Lean statement

Conditional coordinate-sweep moment estimate

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.CoordinateSweeps.conditional_main OAI.CoordinateSweeps.all_grid_moment

Lean statement file: CoordinateSweeps.leanShow on the mapFamily 238. OpenAI lists this as a main result in its Lean catalogue.

Read the cited theorem in the paper 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.

def ConditionalMain : Prop :=  ∃ (r q : ℕ) (zStar : ℝ), 1 ≤ r ∧ 1 ≤ q ∧ 0 < zStar ∧ zStar ≤ 1 / ((2 : ℕ) : ℝ) ∧    ∀ (G : Grid), G.Allowed r → ∀ (h : ℕ) (H : G.Holes h) (hf : H.Feasible),    ∀ z ∈ Set.Icc 0 zStar, ∀ ρ : UnitaryIrrep H.stabilizer,      logMoment (schattenMoment q (H.conditionalAverage hf z ρ)) ≤        ((-c G.size * Real.log ρ.dimension + e G.size * h * Real.log G.size - H.cost : ℝ) : EReal)⋮theorem all_grid_moment {r : ℕ} (hr : ScaleLarge r) (hrS : SparseScalar.ScaleReady r)    (hS : ∀ G : Grid, G.Allowed r → ScalarInduction.MainSizeReady SparseScalar.q G.size)    {zStar : ℝ} (hζ : SparsePerturb r zStar) {z : ℝ} (hz : z∈Set.Icc 0 zStar) :    ∀ G : Grid, G.Allowed r → MomentBound G SparseScalar.q z := by  sorry⋮theorem conditional_main : ConditionalMain := 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.

structure OAI.CoordinateSweeps.FiniteLaw line 19

structure FiniteLaw (Ω : Type*) where  mass : Ω → ℝ  nonneg : ∀ x, 0 ≤ mass x  total : ∑ᶠ x, mass x = 1

definition OAI.CoordinateSweeps.FiniteLaw.uniform line 41

def uniform (Ω : Type*) [Fintype Ω] [Nonempty Ω] : FiniteLaw Ω where  mass _ := (Fintype.card Ω : ℝ)⁻¹  nonneg _ := inv_nonneg.mpr (Nat.cast_nonneg _)  total := by    rw [finsum_eq_sum_of_fintype]    simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]    exact mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero)

definition OAI.CoordinateSweeps.FiniteLaw.map line 49

def map (μ : FiniteLaw Ω) (f : Ω → X) : FiniteLaw X where  mass y := ∑ x, if f x = y then μ x else 0  nonneg y := Finset.sum_nonneg (fun x _ => by split_ifs; exact μ.nonneg x; exact le_rfl)  total := by    rw [finsum_eq_sum_of_fintype]    classical    rw [Finset.sum_comm]    simp [μ.sum_mass]

abbreviation OAI.CoordinateSweeps.Cube line 60
```
abbrev Cube (d : ℕ) := Fin d → Bool
```

definition OAI.CoordinateSweeps.pairSwitch line 62

def pairSwitch {L : Type*} (bits : L → Bool) : Equiv.Perm (Bool × L) where

Show the remaining 10 lines

  toFun x := (Bool.xor x.1 (bits x.2), x.2)  invFun x := (Bool.xor x.1 (bits x.2), x.2)  left_inv point := by    rcases point with ⟨bit, label⟩    change ((bit ^^ bits label) ^^ bits label, label) = (bit, label)    cases bit <;> cases bits label <;> simp  right_inv point := by    rcases point with ⟨bit, label⟩    change ((bit ^^ bits label) ^^ bits label, label) = (bit, label)    cases bit <;> cases bits label <;> simp

definition OAI.CoordinateSweeps.binaryLayer line 74

def binaryLayer {d : ℕ} (j : Fin d) (bits : ({k : Fin d // k ≠ j} → Bool) → Bool) :    Equiv.Perm (Cube d) :=  (Equiv.funSplitAt j Bool).trans ((pairSwitch bits).trans (Equiv.funSplitAt j Bool).symm)

definition OAI.CoordinateSweeps.orderedProduct line 79

def orderedProduct {G : Type*} [Monoid G] {d : ℕ} (g : Fin d → G) : G :=  (List.ofFn g).reverse.prod

