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.MinUncut.main
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theoremmain : Nonempty Reduction := bysorry
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.
structureOutputwhere vertices : ℕ adjacent : Fin vertices → Fin vertices → Bool symmetric : ∀ u v, adjacent u v = adjacent v u loopless : ∀ u, adjacent u u = false threshold : ℕ threshold_pos : 1 ≤ threshold
defOutput.bits (G : Output) : List Bool := nameBits G.vertices ++ ((List.finRange G.vertices).flatMap fun u => (List.finRange G.vertices).map fun v => G.adjacent u v) ++ nameBits G.threshold
reduce : ℕ → Formula → Output machine : Turing.TM2Computable inputBits Output.bits (fun p => reduce p.1 p.2) finiteAlphabet : ∀ k : machine.tm.K, Finite (machine.tm.Γ k) time : ℕ → Polynomial ℕ runtime : ∀ K φ, Turing.TM2OutputsInTime machine.tm ((inputBits (K, φ)).map machine.inputAlphabet.symm) (some (((reduce K φ).bits).map machine.outputAlphabet.symm)) ((time K).eval (formulaBits φ).length) size : ℕ → Polynomial ℕ output_size : ∀ K φ, (reduce K φ).vertices + (reduce K φ).bits.length ≤ (size K).eval (formulaBits φ).length yes : ∀ K, 2 ≤ K → ∀ φ, φ.Satisfiable → (reduce K φ).opt ≤ (reduce K φ).threshold no : ∀ K, 2 ≤ K → ∀ φ, ¬ φ.Satisfiable → K * (reduce K φ).threshold < (reduce K φ).opt
Possible paper match, with no recorded review
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: uncut, length, formula, bit, size, opt, satisfiable, finite and four more. Found in the paper’s title, which its main theorem states: min and cut. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: encoded, one, unsatisfiable, sat, simple, unweighted, constants and deterministic.
Statements in Constant-factor hardness of Min-UnCut come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1.1 of Constant-factor hardness of Min-UnCut
Theorem
For every fixed integer K\ge2, there is a deterministic polynomial-time many-one reduction mapping a 3SAT formula \varphi to a pair (G,k), where G is an explicit finite simple undirected unweighted graph and k\ge1 is a binary-encoded integer, such that
For each fixed K, the graph size, full construction work, and bit complexity are polynomial in the bit length of \varphi. All constants are effective; no advice or oracle is supplied to the reduction.
For every fixed integer $K\ge2$, there is a deterministic polynomial-time
many-one reduction mapping a $3$SAT formula $\varphi$ to a pair $(G,k)$, where
$G$ is an explicit finite simple undirected unweighted graph and $k\ge1$ is
a binary-encoded integer, such that
\[
\begin{aligned}
\varphi\text{ satisfiable}&\quad\Longrightarrow\quad
\OPT_{\rm uncut}(G)\le k,\\
\varphi\text{ unsatisfiable}&\quad\Longrightarrow\quad
\OPT_{\rm uncut}(G)>Kk.
\end{aligned}
\]
For each fixed $K$, the graph size, full construction work, and bit complexity
are polynomial in the bit length of $\varphi$. All constants are effective;
no advice or oracle is supplied to the reduction.
Other possible paper matches, closest wording first
Theorem 2.2 of Constant-factor hardness of Min-UnCut. Words of the Lean statement, its names split into words, that also appear in the theorem: assignment, length, nonempty, list, bit, inputs and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unsatisfiable, sat, gap and deterministic.
Theorem 1.1 of The Factor-Two Hardness Threshold for Vertex Cover. Words of the Lean statement, its names split into words, that also appear in the theorem: vertices, formula, output, lists, satisfiable, finite and four more. Found in the paper’s title, which its main theorem states: threshold. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unsatisfiable, sat, gap, simple, unweighted and deterministic.
Theorem 2.1 of The Factor-Two Hardness Threshold for Vertex Cover. Words of the Lean statement, its names split into words, that also appear in the theorem: formulas, nonempty, satisfiable, alphabets, reduction, time and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: give, unsatisfiable, sat, instances, hardness, constant and one more.
Corollary 1.2 of Constant-factor hardness of Min-UnCut. Words of the Lean statement, its names split into words, that also appear in the theorem: clause, time and polynomial. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: approximation, factor and deterministic.
Proposition 6.1 of Constant-factor hardness of Min-UnCut. Words of the Lean statement, its names split into words, that also appear in the theorem: positive, bits, list, yes, alphabet and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: give, greater, constructed, gap and constants.
