Symplectic ball packing in higher dimensions

The explanation

Take a six-dimensional ball of capacity 1, the target. Capacity is π times the squared radius. Try to fit two smaller balls of capacity 0.6 inside it. Their volumes compare as cubes in six dimensions, so together they need 2 × 0.6³ = 0.432 of the target's volume. By volume alone they fit. But a symplectic map must keep a special kind of area, not just volume. Under that rule, two balls fit only if their capacities add to less than 1. Here 0.6 + 0.6 = 1.2, so they do not fit. Now try eight balls of capacity 0.49. Each pair adds to 0.98, below 1. The volumes add to 8 × 0.49³, about 0.941. The paper says these eight balls do fit, filling about 94% of the volume. Nine such balls would need about 1.059, too much.

The setting is a symplectic space, a space with a rule for measuring signed area in pairs of directions. A symplectic embedding is a map that keeps that rule. The question is when several balls embed this way, without overlap, inside a bigger ball. The first test is volume. The second is Gromov's two-ball obstruction: any two capacities must add to less than the targetSource 1. In dimension four, other obstructions also matter, so these two tests are not enough thereSource 2. Siegel and Yao conjectured that in higher dimensions they are enoughSource 3.

Earlier work showed volume suffices in special regimes. Biran showed this for many equal balls in dimension four. Buse and Hind extended it to higher dimensionsSource 4. Buse, Hind and Opshtein did the same for unequal balls when all are very smallSource 5.

The paper proves the full criterion for n at least 3. Balls of capacities R1 to Rk fit in the open ball of capacity R under two tests. The n-th powers of the capacities must add to less than Rⁿ. Each pair must add to less than R. The paper proves these two tests are exactly rightSource 6. The key obstacle is that the volume test is only an average over a region. It does not say the leftover area is positive at every point, so the balls cannot simply be stackedSource 7. The paper fixes this with a Hamiltonian comparison, a controlled motion from classical mechanics that rearranges the leftover areaSource 8. It builds its models over a curved surface with a handle, using tools from algebraic geometry. The proof splits into three cases, by whether one capacity is at least half the target and whether n is 3Source 9.

How sure is this? Lean, a computer proof checker, has formal statements of the full criterion for every n at least 3. It also states the necessity direction alone. So the paper proves this result, as far as the Lean statements match the paper. No specialist review is published.

What it doesn't show. This does not cover dimension four, where extra obstructions are known to matter. It also does not give an explicit formula for the embeddings.