Lean statement

Polynomial mixing and connectivity for every graphical degree sequence

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.Problem315.switch_chain_main OAI.Problem315.switch_chain_tv_bound OAI.Problem315.switch_connectivity

Lean statement file: SwitchChain.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 131 link it.

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

The Lean statement

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

theorem switch_chain_main :    ∀ (n : Nat) (d : Fin n → Nat), 4 ≤ n → (∀ v, d v ≤ n - 1) → Graphical n d → ((∃ h : (∃ t : Nat, MixedAt n d t), mixingTime n d h ≤ 2 * n ^ 8) ∧ (stateCount n d = 1 → (MixedAt n d 0 ∧ ∀ h : (∃ t : Nat, MixedAt n d t), mixingTime n d h = 0)) ∧ (1 < stateCount n d → hasSpectralGapAtLeast n d ((1 : ℝ) / (24 * (n : ℝ) ^ 2 * (Nat.choose n 4 : ℝ))))) := by  sorry⋮theorem switch_chain_tv_bound :    ∀ (n : Nat) (d : Fin n → Nat), 4 ≤ n → (∀ v, d v ≤ n - 1) → Graphical n d → 1 < stateCount n d → ∀ (t : Nat) (G : GraphState n d), tvDistanceFrom n d t G ≤ Real.sqrt (stateCount n d : ℝ) / 2 * Real.exp (- (t : ℝ) / (24 * (n : ℝ) ^ 2 * (Nat.choose n 4 : ℝ))) := by  sorry⋮theorem switch_connectivity :    ∀ (n : Nat) (d : Fin n → Nat), 4 ≤ n → (∀ v, d v ≤ n - 1) → Graphical n d → ∀ G H : GraphState n d, Relation.ReflTransGen (SwitchStep n d) G H := 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.

definition OAI.Problem315.graphDegree line 11

def graphDegree {n : Nat} (G : SimpleGraph (Fin n)) (v : Fin n) : Nat := by  classical  exact (Finset.univ.filter (fun w => G.Adj v w)).card

definition OAI.Problem315.Realizes line 15

def Realizes (n : Nat) (d : Fin n → Nat) (G : SimpleGraph (Fin n)) : Prop :=  ∀ v, graphDegree G v = d v

abbreviation OAI.Problem315.GraphState line 18

abbrev GraphState (n : Nat) (d : Fin n → Nat) :=  {G : SimpleGraph (Fin n) // Realizes n d G}

definition OAI.Problem315.Graphical line 21

def Graphical (n : Nat) (d : Fin n → Nat) : Prop :=  Nonempty (GraphState n d)

abbreviation OAI.Problem315.FourSet line 24

abbrev FourSet (n : Nat) := {S : Finset (Fin n) // S.card = 4}

definition OAI.Problem315.IsPerfectMatchingOn line 26

def IsPerfectMatchingOn {n : Nat} (S : Finset (Fin n))    (M : SimpleGraph (Fin n)) : Prop :=  ∀ v, (v ∈ S → ∃! w, M.Adj v w) ∧    (v ∉ S → ∀ w, ¬ M.Adj v w)

abbreviation OAI.Problem315.PerfectMatching line 31

abbrev PerfectMatching {n : Nat} (S : FourSet n) :=  {M : SimpleGraph (Fin n) // IsPerfectMatchingOn S.1 M}

abbreviation OAI.Problem315.SwitchProposal line 34

abbrev SwitchProposal (n : Nat) :=  Σ S : FourSet n,    {p : PerfectMatching S × PerfectMatching S // p.1 ≠ p.2}

definition OAI.Problem315.removedMatching line 38

def removedMatching {n : Nat} (p : SwitchProposal n) : SimpleGraph (Fin n) :=  p.2.val.1.val

definition OAI.Problem315.addedMatching line 41

def addedMatching {n : Nat} (p : SwitchProposal n) : SimpleGraph (Fin n) :=  p.2.val.2.val

definition OAI.Problem315.validSwitch line 44

def validSwitch {n : Nat} (G : SimpleGraph (Fin n))    (p : SwitchProposal n) : Prop :=  (∀ ⦃u v⦄, (removedMatching p).Adj u v → G.Adj u v) ∧  (∀ ⦃u v⦄, (addedMatching p).Adj u v → ¬ G.Adj u v)

definition OAI.Problem315.replaceEdges line 49

def replaceEdges {n : Nat} (G M N : SimpleGraph (Fin n)) :    SimpleGraph (Fin n) where

