ProofAtlas

Plane topology · symplectic geometry · persistence · Jordan curves

Inscribed Square Problem

Collaboration beta

Modern Floer and persistence tools prove broad rectangle results and important special cases; the arbitrary Jordan-curve square remains open.

p1,p2,p3,p4∈γform a square
Known results and sources
Problem-first open-research thumbnail for Inscribed Square Problem, showing an unresolved mathematical structure without a completion mark or navigation arrow.
Must every Jordan curve inscribe a square? The page presents source-reported partial structure without claiming a proof.

Research problem

Exact mathematical statement

Every planar Jordan curve contains four distinct points that form a nondegenerate square.

p1,p2,p3,p4∈γform a squarep_1,p_2,p_3,p_4\in\gamma\quad\text{form a square}

This remains open; the source reports partial results and route constraints, not a complete proof.

Problem infographic

Problem at a glance

Three-panel source-bound explainer for Inscribed Square Problem: exact target, strongest retained result, and the unresolved closing bridge.
The source separates the exact target, retained partial results, and the still-open bridge.

Current mathematical picture

Where work on Inscribed Square Problem stands

Open problem

The cumulative v20 source reports quantitative rank-two analytic, target-current, radial/capping, self-stress, and contour-shell advances while explicitly retaining the Inscribed Square Problem as open. Its global carrier, typed current-transfer, target-to-source localization, transition-classification, and formal-verification interfaces remain unresolved.

Leading routeTwo-center analytic corridor

Source-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.

Route status · Active route
Main reductionElegant contour-shell engine

The source reports a lossless lowering measure and explicit shell currents; B0/B1 must still retain controlled-sector mass and accumulate defects into a valid relation, capping/stress transition, or excluded zero model.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeB0/B1: retain controlled-sector contour-shell mass and accumulate its defects into a valid relation or excluded model.Task status · Ready to work on
Latest mathematical updatev20 repairs and narrows the four-engine Inscribed Square frontier

The cumulative v20 source reports quantitative analytic and finite-dimensional advances while preserving the exact open target and naming the unresolved carrier, transfer, localization, transition, and formal interfaces.

Retained source record

Work mapped so far

Inscribed Square Problem in numbers

3.9kretained lines of mathematical investigation1,013 in the current working snapshot
Argument development
3,215 · 83%
Explored or eliminated routes
82 · 2%
Computational analysis
27 · 1%
Open obligations
248 · 6%
Definitions and setup
307 · 8%
10inventoried working statements6routes investigated7open questions7contribution-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.

Status checkedAug 21, 2026
Source recordedAug 27, 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

22 selected steps

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

22 selected steps

Scroll horizontally to explore the route

Working route overview for Inscribed Square ProblemA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Inscribed Square Problem remains open — Depends on missing premiseInscribed Square Problemremains openElegant contour-shell engine — Depends on missing premiseElegant contour-shell engineFour live engines and one formal-upstream obligation — Depends on missing premiseFour live engines and oneformal-upstream obligationRadial/capping carrier and typed-transfer engine — Depends on missing premiseRadial/capping carrier andtyped-transfer engineSource-localized self-stress/model engine — Depends on missing premiseSource-localizedself-stress/model engineAN-22--AN-29 quantitative corridor and target-current packet — Depends on missing premiseAN-22--AN-29 quantitativecorridor and target-currentpacketGlobal closure interfaces remain unresolved — Depends on missing premiseGlobal closure interfacesremain unresolvedv20 implication firewall and retired shortcuts — Depends on missing premisev20 implication firewall andretired shortcutsSquare-free contradiction formulation — Depends on missing premiseSquare-free contradictionformulationUnmarked persistence route — Depends on missing premiseUnmarked persistence routeTwo-center analytic corridor — activeTwo-center analytic corridorRadial alternating end transfer — activeRadial alternating endtransferTwo-center self-stress/model route — activeTwo-center self-stress/modelrouteElegant contour-shell route — activeElegant contour-shell routeRepresentation-level, regularity, and calibrated-moment shortcuts — stoppedRepresentation-level,regularity, andcalibrated-moment…B2a: select a proper oriented elegant carrier or classify every singular continuation obstruction. — OpenB2a: select a properoriented elegant carrier orclassify…RC-21: construct the finite-slab base-times-target three-current, cap/slicing formula, anchor transport, and every boundary face. — OpenRC-21: construct thefinite-slabbase-times-target…B3: convert an isolated later vertex-on-side incidence or endpoint relation into an inscribed boundary square, or into a route already covered by B2b, B4S, or B4T. — OpenB3: convert an isolatedlater vertex-on-sideincidence…B4S: convert hidden/cancellation defects or localized target data into a source-sheet, grid, packet-chain, or stressed-slab theorem. — OpenB4S: converthidden/cancellation defectsor…B0/B1: retain controlled-sector contour-shell mass and accumulate its defects into a valid relation or excluded model. — OpenB0/B1: retaincontrolled-sectorcontour-shell…B4T: classify every compactness failure by a proved transition or escape coordinate. — OpenB4T: classify everycompactness failure by aproved…B6: independently formalize every P* input actually used by a terminal route and audit orientations and compactification faces. — OpenB6: independently formalizeevery P* input actually usedby…
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.

