Remark: Family 097. The Euclidean Steinitz–Bergström conjecture

Proves that any finite sequence in the Euclidean unit ball of ${\mathbb{R}}^d$ 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 $S_2(d)=\Theta (\sqrt{d})$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.