Proposition 2.2 of A Direct Proof of Optimal Max-Cut Hardness. Words of the Lean statement, its names split into words, that also appear in the theorem: map, output, input, size, yes, satisfied and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: deterministic and constant.
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
Atlas has no report of an independent group using Comparator to verify OpenAI’s proof of this Lean statement at the version shown here. 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.)).
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 102’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: None found; all five hardness statements are formalized as polynomial-time (Mathlib TM2, finite stack alphabets) Karp gap reductions from binary 3SAT (or an arbitrary NP verifier language), which is the standard NP-hardness encoding. Cone is ~190k lines and uses 'Std'/'Lean' (core) only; the OptimalMaxCut axiom_files flag is a comment-only false positive.
Definitions the group asks readers to compare with the paper: Complexity notions are Mathlib's Turing.TM2ComputableInPolyTime (note: Mathlib's FinTM2 only forces the input stack alphabet to be finite, so every challenge adds finiteAlphabet : ∀ k, Finite (computation.tm.Γ k)), outputs are measured in explicit unary/binary encodings (gameBits writes every number in unary); alphaGW is defined as an sInf over ρ ∈ [-1,1) (equals the GW constant, checked); InNP is the certificate definition via NPVerifier with a polynomial witnessBound
OpenAI’s own note on what this Lean statement covers
“The formalization constructs a deterministic reduction from encoded $3$-SAT formulas to finite simple unweighted Min-UnCut instances with arbitrarily large fixed multiplicative gaps. For every integer $K\ge2$, satisfiable formulas give optimum at most the output threshold, while unsatisfiable formulas give optimum strictly greater than $K$ times that threshold. Runtime and output length are polynomial for each fixed $K$, establishing hardness for every fixed approximation factor greater than one.”
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
130 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: 34 to its own declarations, 60 to Mathlib’s documentation. 7 declared names also link to the same declaration in OpenAI’s library. 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 \varepsilon,\delta\in(0,1/2), there are an integer s\ge1 and a deterministic polynomial-time reduction from \mathrm{3SAT} to explicit unweighted Unique Games instances G_\varphi over K=\F_2^s such that
The graph is simple and bipartite, and every constraint is a translation a(v_e)=a(u_e)+c_e of K. The alphabet, the degree of the running-time polynomial, and its constants depend only on \varepsilon,\delta.
For every fixed \(\varepsilon,\delta\in(0,1/2)\), there are an integer
\(s\ge1\) and a deterministic polynomial-time reduction from
\(\mathrm{3SAT}\) to explicit unweighted Unique Games instances
\(G_\varphi\) over \(K=\F_2^s\) such that
\[
\begin{aligned}
\varphi\text{ satisfiable}
&\quad\Longrightarrow\quad \val(G_\varphi)\ge1-\varepsilon,\\
\varphi\text{ unsatisfiable}
&\quad\Longrightarrow\quad \val(G_\varphi)\le\delta.
\end{aligned}
\]
The graph is simple and bipartite, and every constraint is a translation
\(a(v_e)=a(u_e)+c_e\) of \(K\). The alphabet, the degree of the
running-time polynomial, and its constants depend only on
\(\varepsilon,\delta\).
For every fixed $\alpha$ with $\GW<\alpha\le1$, it is NP-hard to approximate
Max-Cut within a factor of $\alpha$. This holds for simple unweighted graphs.
Theorem
Fix a rational t\in(0,1) and a constant 0<\eps<(t-B(t))/4. For graphs equipped with a rational probability distribution on edges, it is NP-hard to distinguish
Fix a rational $t\in(0,1)$ and a constant
$0<\eps<(t-B(t))/4$. For graphs equipped with a rational probability
distribution on edges, it is NP-hard to distinguish
\begin{align*}
\textnormal{YES:}\quad&\Val\ge\frac{1+t}{2}-\eps,\\
\textnormal{NO:}\quad&\Val\le\frac{1+B(t)}{2}+\eps,
\end{align*}
where $\Val$ is the maximum cut weight.
For every fixed integer m\geq4, there is a deterministic polynomial-time reduction from \mathrm{3SAT} to finite simple undirected unweighted graphs with the following guarantees:
\begin{array}{ll}
\text{the formula is satisfiable}
&\Longrightarrow\quad
\tau(G)<\left(\dfrac12+\dfrac1m\right)|V(G)|,\\[6pt]
\text{the formula is unsatisfiable}
&\Longrightarrow\quad
\tau(G)>\left(1-\dfrac1m\right)|V(G)|.
\end{array}
The output lists all vertices and edges explicitly.
