Symplectic ball packing in higher dimensions

1 · The essentials

This result asks when several balls can fit without overlap inside a larger ball while respecting a particular geometric structure. The allowed ways of fitting them must preserve that structure, not merely keep the balls from overlapping. In spaces of dimension six or higher with an even number of dimensions, the answer is given by two strict size tests. First, the combined volume of the smaller balls must be less than the volume of the larger one. Second, the sizes of every two smaller balls must add to less than the size of the larger ball, using a size measure called capacity: pi times the radius squared. Together, these tests are both necessary and sufficient for any finite number of balls with positive sizes. The smaller balls include their boundaries and must fit strictly inside the larger ball.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, think of placing several objects into a container under special handling rules: having enough total room is one check, while checking every pair of objects is another. Here those two checks completely decide whether a placement is possible, but the special geometric rules are not the rules for packing ordinary physical objects.

Why it matters

The result gives an exact criterion rather than only a necessary restriction or a sufficient recipe. The manuscript identifies it as a resolution of Siegel and Yao’s Conjecture A. Its stated scope includes arbitrary positive capacities and any finite number of balls in every even dimension at least six.

2 · Going further

The precise problem is to find disjoint symplectic embeddings of finitely many closed standard balls of dimension $2 n$ into an open ball of the same dimension. For integers $n \geq 3$ and $k \geq 1$ , let the source capacities be $R_{1}, \dots, R_{k} > 0$ and the target capacity be $R > 0$ . Capacity is normalized as $π$ times the squared Euclidean radius, and each embedding must be defined on a neighborhood of its closed source ball.

The criterion is exactly $\sum_{i = 1}^{k} R_{i}^{n} < R^{n}$ together with $R_{i} + R_{j} < R$ for every distinct pair $i, j$ . The first inequality is the total-volume condition expressed in capacities; the second is a separate pairwise condition. Both inequalities are strict, in accordance with the stated problem of placing closed balls inside an open target. When there is only one source ball, there are no distinct pairs to check.

The manuscript identifies this equivalence as Siegel and Yao’s Conjecture A, and the repository summary describes its resolution for arbitrary capacities in every dimension $2 n \geq 6$ . The supplied abstract states the result but gives no construction or main idea of the argument, so an explanation of the proof mechanism is not available from this source. Only the main manuscript is supplied; no companion manuscripts or relationships among them are described.

Useful notions

Euclidean balls and their radii
Open and closed sets
Strict inequalities and finite sums
Volume in higher dimensions
Symplectic structures and embeddings

What was known before / what is new

Before

The source identifies the packing criterion as Conjecture A of Siegel and Yao. It does not describe earlier partial results, previously settled capacities, or the methods that preceded this manuscript.

New

The stated result makes the volume and pairwise conditions jointly sufficient as well as necessary for all positive capacities in dimensions $2 n \geq 6$ . It covers every finite number of source balls, with embeddings defined on neighborhoods of the closed balls. The main result has a Lean formalization.

Limits and assumptions

The statement assumes an even dimension $2 n$ with $n \geq 3$ , a finite number $k \geq 1$ of standard source balls, and positive source and target capacities. It concerns closed source balls embedded disjointly into an open target ball, with each embedding defined on a neighborhood of its source. Equality in either required inequality does not satisfy the criterion. The main result has a Lean formalization, but the supplied material does not specify its detailed scope, so verification of every explanatory claim should not be inferred. The source gives no result for lower dimensions or other source and target shapes, and it lists no remaining open problems.

Mini glossary

Capacity: For the standard balls in this statement, capacity is $π$ times the squared Euclidean radius.
Symplectic embedding: An embedding that preserves the symplectic geometric structure.
Closed ball: A ball that includes its boundary.
Open ball: A ball that excludes its boundary.
Pairwise condition: The requirement $R_{i} + R_{j} < R$ for each pair of distinct source balls.

Keywords : Symplectic geometry · Ball packing · Capacity

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension $2 n \geq 6$ . Finitely many closed symplectic balls of capacities $R_{1}, \dots, R_{k}$ embed disjointly into an open ball of capacity R exactly when $\sum_{i} R_{i}^{n} < R^{n}$ and $R_{i} + R_{j} < R$ for every distinct pair i, j.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/343.md

Manuscripts in this family

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Symplectic Ball Packings in Higher Dimensions

23/09/2026

To be verified

We prove that, for all integers n ≥ 3 and k ≥ 1 and all positive real capacities $R_{1}, \dots, R_{k}$ , the closed standard symplectic $2 n$ -balls of these capacities embed disjointly into the interior of a ball of capacity R > 0 if and only if $i = 1 \sum k R_{i}^{n} < R^{n}, R_{i} + R_{j} < R (i \neq = j) .$ 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.

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