Prescribed-order Euclidean prefix discrepancy O(sqrt d), independent of length (Bansal-Jiang-Meka-Singla-Sinha Conjecture 6.3)

Theorem 1.1: an absolute constant $C$ such that every finite sequence in the Euclidean unit ball of $\mathbb R^d$, in prescribed order, has one signing with every prefix of norm at most $C\sqrt d$, independent of $N$. This removes the $\sqrt{\log N}$ term from Banaszczyk's bound and is optimal up to the constant. It is existential: no efficient or online algorithm is given, so the algorithmic and online versions of the question remain open.

Theorem 1.1: an absolute constant $C$ such that every finite sequence in the Euclidean unit ball of $\mathbb R^d$, in prescribed order, has one signing with every prefix of norm at most $C\sqrt d$, independent of $N$. This removes the $\sqrt{\log N}$ term from Banaszczyk's bound and is optimal up to the constant. It is existential: no efficient or online algorithm is given, so the algorithmic and online versions of the question remain open.