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.SKRatio.ratio_cutoff
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theoremratio_cutoff (β ε η : ℝ) (hβ0 : 0 ≤ β) (hβhalf : β < 1 / 2) (hε0 : 0 < ε) (hεhalf : ε < 1 / 2) (hη : 0 < η) : Tendsto (fun n : ℕ => disorderLaw β n {g : Disorder n | 1 + η < (mixingTime g ε : ℝ) / (mixingTime g (1 - ε) : ℝ)}) atTop (𝓝 0) ∧ (∀ᶠ n : ℕ in atTop, ∀ g : Disorder n, 0 < mixingTime g (1 - ε)) := 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.
defsiteKernel {n : ℕ} (g : Disorder n) (i : Fin n) : Matrix (Spin n) (Spin n) ℝ := fun x y => (if y = x then mass g 0 x / (mass g 0 x + mass g 0 (flip i x)) else0) + (if y = flip i x then mass g 0 (flip i x) / (mass g 0 x + mass g 0 (flip i x)) else0)
No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.
Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: bath, constant, depending, discrete, heat, least, probability, site and three more.
Theorem 1.1 of A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase
Theorem
There is a finite constant C_\beta, depending only on \beta, such that
\Prob_J\left\{\Var_{\mu_0}(f)\le C_\beta\D_0(f)
\text{ for every }f:\{-1,1\}^n\longrightarrow\R\right\}
\longrightarrow1.
Consequently, the discrete-time heat-bath chain that chooses a uniform site has spectral gap at least 1/(C_\beta n) with probability tending to one.
There is a finite constant $C_\beta$, depending only on $\beta$, such that
\[
\Prob_J\left\{\Var_{\mu_0}(f)\le C_\beta\D_0(f)
\text{ for every }f:\{-1,1\}^n\longrightarrow\R\right\}
\longrightarrow1.
\]
Consequently, the discrete-time heat-bath chain that chooses a uniform site
has spectral gap at least $1/(C_\beta n)$ with probability tending to one.
Other possible paper matches, closest wording first
Proposition A.2 of Critical slowing down in the Sherrington–Kirkpatrick model. Words of the Lean statement, its names split into words, that also appear in the theorem: discrete, coupling, kernel, spin, site, mixing and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bath and heat.
Proposition 9.12 of Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase. Words of the Lean statement, its names split into words, that also appear in the theorem: update, total, variation, spin, cutoff, site and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: single, beta and probability.
Proposition G.5 of Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase. Words of the Lean statement, its names split into words, that also appear in the theorem: update, discrete, cutoff and site. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: single, bath and heat.
Corollary 2 of A typical-start upper bound for low-temperature SK Glauber dynamics. Words of the Lean statement, its names split into words, that also appear in the theorem: total, variation, mixing, time, exp and pi. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: model and beta.
Proposition A.1 of Critical slowing down in the Sherrington–Kirkpatrick model. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, kernel, mixing, measure, time and law.
Theorem 14.2 of Critical mixing in the Sherrington–Kirkpatrick model. Words of the Lean statement, its names split into words, that also appear in the theorem: site, real, function and disorder. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dynamics, support, bath, heat, tending and probability.
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
two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; 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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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 227’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.
Finds part of it stated, or a narrower or weaker statementTheir term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Stated: stretched-exponential barrier for β > 1 (SK.main : Tendsto continuousBadMass atTop (𝓝 1) ∧ Tendsto discreteBadMass atTop (𝓝 1) at time exp(n^(1/10000)), expected bad Gibbs mass over disorder); high-temperature spectral gap sk_main (0<β<1: Poincaré constant C and gap ≥ 1/(Cn)) as a supporting result; ratio cutoff ratio_cutoff only for 0 ≤ β < 1/2. Critical case is one-sided: critical_mixing_bounds gives n^(2/3-ε) ≤ continuousMixingTime ≤ exp(εn) (discrete n^(5/3-ε) ≤ ... ≤ exp(εn)), so the headline 'mixing time n^{2/3+o(1)} at β=1' has no n^{2/3+ε} upper bound in Lean. Not stated: cutoff for 1/2 <= β < 1 and its log n location, and the critical autocorrelation-process limits (Gaussian/Rademacher, quench relaxing to stationary).
Definitions the group asks readers to compare with the paper: Custom 'mixing time': continuousMixingTime W := sInf {t | 0 ≤ t ∧ ∀ x, continuousDistance W t x ≤ 1/4} (matrix exponential kernel NormedSpace.exp (t • generator W)), discreteMixingTime W := sInf {k : ℕ | ∀ x, discreteDistance W k x ≤ 1/4}, SKRatio.mixingTime g ε := sInf {k | discreteDistance g k ≤ ε}: standard worst-start total-variation mixing (sets are nonempty so sInf is a true min), but hand-written; SKBarriers uses its own heatBath kernel and a Poissonized continuousKernel (series in k).
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: Not stated: the critical autocorrelation universality (clause 4, the titular result); the n^{2/3+o(1)} upper bound at β = 1; the stretched-exponential upper bound for β > 1; cutoff for 1/2 ≤ β < 1, its log n location and continuous time (this last part is declared). Stated: the high-temperature spectral gap (supporting), discrete ratio cutoff for β < 1/2, critical lower bounds with an e^{εn} upper bound, the β > 1 barrier, and critical covariance/linear-test scaling.
