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. (Lean)
Manuscript
Symplectic Ball Packings in Higher Dimensions
September 23, 2026 · source · cite as \cite{OAI:Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026}
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.
Result 343 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md