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Symplectic geometry · embeddings · packing

Higher-Dimensional Symplectic Ball-Packing Conjecture

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For symplectic balls in real dimension at least six, are total volume and the pairwise two-ball nonsqueezing inequalities the only obstructions to packing them strictly into a larger ball?

⨆iB2n(ci)↪B2n(C)⇔∑icin<Cnandci+cj<C (i≠j)
Known results and sources
Colored phase-space balls approach one larger target ball above strict volume and two-ball constraint diagrams, with a question mark at the embedding step.
The higher-dimensional conjecture asks whether strict volume and pairwise nonsqueezing inequalities completely characterize symplectic ball packings.

Research problem

Exact mathematical statement

For n≥3n\ge 3, define

B2n(c)={z∈ℂn:π∑j=1n|zj|2<c}.B^{2n}(c)=\left\{z\in\mathbb C^n: \pi\sum_{j=1}^n|z_j|^2<c\right\}.

Given positive capacities c1,…,ck,Cc_1,…,c_k,C, does a strict symplectic packing

⨆i=1kB2n(ci)↪B2n(C)\bigsqcup_{i=1}^k B^{2n}(c_i)\hookrightarrow B^{2n}(C)

exist if and only if ∑icin<Cn\sum_i c_i^n<C^n and ci+cj<Cc_i+c_j<C for every i≠ji\ne j? Here strict means that after an arbitrarily small shrink, the embedding extends to neighborhoods of the corresponding closed balls. Simultaneous capacity scaling is by conjugation with conformal Euclidean dilations; Euclidean dilation itself is not symplectic.

Problem infographic

Problem at a glance

A landscape diagram sends multiple action-angle balls through strict volume and pairwise gates toward a target ball while leaving the final higher-dimensional packing conjectural.
The two displayed inequalities are necessary; the open question is whether they suffice for every strict packing in dimensions at least six.

Current mathematical picture

Where work on Higher-Dimensional Symplectic Ball-Packing Conjecture stands

Open conjecture

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

Useful failureSource-reported limitation

Volume matching, full-measure free-locus packing, and an equimeasurable shell profile do not by themselves provide a connected symplectic embedding or restore Liouville-period-compatible seams. The decisive missing theorem is a relative anisotropic strip-folding or direct singular baker construction that turns the actual two-block source remainder into one connected strict embedding with matched flux and seams.

Route status · Narrowed route
Main reductionConnected benchmark implication

The current work proves that the connected elongated-ellipsoid embedding for every strict R implies the sharp nine-equal-ball packing after aggregation and conformal rescaling.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeConstruct the exact endpoint block rearrangement or singular baker map.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

Higher-Dimensional Symplectic Ball-Packing Conjecture in numbers

3.7kretained lines of mathematical investigation3,691 in the current working snapshot
Argument development
3,085 · 84%
Explored or eliminated routes
139 · 4%
Computational analysis
79 · 2%
Open obligations
213 · 6%
Definitions and setup
175 · 5%
8selected 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.

Proposed byKyler Siegel, Yuan YaoProposed2023Status 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 Higher-Dimensional Symplectic Ball-Packing ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Volume and two-ball nonsqueezing should be the only higher-dimensional ball-packing obstructions — Depends on missing premiseVolume and two-ballnonsqueezing should be theonly…Connected benchmark implication — Depends on missing premiseConnected benchmarkimplicationCurrent reduction — Depends on missing premiseCurrent reductionExact conjectural criterion — Depends on missing premiseExact conjectural criterionClosing target — Depends on missing premiseClosing targetFull measure is not whole balls — Depends on missing premiseFull measure is not wholeballsNo benchmark equivalence — Depends on missing premiseNo benchmark equivalenceReported k≤2^n range — Depends on missing premiseReported k≤2^n rangeSource-reported limitation — stoppedSource-reported limitationConstruct the exact endpoint block rearrangement or singular baker map. — OpenConstruct the exact endpointblock rearrangement orsingular…Sew the finite strips, corridors, seams, and coordinate axes. — OpenSew the finite strips,corridors, seams, andcoordinate…Prove global injectivity and then separate benchmark scope from the weighted problem. — OpenProve global injectivity andthen separate benchmarkscope…
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 routeSource-reported limitation

Volume matching, full-measure free-locus packing, and an equimeasurable shell profile do not by themselves provide a connected symplectic embedding or restore Liouville-period-compatible seams. The decisive missing theorem is a relative anisotropic strip-folding or direct singular baker construction that turns the actual two-block source remainder into one connected strict embedding with matched flux and seams.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Construct the exact endpoint block rearrangement or singular baker map.Suggested move: Write the two actual source blocks and strict target bands in action-angle coordinates, then build a finite cut-and-stack with matched Liouville periods and a restored connected face.
Ready to work on
02
Sew the finite strips, corridors, seams, and coordinate axes.Suggested move: Integerize the layer data, use the exact nine-sector ledger and whole-parcel suspension, and match each axis, facet, seam, junction, and bulk overlap before applying relative interpolation.
Ready to work on
03
Prove global injectivity and then separate benchmark scope from the weighted problem.Suggested move: Exhibit a recoverable target label or disjoint action-corridor invariant with quantitative strict buffers; only afterward develop the additional adaptive weighted sewing needed for arbitrary capacities.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The volume-plus-two-ball criterion remains Conjecture A for n at least 3. The cited peer-reviewed paper supplies evidence and stabilized theorems, not a proof of the unstabilized many-ball statement.

[2]
External progress

What the literature has established

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

  1. Peer reviewedThe journal version states the exact criterion as Conjecture A, proves stabilized packing results, and explains why the standard pseudoholomorphic-curve obstruction paradigm is not expected to yield stronger…[2]
  2. PreprintSiegel and Yao posted the conjecture that higher-dimensional ball packing is controlled entirely by volume and Gromov’s two-ball obstruction.[1]
2 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHigher-dimensional symplectic ball-packing conjecture
Related problemfour-dimensional symplectic ball packing

In real dimension four, refined packing obstructions exist and the general problem is computed or reduced to combinatorics; this contrasts with the proposed flexibility for n at least 3.

[2]
Solved special casestabilized four-dimensional ball packings

Siegel and Yao prove an equivalence between four-dimensional packings and their products with a closed symplectic surface, plus a stabilized two-ball theorem.

[2]

Formalization opportunities

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

  • Formalization targetA problem-level formal statement needs symplectic manifolds and embeddings, standard balls with capacity normalization, disjoint unions, strict interior or neighborhood-of-closed conventions, and volume preservation.
  • Formalization targetFormal proof routes would require nonsqueezing, symplectic blowups, pseudoholomorphic curve theory, and substantial high-dimensional embedding infrastructure.
  • Formalization targetNo scoped public problem-level formalization was identified.

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 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

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 statements3 proposed statements3 open questions1 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction2 of 82
  • lemma2 of 82
  • equivalence1 of 81
  • negative result2 of 82
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
  • retained route statementVolume and two-ball nonsqueezing should be the only higher-dimensional ball-packing obstructions
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact conjectural criterionintermediate
  • retained route statementConnected benchmark implicationintermediate
  • retained route statementNo benchmark equivalenceintermediate
  • retained route statementReported k≤2^n rangeintermediate
  • retained route statementFull measure is not whole ballsintermediate
  • 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 failureSource-reported limitationreported failure
  • Research targetConstruct the exact endpoint block rearrangement or singular baker map.open
  • Research targetSew the finite strips, corridors, seams, and coordinate axes.open
  • Research targetProve global injectivity and then separate benchmark scope from the weighted problem.open
  • Research targetRelative anisotropic strip foldingsuperseded
  • Narrowed routeSource-reported limitationVolume matching, full-measure free-locus packing, and an equimeasurable shell profile do not by themselves provide a connected symplectic embedding or restore Liouville-period-compatible seams. The decisive missing theorem is a relative anisotropic strip-folding or direct singular baker construction that turns the actual two-block source remainder into one connected strict embedding with matched flux and seams.
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 bridgeConstruct the exact endpoint block rearrangement or singular baker map.

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.

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Prepared starting pointConstruct the exact endpoint block rearrangement or singular baker map.

Higher-Dimensional Symplectic Ball-Packing Conjecture · ready to start

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Research contextPrepared context for any AI agent

For symplectic balls in real dimension at least six, are total volume and the pairwise two-ball nonsqueezing inequalities the only obstructions to packing them strictly into a larger ball?

  • Exact question and boundaries
  • 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.

  1. 1
    On Symplectic Packing Problems in Higher Dimensionsoriginal source · Kyler Siegel, Yuan Yao · arXiv · 2023-12-20 · ARXIV 2312.13224 · DOI 10.48550/arXiv.2312.13224 · accessed Aug 14, 2026
  2. 2
    On symplectic packing problems in higher dimensionspeer reviewed result · Kyler Siegel, Yuan Yao · Mathematische Annalen 392, 5361–5392 · 2025-07-12 · ARXIV 2312.13224 · DOI 10.1007/s00208-025-03221-7 · accessed Aug 14, 2026

Important qualifications

  • The conjecture was formulated in the reviewed 2023 preprint and 2025 journal article; the proposal year follows the first arXiv posting.
  • The journal article’s Conjecture A uses closed balls embedded into the interior, while the workspace states an equivalent strict neighborhood-of-closed-balls convention after shrink; that alignment requires ordinary statement review.
  • Stabilized packing theorems and the dimension-four classification are neighboring results, not a proof of Conjecture A.
  • The scoped search found no public problem-level formalization or canonical computation; it was not exhaustive.
  • The submitted packet and its URLs were excluded from external authority and were not fetched.

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