OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed \(0\lt \delta\lt 1/3\), a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size \(\delta n\) otherwise, where n is the number of vertices. (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.
For every fixed real 0<\delta<1/3, there is a deterministic polynomial-time algorithm R_\delta that maps each explicitly encoded Boolean 3-CNF formula \phi to a nonempty finite simple undirected unweighted graph G_\phi such that
\begin{array}{ll}
\phi\text{ is satisfiable}
&\Longrightarrow G_\phi\text{ is three-colorable},\\[2pt]
\phi\text{ is unsatisfiable}
&\Longrightarrow \alpha(G_\phi)<\delta\,|V(G_\phi)|.
\end{array}
The running time and explicit output size are polynomial in the ordinary binary encoding length of \phi. Their constants and polynomial degree may depend on the fixed \delta, but not on \phi. The coloring in the first implication covers every vertex and is not supplied with the graph.
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every fixed pair of integers 3\le k\le c, it is NP-hard to distinguish k-colorable graphs from graphs that are not c-colorable.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
Reported proof verification
One Lean statement in the release; one independent group reports that Comparator accepted a run for this file at the version shown here.
We prove that, for every fixed \(0\lt \delta\lt 1/3\), it is NP-hard to distinguish three-colorable graphs from graphs in which every independent set has fewer than δ times the number of vertices. Consequently, for every fixed integer c ≥ 3, finding a proper c-coloring of a three-colorable graph is NP-hard.
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.
For every fixed real 0<\delta<1/3, there is a deterministic polynomial-time algorithm R_\delta that maps each explicitly encoded Boolean 3-CNF formula \phi to a nonempty finite simple undirected unweighted graph G_\phi such that
\begin{array}{ll}
\phi\text{ is satisfiable}
&\Longrightarrow G_\phi\text{ is three-colorable},\\[2pt]
\phi\text{ is unsatisfiable}
&\Longrightarrow \alpha(G_\phi)<\delta\,|V(G_\phi)|.
\end{array}
The running time and explicit output size are polynomial in the ordinary binary encoding length of \phi. Their constants and polynomial degree may depend on the fixed \delta, but not on \phi. The coloring in the first implication covers every vertex and is not supplied with the graph.
The words they share: 1/3, real, undirected, three, length, unweighted, vertex, finite, satisfiable, coloring, maps, size, binary, simple, time, polynomial, output, explicit, formula, nonempty, graph, independent, delta, unsatisfiable, deterministic.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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.
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:Independent-set gap reduction stated for every 0<delta<1/3; the ‘NP-hard to c-color a 3-colorable graph’ corollary is not separately stated (one-line consequence).
Definitions the group asks readers to compare with the paper:Same Mathlib TM2ComputableInPolyTime + finiteAlphabets encoding; reduce is only constrained on formulaBits φ inputs (soundness/completeness quantify over φ : Formula, not all bit strings); the proof cone contains opaque parameter constants (e.g. degreeData, fixedWalkParameter), which are sound (arbitrary inhabitants, not axioms)
“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 formalized result shows hardness of finding large independent sets even under a three-colorability promise. For every fixed $0<\delta<1/3$, a deterministic polynomial-time reduction maps binary 3SAT formulas to nonempty finite simple graphs. Satisfiable formulas produce three-colorable graphs; unsatisfiable formulas produce graphs whose largest independent set has fewer than $\delta$ times the number of vertices. The complete adjacency-matrix output is included in the runtime bound.
AI-drafted comparisons, with no recorded human review
Independent-set hardness in three-colorable graphsIndependentSets
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for fixed δ in (0, 1/3), a polynomial-time reduction from 3SAT producing three-colourable graphs from satisfiable formulas and graphs with no independent set of δ times the vertex count from unsatisfiable ones. The colouring corollary is not part of the statement.
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.
Independent-set hardness in three-colorable graphsIndependentSets
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“REFUSED”
Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
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
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 proof 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 106The 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 did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
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.