Lean statement

Limiting satisfiability threshold for every fixed $k\ge3$

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.FixedClauseThreshold.main

Lean statement file: FixedClauseThreshold.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 235 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem main (k : ℕ) (hk : 3 ≤ k) : HasLimitingThreshold k := by  sorry

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.

abbreviation OAI.FixedClauseThreshold.Assignment line 21
```
abbrev Assignment (n : ℕ) := Fin n → Bool
```

abbreviation OAI.FixedClauseThreshold.ProperClause line 23

abbrev ProperClause (n k : ℕ) :=  { C : Fin n → Option Bool // (Finset.univ.filter fun i => (C i).isSome).card = k }

abbreviation OAI.FixedClauseThreshold.Formula line 26

abbrev Formula (n k m : ℕ) := Fin m → ProperClause n k

definition OAI.FixedClauseThreshold.SatisfiesClause line 28

def SatisfiesClause {n k : ℕ} (σ : Assignment n) (C : ProperClause n k) : Prop :=  ∃ i : Fin n, C.val i = some (σ i)

definition OAI.FixedClauseThreshold.Satisfies line 31

def Satisfies {n k m : ℕ} (σ : Assignment n) (F : Formula n k m) : Prop :=  ∀ j, SatisfiesClause σ (F j)

definition OAI.FixedClauseThreshold.Satisfiable line 34

def Satisfiable {n k m : ℕ} (F : Formula n k m) : Prop :=  ∃ σ : Assignment n, Satisfies σ F

definition OAI.FixedClauseThreshold.properSATProbability line 37

noncomputable def properSATProbability (n k m : ℕ) : ℝ := by  classical  exact (Fintype.card { F : Formula n k m // Satisfiable F } : ℝ) /    (Fintype.card (Formula n k m) : ℝ)

definition OAI.FixedClauseThreshold.HasLimitingThreshold line 42

def HasLimitingThreshold (k : ℕ) : Prop :=  ∃ α : ℝ, 0 < α ∧    (∀ c : ℝ, 0 ≤ c → c < α →      Filter.Tendsto (fun n : ℕ => properSATProbability n k ⌊c * (n : ℝ)⌋₊)        Filter.atTop (nhds 1)) ∧    (∀ c : ℝ, α < c →      Filter.Tendsto (fun n : ℕ => properSATProbability n k ⌊c * (n : ℝ)⌋₊)        Filter.atTop (nhds 0))

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: proper, clauses and integer. Found in the paper’s title, which its main theorem states: limiting, threshold and satisfiability. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sampled, replacement, independent, signed and alpha.

Statements in A Limiting Satisfiability Threshold for Every Fixed Clause Size come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of A Limiting Satisfiability Threshold for Every Fixed Clause Size

Theorem

For every fixed integer k\ge3, there is a constant \alpha_k\in(0,\infty) such that, for independent proper uniformly signed k-clauses sampled with replacement,

\lim_{n\to\infty}P_n(\lfloor cn\rfloor) = \begin{cases} 1,&0\le c<\alpha_k,\\ 0,&c>\alpha_k. \end{cases}

Section 1, The limiting-threshold problem Read it in the paper TeX source, line 31

Show the TeX

For every fixed integer \(k\ge3\), there is a constant
\(\alpha_k\in(0,\infty)\) such that, for independent proper uniformly
signed \(k\)-clauses sampled with replacement,
\[
 \lim_{n\to\infty}P_n(\lfloor cn\rfloor)
 =
 \begin{cases}
  1,&0\le c<\alpha_k,\\
  0,&c>\alpha_k.
 \end{cases}
\]