Active routeTwo-center analytic corridor

Source-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.

Route status · Active route
Active routeRadial alternating end transfer

Source-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.

Route status · Active route
Active routeTwo-center self-stress/model route

Source-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.

Route status · Active route
Active routeElegant contour-shell route

Source-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.

Route status · Active route
Active routeR-E continuous-lift bypass

Source-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.

Route status · Active route

Explored alternatives

Other routes

1 recorded
Useful but insufficientMarked-endpoint vineyard route

The marked-endpoint problem is not the controlling obstruction, and the degree-one vineyard shortcut is quarantined. Unmarked fixed-action persistence, proper-ray recurrence, and direct microlocal or angular routes remain viable.

Route status · Useful but insufficient

Route statements and reductions

Statements the next route can inspect and build on

Route statementFour live engines and one formal-upstream obligation

The source organizes the open frontier into rank-two analytic, radial/capping, self-stress/model, and elegant contour-shell engines, together with an upstream formal-compatibility obligation.

Source-reported route statement · dependencies incomplete
Route statementGlobal closure interfaces remain unresolved

The source reports that a successful terminal route still needs at least one genuinely new global theorem, including carrier extraction or typed current transfer, target-to-source localization, transition classification, or the stated formal closures.

Source-reported route statement · dependencies incomplete
Route statementAN-22--AN-29 quantitative corridor and target-current packet

The source labels AN-22--AN-24 and AN-26--AN-29 as proved at their stated scopes and AN-25 as a proved reduction, yielding little-Besov/VMO regularity, deep packet corridors, a network-or-spiral reduction, and compact target-current/multipole data without global closure.

Source-reported route statement · dependencies incomplete
Route statementRadial/capping carrier and typed-transfer engine

The source retains an odd relative radial class and a conditional carrier-end cocycle, but B2a and RC-21 still require a proper elegant carrier, a base-times-target three-current, cap/slicing identities, anchor transport, and all boundary faces.

Source-reported route statement · dependencies incomplete
Route statementSource-localized self-stress/model engine

The source reports exact finite-dimensional stress data and model exclusions while retaining source packet-chain localization and propagation under source, frame, phase, and scale control as unresolved.

Source-reported route statement · dependencies incomplete
Route statementElegant contour-shell engine

The source reports a lossless lowering measure and explicit shell currents; B0/B1 must still retain controlled-sector mass and accumulate defects into a valid relation, capping/stress transition, or excluded zero model.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

7 featured tasks
01
B0/B1: retain controlled-sector contour-shell mass and accumulate its defects into a valid relation or excluded model.Suggested move: B0/B1: retain controlled-sector contour-shell mass and accumulate its defects into a valid relation or excluded model.
Ready to work on
02
B2a: select a proper oriented elegant carrier or classify every singular continuation obstruction.Suggested move: B2a: select a proper oriented elegant carrier or classify every singular continuation obstruction.
Ready to work on
03
B4S: convert hidden/cancellation defects or localized target data into a source-sheet, grid, packet-chain, or stressed-slab theorem.Suggested move: B4S: convert hidden/cancellation defects or localized target data into a source-sheet, grid, packet-chain, or stressed-slab theorem.
Ready to work on
04
B4T: classify every compactness failure by a proved transition or escape coordinate.Suggested move: B4T: classify every compactness failure by a proved transition or escape coordinate.
Ready to work on
05
B6: independently formalize every P* input actually used by a terminal route and audit orientations and compactification faces.Suggested move: B6: independently formalize every P* input actually used by a terminal route and audit orientations and compactification faces.
Ready to work on
06
RC-21: construct the finite-slab base-times-target three-current, cap/slicing formula, anchor transport, and every boundary face.Suggested move: RC-21: construct the finite-slab base-times-target three-current, cap/slicing formula, anchor transport, and every boundary face.
Ready to work on
07
B3: convert an isolated later vertex-on-side incidence or endpoint relation into an inscribed boundary square, or into a route already covered by B2b, B4S, or B4T.Suggested move: B3: convert an isolated later vertex-on-side incidence or endpoint relation into an inscribed boundary square, or into a route already covered by B2b, B4S, or B4T.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 21, 2026
Current statusOpen problem

Recent Floer-theoretic work proves rectangle and square results under substantial regularity or geometric hypotheses; the full arbitrary-Jordan-curve problem remains open.

