OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Proves that any finite sequence in the Euclidean unit ball of ℝ^(d) admits signs keeping every partial sum within C\sqrt d, independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order S_2(d)=Θ(\sqrt d).
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MANUSCRIPT 01
The Euclidean Steinitz–Bergström theorem
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝ^(d) has one signing for which every signed prefix has norm at most C\sqrt d, with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its…
Full source abstract
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝ^(d) has one signing for which every signed prefix has norm at most C\sqrt d, with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, S_2(d)=Θ(\sqrt d), and resolves the Euclidean Steinitz–Bergström conjecture.
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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.