Independent-set hardness in three-colorable graphs
Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorem selected for verification:OAI.LargeIndependentSets.mainTheoremReal
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Definitions needed to read it
Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.
structureGraphwhere vertices : ℕ nonempty : 0 < vertices adj : Fin vertices → Fin vertices → Bool symm : ∀ u v, adj u v = adj v u loopless : ∀ u, adj u u = false
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: 1/3, real, undirected, three, length, unweighted, vertex, finite and 13 more. Found in the paper’s title, which its main theorem states: independent. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: delta, unsatisfiable and deterministic.
Theorem 1.1 of Hardness of finding large independent sets in three-colorable graphs
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.
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.
Other possible paper matches, closest wording first
Proposition 3.1 of Hardness of finding large independent sets in three-colorable graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: length, unweighted, finite, alphabets, size, binary and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: deterministic.
Theorem 2.1 of Hardness of finding large independent sets in three-colorable graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: completeness, reduction, satisfiable, alphabets, lists, time and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sat, unsatisfiable and deterministic.
Proposition 6.1 of Hardness of finding large independent sets in three-colorable graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: eq, bounds, vertex, map, lists, output and three more.
Corollary 1.2 of Hardness of finding large independent sets in three-colorable graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: colorable, integers and graphs.
Theorem 2.2 of Hardness of finding large independent sets in three-colorable graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: positive, integer and independent.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
One independent group reports that Comparator accepted a run for this Lean statement file: MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
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 (.github/workflows/openai-math-kernel.yml), one j…
No verdict reachedTheir term: “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.)).
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 106’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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. OAI/Combinatorics/IndependentSets/PCP/TableIteration.lean:38 degreeData, Gap.lean:128 fixedWalkParameter), which are sound (arbitrary inhabitants, not axioms)
OpenAI’s own note on what this Lean statement covers
“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.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
84 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 27 to its own declarations, 54 to Mathlib’s documentation. 15 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
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.
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.
Corollary
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.
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
15 local vocabulary declarations
Lines of Lean, without comments or blank lines
50
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Independent-set hardness in three-colorable graphs
Family 106: Hardness of coloring three-colorable graphs
Candidate pairing
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.
Hardness of finding large independent sets in three-colorable graphs
TeX label thm:main, build/source/sections/introduction.tex, line 12
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.