we-solve (beta)

CS-158-A1 · revision r1 · on CS-158

OpenAI claim summary for CS-158

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 110; 2026-09-24).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Randomized k-server with squared-logarithmic ratio
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Randomized k-server with squared-logarithmic ratio

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 110; 2026-09-24), the asserted result is that randomized $k$-server on every metric has competitive ratio of order $\log^2(k+1)$ against oblivious requests, using one policy for every finite sequence.

Mathematical statement of the claimed result

Manuscript 1.
For each $k\ge2$, arbitrary metric $(X,d)$ with at least $k+1$ points, and initial server tuple $s\in X^k$, the claim provides one randomized online $k$-server policy $A$ and finite $B\ge0$ such that every fixed finite request sequence $\sigma$ satisfies \[ \mathbb E\operatorname{cost}_{A,s}(\sigma) \le C(\log(k+1))^2\operatorname{OPT}_{X,d,k,s}(\sigma)+B, \] with absolute $C$. The adversary is oblivious to policy randomness. Additive $B$ may depend on the metric, $k,s$ but not on the sequence or horizon; distinct initial positions allow $B=0$. No computational-efficiency bound is asserted by this existence statement.

Outline of the argument

For manuscript 1 (family 110; 2026-09-24), a hidden comparator is tracked by its posterior measure, separating changes of true metric mass from allocation costs. Protected anchors and a tree of partition labels control travel and edits, with two telescoping logarithmic budgets accounting for the final ratio.

Formal verification

For manuscript 1 (family 110; 2026-09-24), OpenAI lists Lean statements for this assertion: randomized $k$-server on every metric has competitive ratio of order $\log^2(k+1)$ against oblivious requests, using one policy for every finite sequence.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are KServer.lean, UniformKServer.lean.

Their named declarations include OAI.KServer.main_theorem, OAI.UniformKServer.uniform_squared_logarithmic_main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources