Lean statement

Equilibrium starts, covariance, and mixing lower bounds

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.

Theorems selected for verification: OAI.CriticalSK.realized_equilibrium_initial_states OAI.CriticalSK.covariance_and_linear_tests OAI.CriticalSK.critical_mixing_lower_bounds

Lean statement file: CriticalSK.leanShow on the mapFamily 227. OpenAI lists this as a main result in its Lean catalogue.

See the closest candidates 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 realized_equilibrium_initial_states :    (∀ t : ℕ → ℝ, (∀ n, 0 ≤ t n) →      Tendsto (fun n => t n / (n : ℝ) ^ ((2 : ℝ) / 3)) atTop (𝓝 0) →      ∀ δ : ℝ, 0 < δ →        Tendsto (fun n => (disorderLaw n {W |          δ ≤ |continuousGoodMass W (t n) - 1|}).toReal) atTop (𝓝 0)) ∧    (∀ k : ℕ → ℕ,      Tendsto (fun n => (k n : ℝ) / (n : ℝ) ^ ((5 : ℝ) / 3)) atTop (𝓝 0) →      ∀ δ : ℝ, 0 < δ →        Tendsto (fun n => (disorderLaw n {W |          δ ≤ |discreteGoodMass W (k n) - 1|}).toReal) atTop (𝓝 0)) := by  sorry⋮theorem covariance_and_linear_tests (M : ℕ → ℝ)    (hMpos : ∀ n, 0 < M n) (hM : Tendsto M atTop atTop) :    Tendsto (fun n : ℕ => (disorderLaw n {W |      (n : ℝ) ^ ((2 : ℝ) / 3) / M n ≤ covarianceNorm W ∧      covarianceNorm W ≤ M n * (n : ℝ) ^ ((2 : ℝ) / 3) ∧      (n : ℝ) ^ ((2 : ℝ) / 3) / M n ≤ linearRayleigh W ∧      linearRayleigh W ≤ M n * (n : ℝ) ^ ((2 : ℝ) / 3)}).toReal) atTop (𝓝 1) := by  sorry⋮theorem critical_mixing_lower_bounds (ε : ℝ) (hε : 0 < ε) :    Tendsto (fun n => (disorderLaw n).real {W |      (n : ℝ) ^ (2 / 3 - ε) ≤ continuousMixingTime W}) atTop (𝓝 1) ∧    Tendsto (fun n => (disorderLaw n).real {W |      (n : ℝ) ^ (5 / 3 - ε) ≤ (discreteMixingTime W : ℝ)}) 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.CriticalSK.Spin line 20
```
abbrev Spin (n : ℕ) := Fin n → Bool
```

abbreviation OAI.CriticalSK.Edge line 22

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

abbreviation OAI.CriticalSK.Disorder line 24

abbrev Disorder (n : ℕ) := Edge n → ℝ

definition OAI.CriticalSK.spinValue line 26

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

definition OAI.CriticalSK.hamiltonian line 28

def hamiltonian {n : ℕ} (W : Disorder n) (x : Spin n) : ℝ :=  ∑ e : Edge n, W e * spinValue (x e.val.1) * spinValue (x e.val.2)

definition OAI.CriticalSK.partition line 31

def partition {n : ℕ} (W : Disorder n) : ℝ :=  ∑ x : Spin n, Real.exp (hamiltonian W x)

definition OAI.CriticalSK.gibbs line 34

def gibbs {n : ℕ} (W : Disorder n) (x : Spin n) : ℝ :=  Real.exp (hamiltonian W x) / partition W

definition OAI.CriticalSK.disorderLaw line 37

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

definition OAI.CriticalSK.sameExcept line 40

def sameExcept {n : ℕ} (i : Fin n) (x y : Spin n) : Prop :=  ∀ j, j ≠ i → y j = x j

definition OAI.CriticalSK.siteKernel line 46

def siteKernel {n : ℕ} (W : Disorder n) (i : Fin n) : Matrix (Spin n) (Spin n) ℝ :=  fun x y => if sameExcept i x y then    gibbs W y / (∑ z : Spin n, if sameExcept i x z then gibbs W z else 0)  else 0

definition OAI.CriticalSK.discreteKernel line 51

def discreteKernel {n : ℕ} (W : Disorder n) : Matrix (Spin n) (Spin n) ℝ :=  (n : ℝ)⁻¹ • ∑ i : Fin n, siteKernel W i

definition OAI.CriticalSK.generator line 54

def generator {n : ℕ} (W : Disorder n) : Matrix (Spin n) (Spin n) ℝ :=  ∑ i : Fin n, (siteKernel W i - 1)

definition OAI.CriticalSK.continuousKernel line 57

def continuousKernel {n : ℕ} (W : Disorder n) (t : ℝ) :    Matrix (Spin n) (Spin n) ℝ :=  NormedSpace.exp (t • generator W)

definition OAI.CriticalSK.totalVariation line 61

def totalVariation {n : ℕ} (W : Disorder n) (p : Spin n → ℝ) : ℝ :=  (1 / 2 : ℝ) * ∑ y : Spin n, |p y - gibbs W y|

definition OAI.CriticalSK.continuousDistance line 64

def continuousDistance {n : ℕ} (W : Disorder n) (t : ℝ) (x : Spin n) : ℝ :=  totalVariation W (continuousKernel W t x)

definition OAI.CriticalSK.discreteDistance line 67

def discreteDistance {n : ℕ} (W : Disorder n) (k : ℕ) (x : Spin n) : ℝ :=  totalVariation W ((discreteKernel W ^ k) x)

definition OAI.CriticalSK.continuousMixingTime line 70

def continuousMixingTime {n : ℕ} (W : Disorder n) : ℝ :=  sInf {t : ℝ | 0 ≤ t ∧ ∀ x, continuousDistance W t x ≤ 1 / 4}

definition OAI.CriticalSK.discreteMixingTime line 73

def discreteMixingTime {n : ℕ} (W : Disorder n) : ℕ :=  sInf {k : ℕ | ∀ x, discreteDistance W k x ≤ 1 / 4}

definition OAI.CriticalSK.mean line 76

def mean {n : ℕ} (W : Disorder n) (f : Spin n → ℝ) : ℝ :=  ∑ x : Spin n, gibbs W x * f x

definition OAI.CriticalSK.variance line 79

def variance {n : ℕ} (W : Disorder n) (f : Spin n → ℝ) : ℝ :=  mean W (fun x => (f x - mean W f) ^ 2)

definition OAI.CriticalSK.dirichlet line 82

def dirichlet {n : ℕ} (W : Disorder n) (f : Spin n → ℝ) : ℝ :=  ∑ i : Fin n, mean W (fun x => (f x - ∑ y : Spin n, siteKernel W i x y * f y) ^ 2)

definition OAI.CriticalSK.linearObservable line 85

def linearObservable {n : ℕ} (a : Fin n → ℝ) (x : Spin n) : ℝ :=  ∑ i : Fin n, a i * spinValue (x i)

definition OAI.CriticalSK.covariance line 88

def covariance {n : ℕ} (W : Disorder n) : Matrix (Fin n) (Fin n) ℝ :=  fun i j => mean W (fun x =>    (spinValue (x i) - mean W (fun y => spinValue (y i))) *    (spinValue (x j) - mean W (fun y => spinValue (y j))))

definition OAI.CriticalSK.covarianceNorm line 93

def covarianceNorm {n : ℕ} (W : Disorder n) : ℝ :=  ‖(covariance W).toEuclideanLin.toContinuousLinearMap‖

3 more definitions are reached the same way; the full file lists them.

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Statements in A typical-start upper bound for low-temperature SK Glauber dynamics come first because the release’s name for this Lean statement shares words with that paper’s title.

Other possible paper matches, closest wording first

Theorem 1.2 of Critical mixing in the Sherrington–Kirkpatrick model. Words of the Lean statement, its names split into words, that also appear in the theorem: lin, equilibrium, 5/3, integer, nonnegative, starts and eleven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sequence and positive.
Theorem 1.1 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: equilibrium, 5/3, integer, realized, 1/4, initial and five more. Found in the paper’s title, which its main theorem states: critical. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sequence.
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, gibbs, start, continuous, mixing and five more.
Theorem 1 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: gibbs, realized, mixing, state, exp, pi and one more. Found in the paper’s title, which its main theorem states: start and bound.
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, kernel, spin, realization, site, mixing and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bath, heat and upper.

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 575 s; peak memory 6.5 GiB).
Reported 11 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 deterministic times $t_n=o(n^{2/3})$ or update counts $k_n=o(n^{5/3})$, the Gibbs mass of realized starting states still farther than $1/4$ from equilibrium tends to one in probability. For every deterministic positive sequence $M_n\to\infty$, the covariance operator norm and the linear Rayleigh supremum, using the unscaled heat-bath Dirichlet form, both lie between $n^{2/3}/M_n$ and $M_n n^{2/3}$ with probability tending to one. The covariance result does not give an upper bound for the full inverse spectral gap.”
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

149 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: 105 to its own declarations, 49 to Mathlib’s documentation. 30 declared names also link to the same declaration in OpenAI’s library. 7 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves realized_equilibrium_initial_states 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 27 of the file’s 28 other declarations; with the theorem itself they span 82 lines. 7 of them are named in the statement itself.