Show the remaining 15 lines

  Adj u v := (G.Adj u v ∧ ¬ M.Adj u v) ∨ N.Adj u v  symm := by    constructor    intro u v h    rcases h with h | h    · left      exact ⟨G.symm.symm u v h.1, fun hm => h.2 (M.symm.symm v u hm)⟩    · right      exact N.symm.symm u v h  loopless := by    constructor    intro v h    rcases h with h | h    · exact G.loopless.irrefl v h.1    · exact N.loopless.irrefl v h

definition OAI.Problem315.proposalResult line 67

def proposalResult {n : Nat} (G : SimpleGraph (Fin n))    (p : SwitchProposal n) : SimpleGraph (Fin n) := by  classical  exact    if validSwitch G p then      replaceEdges G (removedMatching p) (addedMatching p)    else G

definition OAI.Problem315.allGraphStates line 75

def allGraphStates (n : Nat) (d : Fin n → Nat) :    Finset (GraphState n d) := by  classical  exact Finset.univ

definition OAI.Problem315.allProposals line 80

def allProposals (n : Nat) : Finset (SwitchProposal n) := by  classical  exact Finset.univ

definition OAI.Problem315.stateCount line 84

def stateCount (n : Nat) (d : Fin n → Nat) : Nat :=  (allGraphStates n d).card

definition OAI.Problem315.switchKernel line 87

def switchKernel (n : Nat) (d : Fin n → Nat)    (G H : GraphState n d) : ℝ := by  classical  exact    (if G = H then (1 : ℝ) / 2 else 0) +      (((allProposals n).filter          (fun p => proposalResult G.val p = H.val)).card : ℝ) /        (12 * (Nat.choose n 4 : ℝ))

definition OAI.Problem315.kernelPow line 96

def kernelPow (n : Nat) (d : Fin n → Nat) :    Nat → GraphState n d → GraphState n d → ℝ := by

Show the remaining 9 lines

  classical  intro t  induction t with  | zero =>      exact fun G H => if G = H then 1 else 0  | succ t previous =>      exact fun G H =>        Finset.sum (allGraphStates n d)          (fun K => previous G K * switchKernel n d K H)

definition OAI.Problem315.uniformAverage line 108

def uniformAverage (n : Nat) (d : Fin n → Nat)    (f : GraphState n d → ℝ) : ℝ :=  Finset.sum (allGraphStates n d) (fun G => f G) /    (stateCount n d : ℝ)

definition OAI.Problem315.uniformInner line 113

def uniformInner (n : Nat) (d : Fin n → Nat)    (f g : GraphState n d → ℝ) : ℝ :=  uniformAverage n d (fun G => f G * g G)

definition OAI.Problem315.uniformVariance line 117

def uniformVariance (n : Nat) (d : Fin n → Nat)    (f : GraphState n d → ℝ) : ℝ :=  uniformAverage n d    (fun G => (f G - uniformAverage n d f) ^ 2)

definition OAI.Problem315.markovApply line 122

def markovApply (n : Nat) (d : Fin n → Nat)    (f : GraphState n d → ℝ) (G : GraphState n d) : ℝ :=  Finset.sum (allGraphStates n d)    (fun H => switchKernel n d G H * f H)

definition OAI.Problem315.dirichletEnergy line 127

def dirichletEnergy (n : Nat) (d : Fin n → Nat)    (f : GraphState n d → ℝ) : ℝ :=  uniformInner n d f (fun G => f G - markovApply n d f G)

definition OAI.Problem315.hasSpectralGapAtLeast line 131

def hasSpectralGapAtLeast (n : Nat) (d : Fin n → Nat) (γ : ℝ) : Prop :=  ∀ f : GraphState n d → ℝ,    γ * uniformVariance n d f ≤ dirichletEnergy n d f

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

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: spectral, mix, gap, least, graphical and degree. Found in the paper’s title, which its main theorem states: chain and switch. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: binom.

Theorem 1.1 of Polynomial mixing of the switch chain for every graphical degree sequence

Theorem

For every n\ge4 and every graphical degree vector d,

t_{\mathrm{mix}}(d)\le 2n^8.

If |\Omega_d|>1, the spectral gap of P_d is at least [24n^2\binom n4]^{-1}.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

For every $n\ge4$ and every graphical degree vector $d$,
\[
 t_{\mathrm{mix}}(d)\le 2n^8.
\]
If $|\Omega_d|>1$, the spectral gap of $P_d$ is at least
$[24n^2\binom n4]^{-1}$.

Other possible paper matches, closest wording first

Corollary 1.2 of Polynomial mixing of the switch chain for every graphical degree sequence. Words of the Lean statement, its names split into words, that also appear in the theorem: time, bounded, uniform, simple, graphical, graph and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: undirected and polynomial.
Corollary 7.3 of Polynomial mixing of the switch chain for every graphical degree sequence. Words of the Lean statement, its names split into words, that also appear in the theorem: kernel and real.
Corollary 6.2 of Polynomial mixing of the switch chain for every graphical degree sequence. Words of the Lean statement, its names split into words, that also appear in the theorem: energy and graph.
Proposition 5.1 of Polynomial mixing of the switch chain for every graphical degree sequence. Words of the Lean statement, its names split into words, that also appear in the theorem: real.
Proposition 4.1 of Polynomial mixing of the switch chain for every graphical degree sequence. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Corollary 6.3 of Polynomial mixing of the switch chain for every graphical degree sequence. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: binom.

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 131’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.

MathVet and cert-machine disagree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: Mixing time <= 2 n^8 (TV 1/4, worst-case start), spectral gap, TV bound and connectivity are stated; the exact uniform sampler (expected polynomial bit time) is not.
Definitions the group asks readers to compare with the paper: Mixing is defined by hand: kernelPow (matrix powers via sums over allGraphStates), tvDistanceFrom := (1/2) ∑ |P^t(G,H) - 1/|Ω||, MixedAt, mixingTime := Nat.find h (the proof argument h is part of the statement), hasSpectralGapAtLeast via Dirichlet form; standard in content, no Mathlib Markov-chain objects
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The exactly uniform sampler is not stated (also n ≥ 4, a harmless restriction).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves the simple undirected case: for $n\ge4$, the lazy chain that proposes switches on four vertices has total-variation mixing time at distance $1/4$ at most $2n^8$. It also proves switch connectivity, the stated exponential total-variation bound, and a spectral gap of at least $1/(24n^2\binom n4)$ when more than one graph realizes the degree sequence.”
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

171 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: 74 to its own declarations, 127 to Mathlib’s documentation. 31 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

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

It reaches all 28 other declarations of the file; with the theorem itself they span 125 lines. 8 of them are named in the statement itself.

Names in the namespace OAI.Problem315 are listed without it.

