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Symplectic Ball Packings in Higher Dimensions
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Abstract
We prove that, for all integers n ≥ 3 and k ≥ 1 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 $\displaystyle \sum_{i=1}^k R_i^n\lt R^n, \qquad R_i+R_j\lt R\quad(i\ne j).$ 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.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Symplectic Zoominar, Working groups on OpenAI papers (updated 2026-10-19). Announces a working group on higher-dimensional symplectic ball packing, whose seminar page names Dan Cristofaro-Gardiner as organizer, without reporting an assessment of the manuscript.
Formalization
- OpenAI math release, Symplectic ball packing in higher dimensions (Lean scope notes) (lean/docs/343.md, release of 2026-10-06). The release's scope note says the Lean development states and proves the full criterion of Theorem 1.1 (volume and pairwise inequalities, closed balls, $n\ge3$) as well as the necessity direction, via the comparator files
BallPacking.leanandBallPackingNecessity.lean. [AI-generated per source] - OpenAI, lean/OAI/Geometry/BallPacking/Main.lean (github.com/openai/math, commit adc7f12). The top-level Lean file states
main_theorem, the equivalence ofHasPackingandPackingInequalitiesfor $n\ge3$, $k\ge1$, as a wrapper around imported rigidity and packing modules. [AI-generated per source]
Related prior literature mentioned by others
- Kyler Siegel and Yuan Yao, On symplectic packing problems in higher dimensions (Mathematische Annalen 392, 5361–5392, 2025). Identified in Prove2Me's discussion of this manuscript as the source of Conjecture A, this paper formulates the volume-and-two-ball criterion claimed in Theorem 1.1 and proves separate rigidity results for packings stabilized by closed surfaces.
Possibly relevant literature (found by the agent, not checked)
- D. Cristofaro-Gardiner and R. Hind, Boundaries of open symplectic manifolds and the failure of packing stability (arXiv:2307.01140, 2023). The paper builds open domains in $\mathbb{R}^4$ where packing stability fails, which adds context to the introduction's survey of regimes in which volume is the only packing obstruction.
- D. Cant, Hamiltonian linking and Symplectic packing (arXiv:2507.01416, 2025). A short note connecting packings of the standard symplectic ball by two sets with Hamiltonian linking, which relates to the two-ball obstruction used in the necessity direction of Theorem 1.1.
Digests and expositions
- Latent Seminar, Symplectic ball packings in higher dimensions (2026-10-07). Explains the capacity criterion, Hamiltonian rearrangement, surface-handle construction and three geometric cases, explicitly attributing the conclusions to the manuscript and disclaiming independent verification of the complete proofs.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 343
Independent sites; their content is not reviewed here.
Cite
@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 provides Lean files for this family: lean/docs/343.md. The Lean statements have not been checked by people against the paper's claims.
Family 343: Symplectic ball packing in higher dimensions
Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension $2n\ge6$. Finitely many closed symplectic balls of capacities $R_1,\ldots,R_k$ embed disjointly into an open ball of capacity R exactly when $\sum_iR_i^n\lt R^n$ and $R_i+R_j\lt R$ for every distinct pair i, j.
This is the only paper in this family.