OpenAI · 343-01 · 2026-09-23 · Differential geometry
Symplectic Ball Packings in Higher Dimensions
preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026
Family 343. Sharp symplectic ball-packing criteria in higher dimensions.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension . Finitely many closed symplectic balls of capacities embed disjointly into an open ball of capacity exactly when and for every distinct pair .
We prove that, for all integers n ≥ 3 and k ≥ 1 and all positive real capacities , the closed standard symplectic -balls of these capacities embed disjointly into the interior of a ball of capacity R > 0 if and only if 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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 343: Symplectic ball packing in higher dimensions. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.