we-solve (beta)

CS-165-A1 · revision r1 · on CS-165

OpenAI claim summary for CS-165

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 124; 2026-09-24).

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

Manuscript

Contents
  1. Polynomial-time three-machine unit-job scheduling
  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

Polynomial-time three-machine unit-job scheduling

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 124; 2026-09-24), the asserted result is that precedence-constrained unit jobs on three identical machines admit an exact deterministic polynomial-time scheduling algorithm.

Mathematical statement of the claimed result

Manuscript 1.
For every explicitly listed finite precedence DAG on $n\ge1$ nonpreemptive unit-duration jobs and three identical machines, the claimed uniform deterministic algorithm produces a schedule with globally minimum makespan. With a supplied integer deadline $1\le T\le n$, it decides feasibility exactly and constructs a feasible schedule in the affirmative case. If $L$ is total binary input length, both tasks obey \[ \operatorname{time}\le O((L+2)^{150020}) \] on a deterministic multitape Turing machine.

Outline of the argument

For manuscript 1 (family 124; 2026-09-24), feasible schedules are reorganized into recursively marked intervals. Global source, sink, and boundary predicates retain only bounded lists after exchanges and pruning, so a dynamic program enumerates polynomially many descriptions while preserving makespan feasibility.

Formal verification

For manuscript 1 (family 124; 2026-09-24), OpenAI lists Lean statements for this assertion: precedence-constrained unit jobs on three identical machines admit an exact deterministic polynomial-time scheduling algorithm.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are ThreeMachine.lean.

Their named declarations include OAI.ThreeMachine.main_theorem.

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