OpenAI · 110-01 · 2026-09-24 · Theoretical computer science
Squared-logarithmic randomized k-server on arbitrary metrics
preprints/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026
Family 110. Optimal-order randomized -server on arbitrary metrics.
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
Establishes a randomized competitive ratio for -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 at request , with a finite instance-dependent additive movement constant.
We prove that randomized k-server has competitive ratio on every metric space against oblivious request sequences, matching the known worst-case lower bound. For each metric and initial configuration, one policy works for all finite request sequences, including on infinite and unbounded spaces. When the initial server positions are distinct, no additive term is needed.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 110: Optimal-order randomized <i>k</i>-server on arbitrary metrics. 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.