Other possible paper matches, closest wording first

Proposition 4.6 of Computing the Random 3-SAT Threshold. Words of the Lean statement, its names split into words, that also appear in the theorem: formula, probability, assignment, satisfiable, real, clauses and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: tending, variables, one and alpha.
Theorem 4.4 of Computing the Random 3-SAT Threshold. Words of the Lean statement, its names split into words, that also appear in the theorem: limiting, threshold, satisfiability, proper, real and clause. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: density and alpha.
Theorem 5.1 of A Limiting Satisfiability Threshold for Every Fixed Clause Size. Words of the Lean statement, its names split into words, that also appear in the theorem: proper, clause and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: independent, signed and alpha.
Theorem 1.1 of Computing the Random 3-SAT Threshold. Words of the Lean statement, its names split into words, that also appear in the theorem: real, finite and integer. Found in the paper’s title, which its main theorem states: sat and threshold. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: number, one and alpha.
Proposition 4.3 of Computing the Random 3-SAT Threshold. Words of the Lean statement, its names split into words, that also appear in the theorem: limiting, real and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: number, one and alpha.
Proposition A.2 of Variance of the Random k-SAT Hitting Time. Words of the Lean statement, its names split into words, that also appear in the theorem: assignment and integers. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: replacement, uniform and one.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 57 s; peak memory 6.1 GiB).
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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 235’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Stated: FixedClauseThreshold.main (k) (hk : 3 ≤ k) : HasLimitingThreshold k (∃ α > 0, P(SAT) → 1 for c < α and → 0 for c > α with m = ⌊c n⌋ i.i.d. uniform proper clauses), SATComputability.main (3-SAT threshold α with a Nat.Partrec.Code producing rationals q r, |q r - α| ≤ 2^-r) and variance_main. But the variance claim is only Θ(n) for k >= 4: for k = 3 the Lean gives Var(H) ≤ C n ell with ell = log(e n) (an n log n upper bound) and a linear lower bound, so the summary's 'hitting-time variance Θ_k(n) for every fixed k ≥ 3' is not stated for k = 3 (docs: 'They do not show that the logarithmic factor in the k=3 variance bound is necessary').
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: the k = 3 linear upper bound Var(H_n) = O(n) (Lean has O(n log n))

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves that every fixed integer $k\ge3$ has a finite positive threshold $\alpha_k$: for every fixed density $c<\alpha_k$, the satisfiability probability tends to one as the number of variables grows, while for every $c>\alpha_k$ it tends to zero. Clauses use distinct variables with independent uniform signs and are sampled independently with replacement. The statement makes no assertion at the threshold itself.”
Quoted from OpenAI’s Lean note for this family

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

56 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: 15 to its own declarations, 17 to Mathlib’s documentation. 9 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 8 other declarations of the file; with the theorem itself they span 24 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.FixedClauseThreshold are listed without it.

abbrev Assignmentline 21library
abbrev ProperClauseline 23library
abbrev Formulaline 26library
def SatisfiesClauseline 28library
def Satisfiesline 31library
def Satisfiableline 34library
def properSATProbabilityline 37library
def HasLimitingThresholdline 42in the statementlibrary
theorem mainline 51selected theoremlibrary

import Mathlibnamespace OAI/-!Proper clauses choose distinct variables uniformly and give them independent fairsigns. Formulas consist of independent uniform whole-clause draws with replacement.Their satisfiability probability is the literal finite cardinality ratio.For zero clauses this probability is one, including when there are fewer variablesthan literals per clause. When there are fewer variables than literals per clauseand the clause count is positive, the ensemble is empty and its probability is zeroby the convention that division by zero is zero.For each fixed clause size at least three, one positive finite real thresholdgoverns the limits at every fixed density below or above it. There is no assertionat the critical density, no explicit threshold value, and no varying-size claim.-/namespace FixedClauseThresholdabbrev Assignment (n : ℕ) := Fin n → Boolabbrev ProperClause (n k : ℕ) :=  { C : Fin n → Option Bool // (Finset.univ.filter fun i => (C i).isSome).card = k }abbrev Formula (n k m : ℕ) := Fin m → ProperClause n kdef SatisfiesClause {n k : ℕ} (σ : Assignment n) (C : ProperClause n k) : Prop :=  ∃ i : Fin n, C.val i = some (σ i)def Satisfies {n k m : ℕ} (σ : Assignment n) (F : Formula n k m) : Prop :=  ∀ j, SatisfiesClause σ (F j)def Satisfiable {n k m : ℕ} (F : Formula n k m) : Prop :=  ∃ σ : Assignment n, Satisfies σ Fnoncomputable def properSATProbability (n k m : ℕ) : ℝ := by  classical  exact (Fintype.card { F : Formula n k m // Satisfiable F } : ℝ) /    (Fintype.card (Formula n k m) : ℝ)def HasLimitingThreshold (k : ℕ) : Prop :=  ∃ α : ℝ, 0 < α ∧    (∀ c : ℝ, 0 ≤ c → c < α →      Filter.Tendsto (fun n : ℕ => properSATProbability n k ⌊c * (n : ℝ)⌋₊)        Filter.atTop (nhds 1)) ∧    (∀ c : ℝ, α < c →      Filter.Tendsto (fun n : ℕ => properSATProbability n k ⌊c * (n : ℝ)⌋₊)        Filter.atTop (nhds 0))theorem main (k : ℕ) (hk : 3 ≤ k) : HasLimitingThreshold k := by  sorryend FixedClauseThresholdend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