def graphDegreeline 11library
def Realizesline 15library
abbrev GraphStateline 18in the statementlibrary
def Graphicalline 21in the statementlibrary
abbrev FourSetline 24library
def IsPerfectMatchingOnline 26library
abbrev PerfectMatchingline 31library
abbrev SwitchProposalline 34library
def removedMatchingline 38library
def addedMatchingline 41library
def validSwitchline 44library
def replaceEdgesline 49library
def proposalResultline 67library
def allGraphStatesline 75library
def allProposalsline 80library
def stateCountline 84in the statementlibrary
def switchKernelline 87library
def kernelPowline 96library
def uniformAverageline 108library
def uniformInnerline 113library
def uniformVarianceline 117library
def markovApplyline 122library
def dirichletEnergyline 127library
def hasSpectralGapAtLeastline 131in the statementlibrary
def tvDistanceFromline 135in the statementlibrary
def MixedAtline 142in the statementlibrary
def mixingTimeline 145in the statementlibrary
def SwitchStepline 150in the statementlibrary
theorem switch_chain_mainline 155selected theoremlibrary
theorem switch_chain_tv_boundline 159selected theoremlibrary
theorem switch_connectivityline 163selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperatorsnamespace Problem315def graphDegree {n : Nat} (G : SimpleGraph (Fin n)) (v : Fin n) : Nat := by  classical  exact (Finset.univ.filter (fun w => G.Adj v w)).carddef Realizes (n : Nat) (d : Fin n → Nat) (G : SimpleGraph (Fin n)) : Prop :=  ∀ v, graphDegree G v = d vabbrev GraphState (n : Nat) (d : Fin n → Nat) :=  {G : SimpleGraph (Fin n) // Realizes n d G}def Graphical (n : Nat) (d : Fin n → Nat) : Prop :=  Nonempty (GraphState n d)abbrev FourSet (n : Nat) := {S : Finset (Fin n) // S.card = 4}def IsPerfectMatchingOn {n : Nat} (S : Finset (Fin n))    (M : SimpleGraph (Fin n)) : Prop :=  ∀ v, (v ∈ S → ∃! w, M.Adj v w) ∧    (v ∉ S → ∀ w, ¬ M.Adj v w)abbrev PerfectMatching {n : Nat} (S : FourSet n) :=  {M : SimpleGraph (Fin n) // IsPerfectMatchingOn S.1 M}abbrev SwitchProposal (n : Nat) :=  Σ S : FourSet n,    {p : PerfectMatching S × PerfectMatching S // p.1 ≠ p.2}def removedMatching {n : Nat} (p : SwitchProposal n) : SimpleGraph (Fin n) :=  p.2.val.1.valdef addedMatching {n : Nat} (p : SwitchProposal n) : SimpleGraph (Fin n) :=  p.2.val.2.valdef validSwitch {n : Nat} (G : SimpleGraph (Fin n))    (p : SwitchProposal n) : Prop :=  (∀ ⦃u v⦄, (removedMatching p).Adj u v → G.Adj u v) ∧  (∀ ⦃u v⦄, (addedMatching p).Adj u v → ¬ G.Adj u v)def replaceEdges {n : Nat} (G M N : SimpleGraph (Fin n)) :    SimpleGraph (Fin n) where  Adj u v := (G.Adj u v ∧ ¬ M.Adj u v) ∨ N.Adj u v  symm := by    constructor    intro u v h    rcases h with h | h    · left      exact ⟨G.symm.symm u v h.1, fun hm => h.2 (M.symm.symm v u hm)⟩    · right      exact N.symm.symm u v h  loopless := by    constructor    intro v h    rcases h with h | h    · exact G.loopless.irrefl v h.1    · exact N.loopless.irrefl v hdef proposalResult {n : Nat} (G : SimpleGraph (Fin n))    (p : SwitchProposal n) : SimpleGraph (Fin n) := by  classical  exact    if validSwitch G p then      replaceEdges G (removedMatching p) (addedMatching p)    else Gdef allGraphStates (n : Nat) (d : Fin n → Nat) :    Finset (GraphState n d) := by  classical  exact Finset.univdef allProposals (n : Nat) : Finset (SwitchProposal n) := by  classical  exact Finset.univdef stateCount (n : Nat) (d : Fin n → Nat) : Nat :=  (allGraphStates n d).carddef switchKernel (n : Nat) (d : Fin n → Nat)    (G H : GraphState n d) : ℝ := by  classical  exact    (if G = H then (1 : ℝ) / 2 else 0) +      (((allProposals n).filter          (fun p => proposalResult G.val p = H.val)).card : ℝ) /        (12 * (Nat.choose n 4 : ℝ))def kernelPow (n : Nat) (d : Fin n → Nat) :    Nat → GraphState n d → GraphState n d → ℝ := by  classical  intro t  induction t with  | zero =>      exact fun G H => if G = H then 1 else 0  | succ t previous =>      exact fun G H =>        Finset.sum (allGraphStates n d)          (fun K => previous G K * switchKernel n d K H)def uniformAverage (n : Nat) (d : Fin n → Nat)    (f : GraphState n d → ℝ) : ℝ :=  Finset.sum (allGraphStates n d) (fun G => f G) /    (stateCount n d : ℝ)def uniformInner (n : Nat) (d : Fin n → Nat)    (f g : GraphState n d → ℝ) : ℝ :=  uniformAverage n d (fun G => f G * g G)def uniformVariance (n : Nat) (d : Fin n → Nat)    (f : GraphState n d → ℝ) : ℝ :=  uniformAverage n d    (fun G => (f G - uniformAverage n d f) ^ 2)def markovApply (n : Nat) (d : Fin n → Nat)    (f : GraphState n d → ℝ) (G : GraphState n d) : ℝ :=  Finset.sum (allGraphStates n d)    (fun H => switchKernel n d G H * f H)def dirichletEnergy (n : Nat) (d : Fin n → Nat)    (f : GraphState n d → ℝ) : ℝ :=  uniformInner n d f (fun G => f G - markovApply n d f G)def hasSpectralGapAtLeast (n : Nat) (d : Fin n → Nat) (γ : ℝ) : Prop :=  ∀ f : GraphState n d → ℝ,    γ * uniformVariance n d f ≤ dirichletEnergy n d fdef tvDistanceFrom (n : Nat) (d : Fin n → Nat)    (t : Nat) (G : GraphState n d) : ℝ :=  (1 / 2 : ℝ) *    Finset.sum (allGraphStates n d)      (fun H =>        |kernelPow n d t G H - (1 : ℝ) / (stateCount n d : ℝ)|)def MixedAt (n : Nat) (d : Fin n → Nat) (t : Nat) : Prop :=  ∀ G : GraphState n d, tvDistanceFrom n d t G ≤ (1 : ℝ) / 4def mixingTime (n : Nat) (d : Fin n → Nat)    (h : ∃ t : Nat, MixedAt n d t) : Nat := by  classical  exact Nat.find hdef SwitchStep (n : Nat) (d : Fin n → Nat)    (G H : GraphState n d) : Prop :=  ∃ p : SwitchProposal n,    validSwitch G.val p ∧ proposalResult G.val p = H.valtheorem switch_chain_main :    ∀ (n : Nat) (d : Fin n → Nat), 4 ≤ n → (∀ v, d v ≤ n - 1) → Graphical n d → ((∃ h : (∃ t : Nat, MixedAt n d t), mixingTime n d h ≤ 2 * n ^ 8) ∧ (stateCount n d = 1 → (MixedAt n d 0 ∧ ∀ h : (∃ t : Nat, MixedAt n d t), mixingTime n d h = 0)) ∧ (1 < stateCount n d → hasSpectralGapAtLeast n d ((1 : ℝ) / (24 * (n : ℝ) ^ 2 * (Nat.choose n 4 : ℝ))))) := by  sorrytheorem switch_chain_tv_bound :    ∀ (n : Nat) (d : Fin n → Nat), 4 ≤ n → (∀ v, d v ≤ n - 1) → Graphical n d → 1 < stateCount n d → ∀ (t : Nat) (G : GraphState n d), tvDistanceFrom n d t G ≤ Real.sqrt (stateCount n d : ℝ) / 2 * Real.exp (- (t : ℝ) / (24 * (n : ℝ) ^ 2 * (Nat.choose n 4 : ℝ))) := by  sorrytheorem switch_connectivity :    ∀ (n : Nat) (d : Fin n → Nat), 4 ≤ n → (∀ v, d v ≤ n - 1) → Graphical n d → ∀ G H : GraphState n d, Relation.ReflTransGen (SwitchStep n d) G H := by  sorryend Problem315endend 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.

