# Symplectic ball packing in higher dimensions > Proves Siegel and Yao's conjecture: from dimension six, ball packings are limited only by volume and the two-ball test. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-343/ Last updated: 2026-10-14 **Proves Siegel and Yao's conjecture: from dimension six, ball packings are limited only by volume and the two-ball test.** **Trust:** Formal proof listed · **Papers:** 1 · **First paper:** September 23, 2026 · **Field:** Differential geometry · **Entry:** 343 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. A symplectic embedding moves a shape while keeping a special kind of area, not just volume. Gromov found that two balls can fail to fit in a bigger ball even when their volumes are small enough. Siegel and Yao conjectured that, from dimension six up, volume and Gromov's two-ball test are the only limits. This paper proves that conjecture for any number of balls of any sizes. Lean, a computer proof checker, has formal statements of this criterion. ## At a glance - **First step:** 1985 (Mikhail Gromov, whose two-ball obstruction started symplectic packing rigidity) - **Significance:** Major. It gives, with Lean statements, a complete packing criterion for balls in all dimensions from six up, settling a conjecture of Siegel and Yao. - **Trust:** Formal proof listed - **Field:** Differential geometry (1 paper) ## 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 target[1]. In dimension four, other obstructions also matter, so these two tests are not enough there[2]. Siegel and Yao conjectured that in higher dimensions they are enough[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 dimensions[4]. Buse, Hind and Opshtein did the same for unequal balls when all are very small[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 right[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 stacked[7]. The paper fixes this with a Hamiltonian comparison, a controlled motion from classical mechanics that rearranges the leftover area[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 3[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. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Major, with Lean statements of the full criterion. It covers dimension six and up only; dimension four has extra obstructions. No specialist review is published. ## Words used - **Symplectic space:** A space with a rule that measures signed area in pairs of directions. - **Symplectic embedding:** A map of one shape into another that keeps the symplectic area rule exactly. - **Capacity:** For a ball, π times the square of its ordinary radius. - **Packing:** Placing several shapes inside one larger shape with no overlaps. - **Gromov's two-ball obstruction:** Two balls fit symplectically in a ball only if their capacities add to less than the target's. - **Hamiltonian map:** A motion that follows the equations of classical mechanics for some energy function. - **Volume obstruction:** The simple test that the pieces' total volume must be less than the target's volume. ## The original paper - [Symplectic Ball Packings in Higher Dimensions](https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf) - September 23, 2026 - [Lean notes](https://github.com/openai/math/blob/main/lean/docs/343.md) ## Quoted lines from the paper (9) [1] From [Symplectic Ball Packings in Higher Dimensions](https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf): > The first inequality is the volume obstruction. The second is Gromov's two-ball obstruction; it expresses a restriction invisible to volume alone. [2] From [Symplectic Ball Packings in Higher Dimensions](https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf): > In real dimension four, additional obstructions remain essential, so the two inequalities in Theorem … do not give a criterion there. [3] From [Symplectic Ball Packings in Higher Dimensions](https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf): > Theorem … proves that these obstructions are complete in every real dimension at least six, resolving Siegel and Yao's Conjecture A positively. [4] From [Symplectic Ball Packings in Higher Dimensions](https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf): > Buse and Hind extended equal-ball stability to higher-dimensional balls and projective spaces and then to all closed rational symplectic manifolds. [5] From [Symplectic Ball Packings in Higher Dimensions](https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf): > For unequal balls, Buse, Hind, and Opshtein proved strong packing stability on every closed symplectic four-manifold, with all capacities required to be sufficiently small. [6] From [Symplectic Ball Packings in Higher Dimensions](https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf): > We prove that, for all integers \(n\ge3\) and \(k\ge1\) and all positive real capacities \(R_1,\ldots,R_k\), the closed standard symplectic \(2n\)-balls of these capacities embed disjointly into the interior of a ball of capacity \(R>0\) if and only if \[ \sum_{i=1}^k R_i^n This integral condition need not make \(P\) positive at each moment, which is the obstacle to simply stacking the balls. [8] From [Symplectic Ball Packings in Higher Dimensions](https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf): > Section … overcomes that obstacle by Hamiltonian comparison. [9] From [Symplectic Ball Packings in Higher Dimensions](https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf): > The pairwise inequalities leave three cases, summarized in Figure …. ## Other sources - [openai/math, Lean scope note for family 343](https://github.com/openai/math/blob/main/lean/docs/343.md): “The formalization proves Siegel and Yao's ball-packing criterion in every symplectic dimension $2n$ with $n\ge3$.” - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [A counterexample to the nearby Lagrangian conjecture](https://closedform.nishachay.tech/e/openai-math-2026-340/) - Differential geometry - [Katok's entropy rigidity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-339/) - Differential geometry - [Counterexamples to stable-Morse and strong Arnold fixed-point bounds](https://closedform.nishachay.tech/e/openai-math-2026-347/) - Differential geometry - [Yau's uniformization conjecture](https://closedform.nishachay.tech/e/openai-math-2026-338/) - Differential geometry - [Spectral scalar curvature, Urysohn width, and macroscopic dimension](https://closedform.nishachay.tech/e/openai-math-2026-336/) - Differential geometry - [Differential geometry](https://closedform.nishachay.tech/science/mathematics/differential-geometry/) - Field · 29 results - [Distances and packing](https://closedform.nishachay.tech/topic/distances-and-packing/) - Topic · 5 results - [Open since the 1980s](https://closedform.nishachay.tech/open-since/1980s/) - Era · 60 results - [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20343%5D%20)