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Lean statement

Spectral-gap bound throughout the high-temperature SK phase

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.SKGap.sk_main

Lean statement file: SKHighTemperature.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 227 link it.

See the candidate theoremWho ran the proof-verification programReport 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 sk_main (β : ℝ) (hβ0 : 0 < β) (hβ1 : β < 1) : MainStatement β := 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.SKGap.Spin line 11

    abbrev Spin (n : ℕ) := Fin n → Bool

  • abbreviation OAI.SKGap.Edge line 12

    abbrev Edge (n : ℕ) := {p : Fin n × Fin n // p.1 < p.2}

  • abbreviation OAI.SKGap.Disorder line 13

    abbrev Disorder (n : ℕ) := Edge n → ℝ

  • definition OAI.SKGap.spinValue line 15

    def spinValue (b : Bool) : ℝ := if b then 1 else -1

  • definition OAI.SKGap.coupling line 17

    def coupling {n : ℕ} (g : Disorder n) (i k : Fin n) : ℝ :=  if h : i < k then g ⟨(i, k), h⟩ else    if h : k < i then g ⟨(k, i), h⟩ else 0

  • definition OAI.SKGap.hamiltonian line 21

    def hamiltonian {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ :=  (1 / 2 : ℝ) * ∑ i, ∑ k, spinValue (x i) * coupling g i k * spinValue (x k) +    ∑ i, h i * spinValue (x i)

  • definition OAI.SKGap.weight line 25

    def weight {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ :=  Real.exp (hamiltonian g h x)

  • definition OAI.SKGap.partition line 28

    def partition {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) : ℝ :=  ∑ x, weight g h x

  • definition OAI.SKGap.mass line 31

    def mass {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ :=  weight g h x / partition g h

  • definition OAI.SKGap.expectation line 34

    def expectation {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (f : Spin n → ℝ) : ℝ :=  ∑ x, mass g h x * f x

  • definition OAI.SKGap.variance line 37

    def variance {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (f : Spin n → ℝ) : ℝ :=  expectation g h (fun x => (f x - expectation g h f) ^ 2)

  • definition OAI.SKGap.flip line 40

    def flip {n : ℕ} (i : Fin n) (x : Spin n) : Spin n :=  Function.update x i (!(x i))

  • definition OAI.SKGap.conditionalExpectation line 44

    def conditionalExpectation {n : ℕ} (g : Disorder n) (h : Fin n → ℝ)    (i : Fin n) (f : Spin n → ℝ) (x : Spin n) : ℝ :=  (weight g h x * f x + weight g h (flip i x) * f (flip i x)) /    (weight g h x + weight g h (flip i x))

  • definition OAI.SKGap.dirichlet line 50

    def dirichlet {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (f : Spin n → ℝ) : ℝ :=  ∑ i, expectation g h (fun x => (f x - conditionalExpectation g h i f x) ^ 2)

  • definition OAI.SKGap.disorderLaw line 53

    def disorderLaw (β : ℝ) (n : ℕ) : Measure (Disorder n) :=  Measure.pi (fun _ : Edge n => gaussianReal 0 (Real.toNNReal (β ^ 2 / n)))

  • definition OAI.SKGap.poincareEvent line 56

    def poincareEvent (n : ℕ) (C : ℝ) : Set (Disorder n) :=  {g | ∀ f : Spin n → ℝ, variance g 0 f ≤ C * dirichlet g 0 f}

  • definition OAI.SKGap.discreteGap line 60

    def discreteGap {n : ℕ} (g : Disorder n) : ℝ :=  sInf {r : ℝ | ∃ f : Spin n → ℝ, 0 < variance g 0 f ∧    r = dirichlet g 0 f / ((n : ℝ) * variance g 0 f)}

  • definition OAI.SKGap.MainStatement line 64

    def MainStatement (β : ℝ) : Prop :=  ∃ C : ℝ, 0 < C ∧    Tendsto (fun n : ℕ => disorderLaw β n (poincareEvent n C)) atTop (nhds 1) ∧    Tendsto (fun n : ℕ => disorderLaw β n {g | 1 / (C * (n : ℝ)) ≤ discreteGap g})      atTop (nhds 1)

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: discrete, bath, heat, gap, site, uniform and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: chooses, least, spectral, depending, beta, tending, constant and probability.

