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Lean statement

Stretched-exponential barriers from typical equilibrium initial states

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.SK.main

Lean statement file: SKBarriers.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 main (β : ℝ) (hβ : 1 < β) :    Tendsto (continuousBadMass β) atTop (𝓝 1) ∧    Tendsto (discreteBadMass β) atTop (𝓝 1) := 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.SK.Config line 14

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

  • abbreviation OAI.SK.Edge line 15

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

  • abbreviation OAI.SK.Disorder line 16

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

  • definition OAI.SK.spin line 18

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

  • definition OAI.SK.hamiltonian line 20

    def hamiltonian {n : ℕ} (J : Disorder n) (x : Config n) : ℝ :=  (∑ e : Edge n, J e * spin (x e.val.1) * spin (x e.val.2)) / Real.sqrt n

  • definition OAI.SK.partition line 23

    def partition {n : ℕ} (β : ℝ) (J : Disorder n) : ℝ :=  ∑ x : Config n, Real.exp (β * hamiltonian J x)

  • definition OAI.SK.gibbs line 26

    def gibbs {n : ℕ} (β : ℝ) (J : Disorder n) (x : Config n) : ℝ :=  Real.exp (β * hamiltonian J x) / partition β J

  • definition OAI.SK.coupling line 29

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

  • definition OAI.SK.localField line 33

    def localField {n : ℕ} (J : Disorder n) (x : Config n) (i : Fin n) : ℝ :=  (∑ j : Fin n, coupling J i j * spin (x j)) / Real.sqrt n

  • definition OAI.SK.heatBath line 36

    def heatBath {n : ℕ} (β : ℝ) (J : Disorder n) (x y : Config n) : ℝ :=  (∑ i : Fin n, if Function.update x i (y i) = y then    Real.exp (β * spin (y i) * localField J x i) /      (2 * Real.cosh (β * localField J x i)) else 0) / n

  • definition OAI.SK.discreteKernel line 41

    def discreteKernel {n : ℕ} (β : ℝ) (J : Disorder n) : ℕ → Config n → Config n → ℝ  | 0, x, y => if x = y then 1 else 0  | k + 1, x, y => ∑ z : Config n, heatBath β J x z * discreteKernel β J k z y

  • definition OAI.SK.continuousKernel line 45

    def continuousKernel {n : ℕ} (β : ℝ) (J : Disorder n) (t : ℝ)    (x y : Config n) : ℝ :=  ∑' k : ℕ, (Real.exp (-(n : ℝ) * t) * ((n : ℝ) * t) ^ k /    (Nat.factorial k : ℝ)) * discreteKernel β J k x y

  • definition OAI.SK.discreteDistance line 50

    def discreteDistance {n : ℕ} (β : ℝ) (J : Disorder n)    (x : Config n) (k : ℕ) : ℝ :=  (1 / 2 : ℝ) * ∑ y : Config n, |discreteKernel β J k x y - gibbs β J y|

  • definition OAI.SK.continuousDistance line 54

    def continuousDistance {n : ℕ} (β : ℝ) (J : Disorder n)    (x : Config n) (t : ℝ) : ℝ :=  (1 / 2 : ℝ) * ∑ y : Config n, |continuousKernel β J t x y - gibbs β J y|

  • definition OAI.SK.disorderLaw line 58

    def disorderLaw (n : ℕ) : Measure (Disorder n) :=  Measure.pi (fun _ : Edge n => gaussianReal 0 1)

  • definition OAI.SK.kappa line 61

    def kappa : ℝ := 1 / 10000

  • definition OAI.SK.timeScale line 63

    def timeScale (n : ℕ) : ℝ := Real.exp ((n : ℝ) ^ kappa)

  • definition OAI.SK.continuousBadMass line 65

    def continuousBadMass (β : ℝ) (n : ℕ) : ℝ :=  ∫ J : Disorder n, (∑ x : Config n, if 1 / 4 < continuousDistance β J x (timeScale n)    then gibbs β J x else 0) ∂disorderLaw n

  • definition OAI.SK.discreteBadMass line 69

    def discreteBadMass (β : ℝ) (n : ℕ) : ℝ :=  ∫ J : Disorder n, (∑ x : Config n, if 1 / 4 < discreteDistance β J x ⌊timeScale n⌋₊    then gibbs β J x else 0) ∂disorderLaw n

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: gibbs, 1/4, mass, time, law and pi. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: obstruction, initial, states, tend, beta, one and probability.

