Siegel-Yao Conjecture A: symplectic packings of balls into a ball in dimension at least six

Theorem 1.1: for $n\ge3$ , $k\ge1$ and positive $R,R_1,\dots,R_k$ , the closed balls $B^{2n}(R_i)$ embed disjointly and symplectically into $\mathrm{Int}\,B^{2n}(R)$ if and only if $\sum_iR_i^n<R^n$ and $R_i+R_j<R$ for $i\ne j$ . The construction uses a positive-genus surface base with toric fibres and a Hamiltonian comparison lemma. Not shown: anything in dimension four, where more obstructions exist, or the stabilized packing problems Siegel and Yao also study.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Siegel-Yao Conjecture A: symplectic packings of balls into a ball in dimension at least six · Siegel-Yao ball packing conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Siegel-Yao Conjecture A: symplectic packings of balls into a ball in dimension at least six

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Conjecture A as the manuscript cites it; the theorem states the conjectured criterion as an equivalence for all $n\ge3$ , $k\ge1$ and positive capacities. Lean: the challenge BallPacking (theorem OAI.HigherDimensionalBallPacking.main_theorem, solution module OAI.Geometry.BallPacking.Main) exists at the pinned commit but is not in the formalization catalogue; its statement was read here. It defines smooth symplectic embeddings on open neighbourhoods of the closed balls with disjoint images in the open target ball and states the iff with the two inequalities, which is the headline. Not rebuilt here.

Source ↗

Siegel-Yao Conjecture A: symplectic packings of balls into a ball in dimension at least six

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean formalization: BallPacking/Main.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Siegel-Yao, On symplectic packing problems in higher dimensions (Math. Ann. 2025)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean formalization: BallPacking/Main.lean

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

This event attributed to Unreleased internal OpenAI model

Siegel-Yao Conjecture A: symplectic packings of balls into a ball in dimension at least six evidence for this event

Lean formalization: BallPacking/Main.lean evidence for this event

VibeMathed record: Siegel-Yao Conjecture A: symplectic packings of balls into a ball in dimension at least six evidence for this event

Siegel-Yao, On symplectic packing problems in higher dimensions (Math. Ann. 2025) evidence for this event

Siegel-Yao Conjecture A: symplectic packings of balls into a ball in dimension at least six parent of this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph