Result 343, Differential geometry

Symplectic ball packing in higher dimensions

Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension 2n≥62n\ge6. Finitely many closed symplectic balls of capacities R1,…,RkR_1,\ldots,R_k embed disjointly into an open ball of capacity R exactly when ∑iRin<Rn\sum_iR_i^n\lt R^n and Ri+Rj<RR_i+R_j\lt R for every distinct pair i, j.

Lean formalization Proof

The bigger picture

Why it matters

Symplectic packing asks how much can fit inside a ball when deformations must preserve a prescribed area-measuring structure. The manuscript claims that, in even dimensions six and higher, two types of numerical checks completely decide ball packing.

What changes?

The manuscript reports an exact criterion in dimension 2n for every integer n at least three and every finite nonempty collection of positive capacities. Capacity is pi times squared radius. Closed symplectic balls embed disjointly inside an open ball precisely when their capacities' nth powers sum to less than the target capacity's nth power, and every pair's capacities sum to less than the target capacity. Each embedding must extend to a neighborhood of its closed source ball.

What does that help mathematicians do?

The claimed criterion separates available volume from a genuinely symplectic restriction. Two balls each with capacity 60 percent of the target pass the volume test in every covered dimension, yet fail the pairwise test. Conversely, the result says that satisfying both tests leaves no further obstruction for these ball packings. The strict inequalities also rule out boundary cases where either test reaches equality.

Are there practical applications?

Its immediate value is foundational: it turns a geometric existence question into explicit numerical checks for this particular packing problem. Researchers can use the claimed criterion to distinguish feasible higher-dimensional ball configurations from impossible ones. The abstract does not describe a computational construction of the embeddings or a physical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Symplectic Ball Packings in Higher Dimensions

September 23, 2026 43 pages

We prove that, for all integers n ≥ 3 and k ≥ 1 and all positive real capacities R1,…,RkR_1,\ldots,R_k, the closed standard symplectic 2n2n-balls of these capacities embed disjointly into the interior of a ball of capacity R > 0 if and only if

∑i=1kRin<Rn,Ri+Rj<R(i≠j).\displaystyle \sum_{i=1}^k R_i^n\lt R^n, \qquad R_i+R_j\lt R\quad(i\ne j).

Capacity is π times the squared Euclidean radius, and each embedding is defined on a neighborhood of its closed source ball. This proves Siegel and Yao's Conjecture A.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026,
  author = {{OpenAI}},
  title = {{Symplectic Ball Packings in Higher Dimensions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf}{OAI:Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/343.md.

Symplectic ball packing in higher dimensions

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves Siegel and Yao's ball-packing criterion in every symplectic dimension 2n2n with n≥3n\ge3. For k≥1k\ge1 closed standard balls of positive capacities R1,…,RkR_1,\ldots,R_k, disjoint symplectic embeddings into the interior of a ball of capacity RR exist exactly when ∑iRin<Rn\sum_i R_i^n<R^n and Ri+Rj<RR_i+R_j<R for all i≠ji\ne j. Capacity is π\pi times squared Euclidean radius, and each embedding is defined on a neighborhood of its closed source ball. The linked statements include both the full equivalence and the necessity direction.

Comparator links

Result Comparator statement
Exact criterion for higher-dimensional symplectic ball packings BallPacking.lean
Necessity of the volume and pairwise capacity inequalities BallPackingNecessity.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.