Polynomial mixing of the switch chain for every graphical degree sequence

Theorem

For every n\ge4 and every graphical degree vector d,

t_{\mathrm{mix}}(d)\le 2n^8.

If |\Omega_d|>1, the spectral gap of P_d is at least [24n^2\binom n4]^{-1}.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

For every $n\ge4$ and every graphical degree vector $d$,
\[
 t_{\mathrm{mix}}(d)\le 2n^8.
\]
If $|\Omega_d|>1$, the spectral gap of $P_d$ is at least
$[24n^2\binom n4]^{-1}$.

Corollary (Exact uniform sampling)

There is a uniform randomized algorithm that, given any finite graphical labeled degree vector d=(d_1,\ldots,d_n), uses unbiased random bits, terminates almost surely, and outputs a graph chosen exactly uniformly from the simple undirected graphs on \{1,\ldots,n\} with labeled degree vector d. Its expected bit running time is bounded by a fixed polynomial in the binary encoding length of d.

Section 1, Introduction Read it in the paper TeX source, line 42

Show the TeX

There is a uniform randomized algorithm that, given any finite graphical
labeled degree vector $d=(d_1,\ldots,d_n)$, uses unbiased random bits,
terminates almost surely, and outputs a graph chosen exactly uniformly
from the simple undirected graphs on $\{1,\ldots,n\}$ with labeled
degree vector $d$. Its expected bit running time is bounded by a fixed
polynomial in the binary encoding length of $d$.

Configuration and file facts

File: lean/ComparatorChallenges/SwitchChain.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 131 link it.
Solution module: OAI.Probability.SwitchChain.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: 28 local vocabulary declarations
Lines of Lean, without comments or blank lines: 125
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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

Plectis Atlas