OpenAI’s own note on what this Lean statement covers
“The formalized supporting result proves ratio cutoff for discrete single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model for $0\le\beta<1/2$. For every $0<\varepsilon<1/2$ and $\eta>0$, the probability over the disorder that $t_{\mathrm{mix}}(\varepsilon)/t_{\mathrm{mix}}(1-\varepsilon)>1+\eta$ tends to zero as $n\to\infty$.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a weaker form of a paper’s theorem”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
74 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: 51 to its own declarations, 27 to Mathlib’s documentation. 17 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.
There is a finite constant $C_\beta$, depending only on $\beta$, such that
\[
\Prob_J\left\{\Var_{\mu_0}(f)\le C_\beta\D_0(f)
\text{ for every }f:\{-1,1\}^n\longrightarrow\R\right\}
\longrightarrow1.
\]
Consequently, the discrete-time heat-bath chain that chooses a uniform site
has spectral gap at least $1/(C_\beta n)$ with probability tending to one.
All convergences are in probability over the disorder. For \beta=0, \lambda(0)=1. For 0<\beta<1, \lambda(\beta) is the equilibrium-operator constant defined in (eq:exact-lambda); it satisfies
For every fixed \(0\leq\beta<1\), there is a deterministic
\(\lambda(\beta)>0\) such that, with
\[
t_n=\frac{\log n}{2\lambda(\beta)},
\]
for every \(\epsilon\in(0,1)\),
\begin{align*}
d_n((1-\epsilon)t_n)&\longrightarrow1,
&
d_n((1+\epsilon)t_n)&\longrightarrow0,\\
d_n^{\rm disc}(\lfloor(1-\epsilon)nt_n\rfloor)&\longrightarrow1,
&
d_n^{\rm disc}(\lceil(1+\epsilon)nt_n\rceil)&\longrightarrow0.
\end{align*}
All convergences are in probability over the disorder.
For \(\beta=0\), \(\lambda(0)=1\).
For \(0<\beta<1\), \(\lambda(\beta)\) is the equilibrium-operator
constant defined in \eqref{eq:exact-lambda}; it satisfies
\[
0<\lambda(\beta)\leq(1-\beta)^2.
\]
For every deterministic sequence $t_n\ge0$ with $t_n=o(n^{2/3})$,
\[
\mu_W\{x:d_{t_n}^{\rm ct}(W,x)>1/4\}
\xrightarrow{\Prob}1.
\]
For every deterministic integer sequence $k_n\ge0$ with $k_n=o(n^{5/3})$,
\[
\mu_W\{x:d_{k_n}^{\rm dt}(W,x)>1/4\}
\xrightarrow{\Prob}1.
\]
The convergence is in probability over the disorder.
Theorem(Covariance and linear tests)
For every positive deterministic sequence M_n\to\infty, with probability tending to one both \|\Sigma\| and \mathcal R_{\rm lin}(\mu_W) belong to
For every positive deterministic sequence $M_n\to\infty$, with probability tending to one both $\|\Sigma\|$ and $\mathcal R_{\rm lin}(\mu_W)$ belong to
\[
\left[\frac{n^{2/3}}{M_n},\ M_n n^{2/3}\right].
\]
Set $\kap=1/10000$. For every fixed $\beta>1$,
\begin{align*}
\PP_{J,\,x_0\sim\pi^J}
\left\{d^J\bigl(x_0,\exp(n^\kap)\bigr)>\frac14\right\}
&\longrightarrow1,\\
\PP_{J,\,x_0\sim\pi^J}
\left\{d_{\mathrm{disc}}^J
\bigl(x_0,\lfloor\exp(n^\kap)\rfloor\bigr)>\frac14\right\}
&\longrightarrow1.
\end{align*}
For every fixed $\beta>1$ and every fixed $a>0$,
\[
\PP_{J,\,x_0\sim\pi^J}\{T^J(x_0)>n^a\}\longrightarrow1,
\qquad
\PP_{J,\,x_0\sim\pi^J}
\{T_{\mathrm{disc}}^J(x_0)>n^a\}\longrightarrow1.
\]
Equivalently, each corresponding Gibbs mass tends to one in probability
over the disorder.
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 227 link it.
Solution module
OAI.Probability.SKRatio.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
16 local vocabulary declarations
Lines of Lean, without comments or blank lines
44
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
Discrete mixing-time ratio cutoff for $\beta<1/2$
Family 227: Critical SK autocorrelation processes and dynamics across the temperature transition
Candidate pairing
Theorem
There is a finite constant C_\beta, depending only on \beta, such that
\Prob_J\left\{\Var_{\mu_0}(f)\le C_\beta\D_0(f)
\text{ for every }f:\{-1,1\}^n\longrightarrow\R\right\}
\longrightarrow1.
Consequently, the discrete-time heat-bath chain that chooses a uniform site has spectral gap at least 1/(C_\beta n) with probability tending to one.
A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase
TeX label thm:main, build/sections/introduction.tex, line 33
The Lean statement says
theoremratio_cutoff (β ε η : ℝ) (hβ0 : 0 ≤ β) (hβhalf : β < 1 / 2) (hε0 : 0 < ε) (hεhalf : ε < 1 / 2) (hη : 0 < η) : Tendsto (fun n : ℕ => disorderLaw β n {g : Disorder n | 1 + η < (mixingTime g ε : ℝ) / (mixingTime g (1 - ε) : ℝ)}) atTop (𝓝 0) ∧ (∀ᶠ n : ℕ in atTop, ∀ g : Disorder n, 0 < mixingTime g (1 - ε)) := 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.