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

abbrev Spinline 20library
abbrev Edgeline 22library
abbrev Disorderline 24library
def spinValueline 26library
def hamiltonianline 28library
def partitionline 31library
def gibbsline 34library
def disorderLawline 37in the statementlibrary
def sameExceptline 40library
def siteKernelline 46library
def discreteKernelline 51library
def generatorline 54library
def continuousKernelline 57library
def totalVariationline 61library
def continuousDistanceline 64library
def discreteDistanceline 67library
def continuousMixingTimeline 70in the statementlibrary
def discreteMixingTimeline 73in the statementlibrary
def meanline 76library
def varianceline 79library
def dirichletline 82library
def linearObservableline 85library
def covarianceline 88library
def covarianceNormline 93in the statementlibrary
def linearRayleighline 98in the statementlibrary
def continuousGoodMassline 103in the statementlibrary
def discreteGoodMassline 107in the statementlibrary
theorem realized_equilibrium_initial_statesline 112selected theoremlibrary
theorem covariance_and_linear_testsline 127selected theoremlibrary
theorem critical_mixing_lower_boundsline 138selected theoremlibrary

Other Lean rules that may affect the statement 1

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance {n : ℕ} (i : Fin n) (x y : Spin n) : Decidable (sameExcept i… line 43

import Mathlibnamespace OAI/-!# Critical SK equilibrium starts, covariance, and mixing lower boundsThe zero-field model has independent Gaussian edge couplings of variance `1 / n`.Continuous heat bath updates each site at rate one; a discrete attempt chooses asite uniformly. Distances and mixing times use total variation threshold `1 / 4`.-/noncomputable sectionopen scoped BigOperators Topology NNReal ENNRealopen MeasureTheory ProbabilityTheory Filternamespace CriticalSKabbrev 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 hamiltonian {n : ℕ} (W : Disorder n) (x : Spin n) : ℝ :=  ∑ e : Edge n, W e * spinValue (x e.val.1) * spinValue (x e.val.2)def partition {n : ℕ} (W : Disorder n) : ℝ :=  ∑ x : Spin n, Real.exp (hamiltonian W x)def gibbs {n : ℕ} (W : Disorder n) (x : Spin n) : ℝ :=  Real.exp (hamiltonian W x) / partition Wdef disorderLaw (n : ℕ) : Measure (Disorder n) :=  Measure.pi fun _ : Edge n => gaussianReal 0 (n : ℝ≥0)⁻¹def sameExcept {n : ℕ} (i : Fin n) (x y : Spin n) : Prop :=  ∀ j, j ≠ i → y j = x jinstance {n : ℕ} (i : Fin n) (x y : Spin n) : Decidable (sameExcept i x y) :=  Classical.propDecidable _def siteKernel {n : ℕ} (W : Disorder n) (i : Fin n) : Matrix (Spin n) (Spin n) ℝ :=  fun x y => if sameExcept i x y then    gibbs W y / (∑ z : Spin n, if sameExcept i x z then gibbs W z else 0)  else 0def discreteKernel {n : ℕ} (W : Disorder n) : Matrix (Spin n) (Spin n) ℝ :=  (n : ℝ)⁻¹ • ∑ i : Fin n, siteKernel W idef generator {n : ℕ} (W : Disorder n) : Matrix (Spin n) (Spin n) ℝ :=  ∑ i : Fin n, (siteKernel W i - 1)def continuousKernel {n : ℕ} (W : Disorder n) (t : ℝ) :    Matrix (Spin n) (Spin n) ℝ :=  NormedSpace.exp (t • generator W)def totalVariation {n : ℕ} (W : Disorder n) (p : Spin n → ℝ) : ℝ :=  (1 / 2 : ℝ) * ∑ y : Spin n, |p y - gibbs W y|def continuousDistance {n : ℕ} (W : Disorder n) (t : ℝ) (x : Spin n) : ℝ :=  totalVariation W (continuousKernel W t x)def discreteDistance {n : ℕ} (W : Disorder n) (k : ℕ) (x : Spin n) : ℝ :=  totalVariation W ((discreteKernel W ^ k) x)def continuousMixingTime {n : ℕ} (W : Disorder n) : ℝ :=  sInf {t : ℝ | 0 ≤ t ∧ ∀ x, continuousDistance W t x ≤ 1 / 4}def discreteMixingTime {n : ℕ} (W : Disorder n) : ℕ :=  sInf {k : ℕ | ∀ x, discreteDistance W k x ≤ 1 / 4}def mean {n : ℕ} (W : Disorder n) (f : Spin n → ℝ) : ℝ :=  ∑ x : Spin n, gibbs W x * f xdef variance {n : ℕ} (W : Disorder n) (f : Spin n → ℝ) : ℝ :=  mean W (fun x => (f x - mean W f) ^ 2)def dirichlet {n : ℕ} (W : Disorder n) (f : Spin n → ℝ) : ℝ :=  ∑ i : Fin n, mean W (fun x => (f x - ∑ y : Spin n, siteKernel W i x y * f y) ^ 2)def linearObservable {n : ℕ} (a : Fin n → ℝ) (x : Spin n) : ℝ :=  ∑ i : Fin n, a i * spinValue (x i)def covariance {n : ℕ} (W : Disorder n) : Matrix (Fin n) (Fin n) ℝ :=  fun i j => mean W (fun x =>    (spinValue (x i) - mean W (fun y => spinValue (y i))) *    (spinValue (x j) - mean W (fun y => spinValue (y j))))def covarianceNorm {n : ℕ} (W : Disorder n) : ℝ :=  ‖(covariance W).toEuclideanLin.toContinuousLinearMap‖/-- The supremum of variance divided by the rate-one Dirichlet form over nonzerolinear spin observables. -/def linearRayleigh {n : ℕ} (W : Disorder n) : ℝ :=  sSup {r : ℝ | ∃ a : Fin n → ℝ, a ≠ 0 ∧    r = variance W (linearObservable a) / dirichlet W (linearObservable a)}/-- Gibbs mass of realized initial states still farther than `1 / 4` from equilibrium. -/def continuousGoodMass {n : ℕ} (W : Disorder n) (t : ℝ) : ℝ :=  ∑ x : Spin n, if 1 / 4 < continuousDistance W t x then gibbs W x else 0/-- Gibbs mass of realized initial states still farther than `1 / 4` after `k` attempts. -/def discreteGoodMass {n : ℕ} (W : Disorder n) (k : ℕ) : ℝ :=  ∑ x : Spin n, if 1 / 4 < discreteDistance W k x then gibbs W x else 0/-- For deterministic nonnegative times `o(n^(2/3))` and attempt counts `o(n^(5/3))`,the Gibbs mass of obstructed initial states tends to one in disorder probability. -/theorem realized_equilibrium_initial_states :    (∀ t : ℕ → ℝ, (∀ n, 0 ≤ t n) →      Tendsto (fun n => t n / (n : ℝ) ^ ((2 : ℝ) / 3)) atTop (𝓝 0) →      ∀ δ : ℝ, 0 < δ →        Tendsto (fun n => (disorderLaw n {W |          δ ≤ |continuousGoodMass W (t n) - 1|}).toReal) atTop (𝓝 0)) ∧    (∀ k : ℕ → ℕ,      Tendsto (fun n => (k n : ℝ) / (n : ℝ) ^ ((5 : ℝ) / 3)) atTop (𝓝 0) →      ∀ δ : ℝ, 0 < δ →        Tendsto (fun n => (disorderLaw n {W |          δ ≤ |discreteGoodMass W (k n) - 1|}).toReal) atTop (𝓝 0)) := by  sorry/-- Both the covariance operator norm and the linear inverse-Rayleigh supremum liebetween `n^(2/3) / M n` and `M n * n^(2/3)` with probability tending to one. -/theorem covariance_and_linear_tests (M : ℕ → ℝ)    (hMpos : ∀ n, 0 < M n) (hM : Tendsto M atTop atTop) :    Tendsto (fun n : ℕ => (disorderLaw n {W |      (n : ℝ) ^ ((2 : ℝ) / 3) / M n ≤ covarianceNorm W ∧      covarianceNorm W ≤ M n * (n : ℝ) ^ ((2 : ℝ) / 3) ∧      (n : ℝ) ^ ((2 : ℝ) / 3) / M n ≤ linearRayleigh W ∧      linearRayleigh W ≤ M n * (n : ℝ) ^ ((2 : ℝ) / 3)}).toReal) atTop (𝓝 1) := by  sorry/-- Worst-start mixing takes at least `n^(2/3-ε)` continuous time and`n^(5/3-ε)` discrete attempts with probability tending to one. -/theorem critical_mixing_lower_bounds (ε : ℝ) (hε : 0 < ε) :    Tendsto (fun n => (disorderLaw n).real {W |      (n : ℝ) ^ (2 / 3 - ε) ≤ continuousMixingTime W}) atTop (𝓝 1) ∧    Tendsto (fun n => (disorderLaw n).real {W |      (n : ℝ) ^ (5 / 3 - ε) ≤ (discreteMixingTime W : ℝ)}) atTop (𝓝 1) := by  sorryend CriticalSKendend 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

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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

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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

File: lean/ComparatorChallenges/CriticalSK.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/MathematicalPhysics/CriticalSK/Rayleigh.lean
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: 28 local vocabulary declarations
Lines of Lean, without comments or blank lines: 84
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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