For every fixed integer \(m\geq4\), there is a deterministic
polynomial-time reduction from \(\mathrm{3SAT}\) to finite simple
undirected unweighted graphs with the following guarantees:
\[
\begin{array}{ll}
\text{the formula is satisfiable}
&\Longrightarrow\quad
\tau(G)<\left(\dfrac12+\dfrac1m\right)|V(G)|,\\[6pt]
\text{the formula is unsatisfiable}
&\Longrightarrow\quad
\tau(G)>\left(1-\dfrac1m\right)|V(G)|.
\end{array}
\]
The output lists all vertices and edges explicitly.
Corollary
For every fixed real \alpha with 1\leq\alpha<2, approximating Vertex Cover within factor \alpha is NP-hard. Thus, unless \mathrm P=\mathrm{NP}, no deterministic polynomial-time algorithm achieves factor 2-\varepsilon for any fixed \varepsilon\in(0,1].
For every fixed real \(\alpha\) with \(1\leq\alpha<2\),
approximating Vertex Cover within factor \(\alpha\) is NP-hard.
Thus, unless \(\mathrm P=\mathrm{NP}\), no deterministic
polynomial-time algorithm achieves factor \(2-\varepsilon\)
for any fixed \(\varepsilon\in(0,1]\).
For every fixed integer K\ge2, there is a deterministic polynomial-time many-one reduction mapping a 3SAT formula \varphi to a pair (G,k), where G is an explicit finite simple undirected unweighted graph and k\ge1 is a binary-encoded integer, such that
For each fixed K, the graph size, full construction work, and bit complexity are polynomial in the bit length of \varphi. All constants are effective; no advice or oracle is supplied to the reduction.
For every fixed integer $K\ge2$, there is a deterministic polynomial-time
many-one reduction mapping a $3$SAT formula $\varphi$ to a pair $(G,k)$, where
$G$ is an explicit finite simple undirected unweighted graph and $k\ge1$ is
a binary-encoded integer, such that
\[
\begin{aligned}
\varphi\text{ satisfiable}&\quad\Longrightarrow\quad
\OPT_{\rm uncut}(G)\le k,\\
\varphi\text{ unsatisfiable}&\quad\Longrightarrow\quad
\OPT_{\rm uncut}(G)>Kk.
\end{aligned}
\]
For each fixed $K$, the graph size, full construction work, and bit complexity
are polynomial in the bit length of $\varphi$. All constants are effective;
no advice or oracle is supplied to the reduction.
Corollary
For every fixed real C>1, minimum 2CNF clause deletion is NP-hard to approximate within factor C. In particular, a deterministic polynomial-time C-approximation would imply \mathsf P=\NP.
For every fixed real $C>1$, minimum $2$CNF clause deletion is NP-hard to
approximate within factor $C$. In particular, a deterministic polynomial-time
$C$-approximation would imply $\mathsf P=\NP$.
For every fixed real constant A\ge1, it is NP-hard to approximate minimum directed feedback vertex set within factor A on unweighted digraphs. More precisely, there is a deterministic polynomial-time reduction from an NP-hard promise problem to instances (G,k) with k\in\N_{>0}. The polynomial and its constants may depend on A. The reduction has the following alternatives:
For every fixed real constant $A\ge1$, it is NP-hard to approximate
minimum directed feedback vertex set within factor $A$ on unweighted
digraphs. More precisely, there is a deterministic polynomial-time
reduction from an NP-hard promise problem to instances $(G,k)$ with
$k\in\N_{>0}$. The polynomial and its constants may depend on $A$.
The reduction has the following alternatives:
\[
\textnormal{YES: }\DFVS(G)\le k,
\qquad
\textnormal{NO: }\DFVS(G)>Ak.
\]
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 102 link it.
Solution module
OAI.Computability.MinUncut.Main
Allowed axioms
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
18 local vocabulary declarations
Lines of Lean, without comments or blank lines
65
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
Arbitrary constant-factor hardness of Min-UnCut
Family 102: The Unique Games Conjecture and optimal approximation thresholds
Candidate pairing
Theorem
For every fixed integer K\ge2, there is a deterministic polynomial-time many-one reduction mapping a 3SAT formula \varphi to a pair (G,k), where G is an explicit finite simple undirected unweighted graph and k\ge1 is a binary-encoded integer, such that
For each fixed K, the graph size, full construction work, and bit complexity are polynomial in the bit length of \varphi. All constants are effective; no advice or oracle is supplied to the reduction.
Constant-factor hardness of Min-UnCut
TeX label thm:main, build/sections/introduction.tex, line 50
The Lean statement says
theoremmain : Nonempty Reduction := bysorry
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.