Statements in Stretched-exponential barriers for typical SK initial states come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem B.1 of Stretched-exponential barriers for typical SK initial states

Theorem (Polynomial-time obstruction from fixed-depth crossings)

For every fixed \beta>1 and every fixed a>0,

\begin{aligned}\PP_{J,\,x_0\sim\pi^J} \left\{d^J(x_0,n^a)>\frac14\right\}&\longrightarrow1, \\ \PP_{J,\,x_0\sim\pi^J} \left\{d_{\mathrm{disc}}^J(x_0,\lfloor n^a\rfloor) >\frac14\right\} &\longrightarrow1.\end{aligned}

Consequently both \PP\{T^J(x_0)>n^a\} and \PP\{T_{\mathrm{disc}}^J(x_0)>n^a\} tend to one under the same joint law. Each assertion is equivalently a statement that the Gibbs mass of the indicated initial states tends to one in probability over J.

Appendix, A fixed number of crossings on polynomial time scales Read it in the paper TeX source, line 12
Show the TeX
For every fixed $\beta>1$ and every fixed $a>0$,
\begin{align}
 \PP_{J,\,x_0\sim\pi^J}
 \left\{d^J(x_0,n^a)>\frac14\right\}&\longrightarrow1,
 \\
 \PP_{J,\,x_0\sim\pi^J}
 \left\{d_{\mathrm{disc}}^J(x_0,\lfloor n^a\rfloor)
                                      >\frac14\right\}
 &\longrightarrow1.
 
\end{align}
Consequently both
$\PP\{T^J(x_0)>n^a\}$ and
$\PP\{T_{\mathrm{disc}}^J(x_0)>n^a\}$ tend to one under the same joint
law. Each assertion is equivalently a statement that the Gibbs mass of
the indicated initial states tends to one in probability over $J$.

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, bath, kernel, spin, heat and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: mixing.
  • Corollary 1.2 of Stretched-exponential barriers for typical SK initial states. Words of the Lean statement, its names split into words, that also appear in the theorem: gibbs, mass, pi and disorder. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: beta, tends, one and probability.
  • 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: 1/10000, gibbs, continuous, time, exp and pi. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: total, variation, typical, model, mixing, state and one more.
  • Theorem 1.1 of A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase. Words of the Lean statement, its names split into words, that also appear in the theorem: discrete, bath, heat and time. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: beta, tending, one and probability.
  • 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: couplings, gibbs, real, field, measure and gaussian. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: expectation, zero, tending, one and probability.
  • Proposition B.4 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: kappa, bath, heat, scaled, fields and exp. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: beta 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

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

“The formalization proves a stretched-exponential mixing obstruction for the zero-field Sherrington–Kirkpatrick model at every fixed inverse temperature $\beta>1$. At time $\exp(n^{1/10000})$, the expected Gibbs mass of initial configurations whose total-variation distance from equilibrium exceeds $1/4$ tends to one. Since this mass lies in $[0,1]$, it also tends to one in probability over the disorder.”

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

92 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: 59 to its own declarations, 26 to Mathlib’s documentation. 20 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names 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.

