proof-assistant search.
  • No intake-packet claim was used as external status authority, and no proof, review, acceptance, credit, publication, or deployment authority is granted.

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