Polynomial-time scheduling on three identical machines
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Let an explicitly listed finite directed acyclic graph specify the precedence constraints on n\ge1 nonpreemptive unit-length jobs on three identical machines. There is a uniform deterministic algorithm that constructs a feasible schedule of minimum makespan. Given also an integer deadline 1\le T\le n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L+2)^{150020}) steps on a deterministic multitape Turing machine, where L is the total binary input length.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We give a uniform deterministic polynomial-time algorithm for scheduling unit-length jobs with arbitrary precedence constraints on three identical parallel machines. The algorithm constructs a schedule of minimum makespan and decides exactly whether all jobs can finish by a specified deadline. The proof reorganizes feasible schedules into intervals whose job sets have descriptions of bounded size. A dynamic program searches a family containing polynomially many such descriptions. Global boundary conditions and simplification of inherited information keep the descriptions bounded throughout the decomposition.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
Let an explicitly listed finite directed acyclic graph specify the precedence constraints on n\ge1 nonpreemptive unit-length jobs on three identical machines. There is a uniform deterministic algorithm that constructs a feasible schedule of minimum makespan. Given also an integer deadline 1\le T\le n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L+2)^{150020}) steps on a deterministic multitape Turing machine, where L is the total binary input length.
The words they share: turing, integer, input, acyclic, length, steps, listed, returns, machine, deadline, feasible, time, finite, decides, exactly, jobs, minimum, binary, precedence, unit, constructs, identical, constraints, makespan, three, deterministic, algorithm, schedule.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
ThreeMachine.lean, the release lists it as a supporting statement.
defReleaseTheorem : Prop := ∃ (k q g : ℕ) (M : Machine k q g) (C : ℕ), 0 < C ∧ ∀ n (G : Instance n), 1 ≤ n → G.acyclic → ∀ deadline : Option ℕ, (∀ T ∈ deadline, 1 ≤ T ∧ T ≤ n) → ∃ out, CorrectOutput G deadline out ∧ M.Produces (encodeInput G deadline) out (C * ((encodeInput G deadline).length + 2) ^ 150020)⋮theoremmain_theorem : ReleaseTheorem := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note:None; the polynomial exponent (150020) is explicit, the machine is fixed before the instance.
Definitions the group asks readers to compare with the paper:Hand-rolled multi-tape Turing machine (Machine k q g, Configuration, Machine.step/run/Produces, tapes via Mathlib Turing.Tape (Fin (g + 3))) instead of Mathlib’s TM2ComputableInPolyTime; the formula’s time bound is on this model’s step count
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
The formalization gives a deterministic polynomial-time algorithm for scheduling nonempty collections of unit-length jobs with arbitrary acyclic precedence constraints on three identical parallel machines. It constructs a schedule of minimum makespan and decides exactly whether a valid specified deadline can be met. The algorithm is one fixed finite machine, and its running time is bounded by a polynomial in the binary input length.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a deterministic polynomial-time algorithm computing a minimum-makespan schedule for unit jobs with precedence constraints on three identical machines.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a algorithm and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 124The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state the headline as OpenAI wrote it.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Re-run through Comparator with Lean’s kernel: accepted (the check took 68 s; peak memory 6.2 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.