import Mathlibnamespace OAIsectionsectionnoncomputable sectionopen scoped BigOperators Topologyopen MeasureTheory ProbabilityTheory Filternamespace SKabbrev Config (n : ℕ) := Fin n → Boolabbrev Edge (n : ℕ) := {p : Fin n × Fin n // p.1 < p.2}abbrev Disorder (n : ℕ) := Edge n → ℝdef spin (b : Bool) : ℝ := if b then 1 else -1def hamiltonian {n : ℕ} (J : Disorder n) (x : Config n) : ℝ :=  (∑ e : Edge n, J e * spin (x e.val.1) * spin (x e.val.2)) / Real.sqrt ndef partition {n : ℕ} (β : ℝ) (J : Disorder n) : ℝ :=  ∑ x : Config n, Real.exp (β * hamiltonian J x)def gibbs {n : ℕ} (β : ℝ) (J : Disorder n) (x : Config n) : ℝ :=  Real.exp (β * hamiltonian J x) / partition β Jdef coupling {n : ℕ} (J : Disorder n) (i j : Fin n) : ℝ :=  if h : i < j then J ⟨(i, j), h⟩ else    if h : j < i then J ⟨(j, i), h⟩ else 0def localField {n : ℕ} (J : Disorder n) (x : Config n) (i : Fin n) : ℝ :=  (∑ j : Fin n, coupling J i j * spin (x j)) / Real.sqrt ndef heatBath {n : ℕ} (β : ℝ) (J : Disorder n) (x y : Config n) : ℝ :=  (∑ i : Fin n, if Function.update x i (y i) = y then    Real.exp (β * spin (y i) * localField J x i) /      (2 * Real.cosh (β * localField J x i)) else 0) / ndef discreteKernel {n : ℕ} (β : ℝ) (J : Disorder n) : ℕ → Config n → Config n → ℝ  | 0, x, y => if x = y then 1 else 0  | k + 1, x, y => ∑ z : Config n, heatBath β J x z * discreteKernel β J k z ydef continuousKernel {n : ℕ} (β : ℝ) (J : Disorder n) (t : ℝ)    (x y : Config n) : ℝ :=  ∑' k : ℕ, (Real.exp (-(n : ℝ) * t) * ((n : ℝ) * t) ^ k /    (Nat.factorial k : ℝ)) * discreteKernel β J k x ydef discreteDistance {n : ℕ} (β : ℝ) (J : Disorder n)    (x : Config n) (k : ℕ) : ℝ :=  (1 / 2 : ℝ) * ∑ y : Config n, |discreteKernel β J k x y - gibbs β J y|def continuousDistance {n : ℕ} (β : ℝ) (J : Disorder n)    (x : Config n) (t : ℝ) : ℝ :=  (1 / 2 : ℝ) * ∑ y : Config n, |continuousKernel β J t x y - gibbs β J y|def disorderLaw (n : ℕ) : Measure (Disorder n) :=  Measure.pi (fun _ : Edge n => gaussianReal 0 1)def kappa : ℝ := 1 / 10000def timeScale (n : ℕ) : ℝ := Real.exp ((n : ℝ) ^ kappa)def continuousBadMass (β : ℝ) (n : ℕ) : ℝ :=  ∫ J : Disorder n, (∑ x : Config n, if 1 / 4 < continuousDistance β J x (timeScale n)    then gibbs β J x else 0) ∂disorderLaw ndef discreteBadMass (β : ℝ) (n : ℕ) : ℝ :=  ∫ J : Disorder n, (∑ x : Config n, if 1 / 4 < discreteDistance β J x ⌊timeScale n⌋₊    then gibbs β J x else 0) ∂disorderLaw nend SKendendendnoncomputable sectionopen scoped BigOperators Topologyopen MeasureTheory ProbabilityTheory Filternamespace SKtheorem main (β : ℝ) (hβ : 1 < β) :    Tendsto (continuousBadMass β) atTop (𝓝 1) ∧    Tendsto (discreteBadMass β) atTop (𝓝 1) := by  sorryend SKendend 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.SKBarriers.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
19 local vocabulary declarations
Lines of Lean, without comments or blank lines
46
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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Plectis Atlas

Stretched-exponential barriers from typical equilibrium initial states

Family 227: Critical SK autocorrelation processes and dynamics across the temperature transition

Candidate pairing

Theorem (Polynomial-time obstruction from fixed-depth crossings)

For every fixed \beta>1 and every fixed a>0,

\begin{aligned}\PP_{J,\,x_0\sim\pi^J} \left\{d^J(x_0,n^a)>\frac14\right\}&\longrightarrow1, \\ \PP_{J,\,x_0\sim\pi^J} \left\{d_{\mathrm{disc}}^J(x_0,\lfloor n^a\rfloor) >\frac14\right\} &\longrightarrow1.\end{aligned}

Consequently both \PP\{T^J(x_0)>n^a\} and \PP\{T_{\mathrm{disc}}^J(x_0)>n^a\} tend to one under the same joint law. Each assertion is equivalently a statement that the Gibbs mass of the indicated initial states tends to one in probability over J.

Stretched-exponential barriers for typical SK initial states

TeX label fixed-thm:polynomial, build/appendices/fixed-depth-crossing.tex, line 12

The Lean statement says

theorem main (β : ℝ) (hβ : 1 < β) :    Tendsto (continuousBadMass β) atTop (𝓝 1) ∧    Tendsto (discreteBadMass β) atTop (𝓝 1) := by  sorry

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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

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

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