Explicit full symplectic packings of the 4-ball

Give one explicit Hamiltonian construction that packs k ≥ 10 equal balls into a 4-ball, filling all but ε of its volume, for every ε > 0.

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1	# Explicit full symplectic packings of the 4-ball
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3	Symplectic maps preserve volume but are far more rigid, so packing balls symplectically is subtle. In dimension four it is known that $k$ equal balls can fill all but an arbitrarily small fraction of a larger ball whenever $k \ge 10$, but the proof gives no picture of such packings. The task is to construct them explicitly, by one algorithm that works for every $k \ge 10$ and every tolerance.
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5	## Problem
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7	Work in $\mathbb{R}^4$ with its standard symplectic form. For $i = 1, \dots, k$ let $B_i$ be the closed unit ball centred at $(3i - 3, 0, 0, 0)$, so that the $B_i$ are disjoint unit balls. Let $B(R)$ be the closed ball of radius $R$ centred at the origin.
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9	Task. Give one algorithm that, for every integer $k \ge 10$ and every $\varepsilon \in (0, 1)$, outputs an explicit smooth Hamiltonian $H_\varepsilon \colon \mathbb{R}^4 \times [0, 1] \to \mathbb{R}$ whose time-1 flow $\phi$ satisfies
10	$$\phi\Bigl(\bigsqcup_{i=1}^{k} B_i\Bigr) \subset B(R) \qquad \text{with} \qquad \frac{k}{R^4} > 1 - \varepsilon,$$
11	so that the $k$ unit balls fill more than a fraction $1 - \varepsilon$ of the volume of $B(R)$.
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13	## Background
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15	McDuff and Polterovich related symplectic packings of the 4-ball to algebraic geometry, including Nagata's conjecture [1]. Biran's work then showed that full packings exist for large numbers of equal balls [2]; the proof uses Seiberg–Witten theory and symplectic inflation and is not constructive. Schlenk's survey stresses this contrast with Euclidean packing: Euclidean packing numbers are usually known only once a maximal packing has been seen, whereas for many known symplectic packing numbers nobody has an idea what a corresponding packing looks like [3].
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17	## What is known
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19	- In dimension four, $k$ equal balls fill the volume of a ball up to any $\varepsilon > 0$ for every $k \ge 10$, by non-explicit methods [1, 2] (as summarised by Epoch's problem write-up).
20	- For $k \le 9$ the optimal packings have explicit constructions; for $k \ge 10$ none are known.
21	- Epoch's write-up notes that a single approximate embedding for one fixed $\varepsilon$ might be found by numerical optimisation, but that one algorithm valid for all $\varepsilon$ is unlikely to have a brute-force solution and would be a significant advance.
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23	## What would count as progress
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25	- An algorithm $(k, \varepsilon) \mapsto H_\varepsilon$, ideally with a proof that the flow embeds the balls as claimed.
26	- Epoch scores submissions with its own verifier: `solution(k, epsilon)` must return the source of a function `Hamiltonian(Q, P, t)` written in PyTorch. The verifier integrates the flow of points sampled on the ball boundaries with a Runge–Kutta method, checks numerically that the flow is symplectic, measures $R$, and tests $k/R^4 > 1 - \varepsilon$ for several $k \ge 10$.
27	- An explicit family for the single case $k = 10$ and all $\varepsilon$ would already be a significant result.
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29	## References
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31	1. D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry, Invent. Math. 115 (1994), 405–429. [doi:10.1007/BF01231766](https://doi.org/10.1007/BF01231766)
32	2. P. Biran, Symplectic packing in dimension 4, Geom. Funct. Anal. 7 (1997), 420–437. [doi:10.1007/s000390050014](https://doi.org/10.1007/s000390050014)
33	3. F. Schlenk, Symplectic embedding problems, old and new, Bull. Amer. Math. Soc. 55 (2018), 139–182. [doi:10.1090/bull/1587](https://doi.org/10.1090/bull/1587)
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35	## Source
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37	Imported from [FrontierMath: Open Problems, symplectic-ball-packing](https://epoch.ai/frontiermath/open-problems/symplectic-ball-packing), curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.