OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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< R^n and R_i+R_j< R for every distinct pair i, j.
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MANUSCRIPT 01
Symplectic Ball Packings in Higher Dimensions
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< R^n, R_i+R_j< R(i\ne j). Capacity is π times the squared…
Full source 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< R^n, R_i+R_j< R(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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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.