The Euclidean Steinitz–Bergström bound

What OpenAI claims

Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within C√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 S₂(d)=Θ(√d). (Lean)

Paper

1. The Euclidean Steinitz–Bergström theorem

   24 September 2026 · Preprint, not peer reviewed · PDF

   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√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, S₂(d)=Θ(√d), and resolves the Euclidean Steinitz–Bergström conjecture.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.