Symplectic ball packing in higher dimensions

What OpenAI claims

Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension 2n ≥ 6. Finitely many closed symplectic balls of capacities R₁,…,R_k embed disjointly into an open ball of capacity R exactly when ∑_iR_iⁿ < Rⁿ and R_i+R_j < R for every distinct pair i, j. (Lean)

Paper

1. Symplectic Ball Packings in Higher Dimensions

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove that, for all integers n ≥ 3 and k ≥ 1 and all positive real capacities R₁,…,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

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.