[1][2]
External progress

What the literature has established

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

  1. PreprintFloer homology and square pegs supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet target.[1]
  2. PreprintThe rectifiable rectangular peg problem supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet target.[2]
2 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusInscribed Square Problem
Related problemInscribed Square Problem

Recent Floer-theoretic work proves rectangle and square results under substantial regularity or geometric hypotheses; the full arbitrary-Jordan-curve problem remains open.

[1][2]

Formalization opportunities

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

  • Formalization targetA formal statement matching the exact public target and all quantifiers.
  • Formalization targetFormal libraries for the principal mathematical structures used by the strongest route.
  • Formalization targetA checked closing argument for the source-identified open bridge.

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

v20 repairs and narrows the four-engine Inscribed Square frontierThe cumulative v20 source reports quantitative analytic and finite-dimensional advances while preserving the exact open target and naming the unresolved carrier, transfer, localization, transition, and formal interfaces.

Changed the research frontierLater mathematical revision

Cumulative v20 source ingested; not a claim of mathematical occurrence time

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

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 removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Claim connections clarified

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 inventory covers all currently cataloged mathematical statements in the research notes.

7 standing statements3 proposed statements7 open questions
Statements by mathematical role10 mapped statements
  • reduction5 of 105
  • theorem candidate1 of 101
  • lemma2 of 102
  • equivalence1 of 101
  • negative result1 of 101
Complete mathematical inventory1 mathematical clusters
v20 recorded four-engine and bypass research frontierThe v20 full snapshot preserves the exact open target, the source-reported analytic and finite-dimensional advances, the four live engines, the independent continuous-lift bypass, the repaired implication boundaries, and every named unresolved global interface.21 displayed rows · 5 routes included
  • retained route statementInscribed Square Problem remains open
  • retained route statementFour live engines and one formal-upstream obligationintermediate
  • retained route statementGlobal closure interfaces remain unresolvedintermediate
  • retained route statementSquare-free contradiction formulationintermediate
  • retained route statementAN-22--AN-29 quantitative corridor and target-current packetintermediate
  • retained route statementv20 implication firewall and retired shortcutsintermediate
  • retained route statementRadial/capping carrier and typed-transfer engineintermediate
  • retained route statementSource-localized self-stress/model engineintermediate
  • retained route statementElegant contour-shell engineintermediate
  • Research targetB2a: select a proper oriented elegant carrier or classify every singular continuation obstruction.open
  • Research targetRC-21: construct the finite-slab base-times-target three-current, cap/slicing formula, anchor transport, and every boundary face.open
  • Research targetB3: convert an isolated later vertex-on-side incidence or endpoint relation into an inscribed boundary square, or into a route already covered by B2b, B4S, or B4T.open
  • Research targetB4S: convert hidden/cancellation defects or localized target data into a source-sheet, grid, packet-chain, or stressed-slab theorem.open
  • Research targetB0/B1: retain controlled-sector contour-shell mass and accumulate its defects into a valid relation or excluded model.open
  • Research targetB4T: classify every compactness failure by a proved transition or escape coordinate.open
  • Research targetB6: independently formalize every P* input actually used by a terminal route and audit orientations and compactification faces.open
  • Active routeTwo-center analytic corridorSource-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.
  • Active routeRadial alternating end transferSource-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.
  • Active routeTwo-center self-stress/model routeSource-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.
  • Active routeElegant contour-shell routeSource-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.
  • Active routeR-E continuous-lift bypassSource-reported live engine with explicit unresolved interfaces; it is not a completed route to the conjecture.
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 bridgeB0/B1: retain controlled-sector contour-shell mass and accumulate its defects into a valid relation or excluded model.

The current research map records this as an open mathematical step.

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.

  • Meet every exact source-stated scope, regularity, normalization, dependency, and boundary condition.
  • Survive independent review without upgrading any source-reported status.

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 pointB0/B1: retain controlled-sector contour-shell mass and accumulate its defects into a valid relation or excluded model.

Inscribed Square Problem · 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

Modern Floer and persistence tools prove broad rectangle results and important special cases; the arbitrary Jordan-curve square remains open.

  • 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

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Sources and references2 cited works · next context review by Nov 21, 2026

The mathematical context was checked on Aug 21, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Floer homology and square pegspreprint · Joshua Evan Greene, Andrew Lobb · arXiv · 2024 · accessed Aug 21, 2026
  2. 2
    The rectifiable rectangular peg problempreprint · Tomohiro Asano, Yuichi Ike · arXiv · 2024 · accessed Aug 21, 2026

Important qualifications

  • This was a bounded status and identity check, not an exhaustive bibliography, priority review, or legal review.
  • Private packet claims were not treated as external authority; submitted links and attachments were not executed or actively rendered.
  • No absence claim is inferred from the bounded search, and recent preprints remain subject to ordinary scholarly review.

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