Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 097-01. The Euclidean Steinitz–Bergström theorem
Manuscript map · lines 4345–4354
OpenAI@2026-10-07#rem:oai-p-097-01
Authors: OpenAI. Family 097.
Title: The Euclidean Steinitz–Bergström theorem.
Abstract: Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝd 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, $S_2(d)={}Theta({}sqrt d)$, and resolves the Euclidean Steinitz–Bergström conjecture.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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