A Limiting Satisfiability Threshold for Every Fixed Clause Size

Theorem

For every fixed integer k\ge3, there is a constant \alpha_k\in(0,\infty) such that, for independent proper uniformly signed k-clauses sampled with replacement,

\lim_{n\to\infty}P_n(\lfloor cn\rfloor) = \begin{cases} 1,&0\le c<\alpha_k,\\ 0,&c>\alpha_k. \end{cases}

Section 1, The limiting-threshold problem Read it in the paper TeX source, line 31

Show the TeX

For every fixed integer \(k\ge3\), there is a constant
\(\alpha_k\in(0,\infty)\) such that, for independent proper uniformly
signed \(k\)-clauses sampled with replacement,
\[
 \lim_{n\to\infty}P_n(\lfloor cn\rfloor)
 =
 \begin{cases}
  1,&0\le c<\alpha_k,\\
  0,&c>\alpha_k.
 \end{cases}
\]

Variance of the Random k-SAT Hitting Time

Theorem

Fix an integer k\ge3, and write

\ell_k(n)=\begin{cases}\log(en),&k=3,\\1,&k\ge4,\end{cases} \qquad U_k=\frac{\log2}{-\log(1-2^{-k})}.

There are constants C_k,c_k>0 such that

\Var(H_n)\le C_kn\ell_k(n)\quad(n\ge k),\qquad \Var(H_n)\ge c_kn\quad\text{for all sufficiently large }n.

For every fixed B>0 there is C_{k,B}<\infty such that

\Var(T_{n,B})\le C_{k,B}n\ell_k(n)\quad(n\ge k).

If B>U_k, then also \Var(T_{n,B})\ge c_kn for all sufficiently large n, with the size cutoff allowed to depend on k and B.

Section 1, Fluctuations of the satisfiability transition Read it in the paper TeX source, line 27

Show the TeX

Fix an integer $k\ge3$, and write
\[
 \ell_k(n)=\begin{cases}\log(en),&k=3,\\1,&k\ge4,\end{cases}
 \qquad U_k=\frac{\log2}{-\log(1-2^{-k})}.
\]
There are constants $C_k,c_k>0$ such that
\[
 \Var(H_n)\le C_kn\ell_k(n)\quad(n\ge k),\qquad
 \Var(H_n)\ge c_kn\quad\text{for all sufficiently large }n.
\]
For every fixed $B>0$ there is $C_{k,B}<\infty$ such that
\[
 \Var(T_{n,B})\le C_{k,B}n\ell_k(n)\quad(n\ge k).
\]
If $B>U_k$, then also $\Var(T_{n,B})\ge c_kn$ for all sufficiently
large $n$, with the size cutoff allowed to depend on $k$ and $B$.

Computing the Random 3-SAT Threshold

Theorem

There is a real number \alpha_3\in(0,\infty) such that, for every fixed real a\ge0,

\lim_{n\to\infty}p(n,\lfloor an\rfloor)= \begin{cases}1,&a<\alpha_3,\\0,&a>\alpha_3.\end{cases}

There is one finite deterministic Turing machine which, on input 1^r for any integer r\ge1, halts with a rational number q_r satisfying |q_r-\alpha_3|\le2^{-r}. Its only input is the unary precision 1^r; it uses no oracle, advice, or noncomputable real constant.

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

There is a real number $\alpha_3\in(0,\infty)$ such that, for every fixed
real $a\ge0$,
\[
 \lim_{n\to\infty}p(n,\lfloor an\rfloor)=
 \begin{cases}1,&a<\alpha_3,\\0,&a>\alpha_3.\end{cases}
\]
There is one finite deterministic Turing machine which, on input $1^r$
for any integer $r\ge1$, halts with a rational number $q_r$ satisfying
$|q_r-\alpha_3|\le2^{-r}$. Its only input is the unary precision $1^r$;
it uses no oracle, advice, or noncomputable real constant.

Configuration and file facts

File: lean/ComparatorChallenges/FixedClauseThreshold.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 235 link it.
Solution module: OAI.Probability.RandomSAT.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: 8 local vocabulary declarations
Lines of Lean, without comments or blank lines: 24
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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