Statements in A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase come first because the release’s name for this Lean statement shares words with that paper’s title.

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.

Section 1, The theorem and the structure of the proof Read it in the paper TeX source, line 33
Show the TeX
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

  • Theorem 6.3 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: unscaled, variances, discrete, bath, heat, gap and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: temperature, high, choosing, least, zero, spectral and six more.
  • 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: bath, heat, rate, gap, site, real and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dynamics, least, spectral, tending and probability.
  • Proposition 4.1 of Functional universality of critical SK autocorrelations. Words of the Lean statement, its names split into words, that also appear in the theorem: variance, valued, real, conditional, functions, disorder and one more. Found in the paper’s title, which its main theorem states: sk. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: inequality, independent and constants.
  • Theorem 19.3 of Critical mixing in the Sherrington–Kirkpatrick model. Words of the Lean statement, its names split into words, that also appear in the theorem: expectation, couplings, form, gibbs, real, conditional and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: zero, field, tending, bound and probability.
  • 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, bath, spin, heat, site and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bound.
  • Proposition 13.6 of Critical mixing in the Sherrington–Kirkpatrick model. Words of the Lean statement, its names split into words, that also appear in the theorem: unscaled, spin, gibbs, rate, gap, conditional and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: inverse, choose, least, cn, spectral, uniformly and four more.

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.

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

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 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.

  • 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: 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).

  • Carlos Toledo’s cert-machine

    Finds part of it stated, or a narrower or weaker statement Their 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.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“For every fixed inverse temperature $0<\beta<1$, the formalization proves a dimension-independent Poincaré inequality for the zero-field Gaussian Sherrington–Kirkpatrick model with probability tending to one over the disorder. Equivalently, the unscaled single-site heat-bath spectral gap stays bounded away from zero. For dynamics that choose one site uniformly at each discrete step, the spectral gap is at least $1/(Cn)$ with probability tending to one, for a constant $C$ depending on $\beta$.”

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

76 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: 56 to its own declarations, 30 to Mathlib’s documentation. 18 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.

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperatorsopen MeasureTheory ProbabilityTheory Filternamespace SKGapabbrev Spin (n : ℕ) := Fin n → Boolabbrev Edge (n : ℕ) := {p : Fin n × Fin n // p.1 < p.2}abbrev Disorder (n : ℕ) := Edge n → ℝdef spinValue (b : Bool) : ℝ := if b then 1 else -1def coupling {n : ℕ} (g : Disorder n) (i k : Fin n) : ℝ :=  if h : i < k then g ⟨(i, k), h⟩ else    if h : k < i then g ⟨(k, i), h⟩ else 0def hamiltonian {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ :=  (1 / 2 : ℝ) * ∑ i, ∑ k, spinValue (x i) * coupling g i k * spinValue (x k) +    ∑ i, h i * spinValue (x i)def weight {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ :=  Real.exp (hamiltonian g h x)def partition {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) : ℝ :=  ∑ x, weight g h xdef mass {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ :=  weight g h x / partition g hdef expectation {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (f : Spin n → ℝ) : ℝ :=  ∑ x, mass g h x * f xdef variance {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (f : Spin n → ℝ) : ℝ :=  expectation g h (fun x => (f x - expectation g h f) ^ 2)def flip {n : ℕ} (i : Fin n) (x : Spin n) : Spin n :=  Function.update x i (!(x i))/-- Conditional expectation under the actual Gibbs law given all other spins. -/def conditionalExpectation {n : ℕ} (g : Disorder n) (h : Fin n → ℝ)    (i : Fin n) (f : Spin n → ℝ) (x : Spin n) : ℝ :=  (weight g h x * f x + weight g h (flip i x) * f (flip i x)) /    (weight g h x + weight g h (flip i x))/-- Unscaled form: each site has a rate-one clock. -/def dirichlet {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (f : Spin n → ℝ) : ℝ :=  ∑ i, expectation g h (fun x => (f x - conditionalExpectation g h i f x) ^ 2)def disorderLaw (β : ℝ) (n : ℕ) : Measure (Disorder n) :=  Measure.pi (fun _ : Edge n => gaussianReal 0 (Real.toNNReal (β ^ 2 / n)))def poincareEvent (n : ℕ) (C : ℝ) : Set (Disorder n) :=  {g | ∀ f : Spin n → ℝ, variance g 0 f ≤ C * dirichlet g 0 f}/-- Rayleigh quotient definition of the uniform-site discrete heat-bath gap. -/def discreteGap {n : ℕ} (g : Disorder n) : ℝ :=  sInf {r : ℝ | ∃ f : Spin n → ℝ, 0 < variance g 0 f ∧    r = dirichlet g 0 f / ((n : ℝ) * variance g 0 f)}def MainStatement (β : ℝ) : Prop :=  ∃ C : ℝ, 0 < C ∧    Tendsto (fun n : ℕ => disorderLaw β n (poincareEvent n C)) atTop (nhds 1) ∧    Tendsto (fun n : ℕ => disorderLaw β n {g | 1 / (C * (n : ℝ)) ≤ discreteGap g})      atTop (nhds 1)theorem sk_main (β : ℝ) (hβ0 : 0 < β) (hβ1 : β < 1) : MainStatement β := by  sorryend SKGapendend 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 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.

