OpenAI · 097-01 · 2026-09-24 · Convex and metric geometry
The Euclidean Steinitz–Bergström theorem
preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026
Family 097. The Euclidean Steinitz–Bergström conjecture.
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
Proves that any finite sequence in the Euclidean unit ball of admits signs keeping every partial sum within , 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 .
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 , 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, , and resolves the Euclidean Steinitz–Bergström conjecture.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 097: The Euclidean Steinitz–Bergström bound. 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.