Remark: Family 110. Optimal-order randomized k -server on arbitrary metrics

Establishes a randomized competitive ratio $O({\log }^2(k+1))$ for $k$-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on infinite unbounded metrics. On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and $\log (t+1)$ at request $t$, with a finite instance-dependent additive movement constant.

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.