Section 1, The theorem and the structure of the proof Read it in the paper TeX source, line 33
Show the TeX
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.

Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase

Theorem

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{aligned}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{aligned}

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

0<\lambda(\beta)\leq(1-\beta)^2.
Section 1, Introduction Read it in the paper TeX source, line 80
Show the TeX
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
\[
 0<\lambda(\beta)\leq(1-\beta)^2.
\]

Critical slowing down in the Sherrington–Kirkpatrick model

Theorem (Realized equilibrium initial states)

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.

Section 1, Introduction Read it in the paper TeX source, line 37
Show the TeX
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

\left[\frac{n^{2/3}}{M_n},\ M_n n^{2/3}\right].
Section 1, Introduction Read it in the paper TeX source, line 71
Show the TeX
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
\[
 \left[\frac{n^{2/3}}{M_n},\ M_n n^{2/3}\right].
\]
Theorem (Critical mixing bounds)

For every fixed \epsilon>0,

\begin{aligned}\Prob\!\left\{n^{2/3-\epsilon}\le t_{\rm mix} \le e^{\epsilon n}\right\}&\longrightarrow1, \\ \Prob\!\left\{n^{5/3-\epsilon}\le k_{\rm mix} \le e^{\epsilon n}\right\}&\longrightarrow1.\end{aligned}
Section 1, Introduction Read it in the paper TeX source, line 89
Show the TeX
For every fixed $\epsilon>0$,
\begin{align}
 \Prob\!\left\{n^{2/3-\epsilon}\le t_{\rm mix}
                   \le e^{\epsilon n}\right\}&\longrightarrow1,
       \\
 \Prob\!\left\{n^{5/3-\epsilon}\le k_{\rm mix}
                   \le e^{\epsilon n}\right\}&\longrightarrow1.
       
\end{align}

Stretched-exponential barriers for typical SK initial states

Theorem

Set \kap=1/10000. For every fixed \beta>1,

\begin{aligned}\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{aligned}
Section 1, Introduction Read it in the paper TeX source, line 62
Show the TeX
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*}
Corollary

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.

Section 1, Introduction Read it in the paper TeX source, line 99
Show the TeX
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.
Configuration and file facts
OpenAI’s Lean catalogue
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.SKGap.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
42
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

Spectral-gap bound throughout the high-temperature SK phase

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

theorem sk_main (β : ℝ) (hβ0 : 0 < β) (hβ1 : β < 1) : MainStatement β := 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.

Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.two independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

18 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/SKHighTemperature.html

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.

Questions to resolve when comparing this statement with the paper

  • Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.

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