abbreviation OAI.CoordinateSweeps.BinaryChoices line 82

abbrev BinaryChoices (d : ℕ) := ∀ j : Fin d, ({k : Fin d // k ≠ j} → Bool) → Bool

definition OAI.CoordinateSweeps.binarySweep line 86

def binarySweep (d : ℕ) (bits : BinaryChoices d) : Equiv.Perm (Cube d) :=  orderedProduct (fun j => binaryLayer j (bits j))

definition OAI.CoordinateSweeps.binaryLaw line 89

def binaryLaw (d : ℕ) : FiniteLaw (Equiv.Perm (Cube d)) :=  (FiniteLaw.uniform (BinaryChoices d)).map (binarySweep d)

structure OAI.CoordinateSweeps.Grid line 92

structure Grid where  b : ℕ  positive : 0 < b  bits : Fin b → ℕ

definition OAI.CoordinateSweeps.Grid.size line 99

def size (G : Grid) : ℕ := ∏ j, 2 ^ G.bits j

abbreviation OAI.CoordinateSweeps.Grid.Slot line 101

abbrev Slot (G : Grid) := ∀ j, Cube (G.bits j)

abbreviation OAI.CoordinateSweeps.Grid.Line line 103

abbrev Line (G : Grid) (j : Fin G.b) := ∀ k : {k : Fin G.b // k ≠ j}, Cube (G.bits k)

abbreviation OAI.CoordinateSweeps.Grid.Choices line 105

abbrev Choices (G : Grid) := ∀ j, G.Line j → Equiv.Perm (Cube (G.bits j))

definition OAI.CoordinateSweeps.Grid.fiberPerm line 107

def fiberPerm {A B : Type*} (p : B → Equiv.Perm A) : Equiv.Perm (A × B) where  toFun x := (p x.2 x.1, x.2)  invFun x := ((p x.2).symm x.1, x.2)  left_inv x := by simp  right_inv x := by simp

definition OAI.CoordinateSweeps.Grid.stage line 113

def stage (G : Grid) (ω : G.Choices) (j : Fin G.b) : Equiv.Perm G.Slot :=  (Equiv.piSplitAt j (fun k => Cube (G.bits k))).trans    ((fiberPerm (ω j)).trans (Equiv.piSplitAt j (fun k => Cube (G.bits k))).symm)

definition OAI.CoordinateSweeps.Grid.boundary line 117

def boundary (G : Grid) (ω : G.Choices) (t : ℕ) : Equiv.Perm G.Slot :=  ((List.ofFn (G.stage ω)).take t).reverse.prod

definition OAI.CoordinateSweeps.Grid.sweep line 120

def sweep (G : Grid) (ω : G.Choices) : Equiv.Perm G.Slot := G.boundary ω G.b

definition OAI.CoordinateSweeps.Grid.Allowed line 123

def Allowed (G : Grid) (r : ℕ) : Prop := ∀ j, r ≤ G.bits j ∧ G.bits j ≤ 2 * r

structure OAI.CoordinateSweeps.Grid.Holes line 126

structure Holes (G : Grid) (h : ℕ) where  path : Fin h → Fin (G.b + 1) → G.Slot  disjoint : ∀ t, Function.Injective (fun i => path i t)  coordinate_step : ∀ i (j : Fin G.b) (k : Fin G.b), k ≠ j →    path i j.succ k = path i j.castSucc k

definition OAI.CoordinateSweeps.Grid.Holes.Compatible line 136

def Compatible (H : G.Holes h) (ω : G.Choices) : Prop :=  ∀ i (t : Fin (G.b + 1)), G.boundary ω t (H.path i 0) = H.path i t

definition OAI.CoordinateSweeps.Grid.Holes.Feasible line 139

def Feasible (H : G.Holes h) : Prop := ∃ ω, H.Compatible ω

definition OAI.CoordinateSweeps.Grid.Holes.lineCount line 141

def lineCount (H : G.Holes h) (j : Fin G.b) (L : G.Line j) : ℕ :=  (Finset.univ.filter (fun i : Fin h =>    (fun k : {k : Fin G.b // k ≠ j} => H.path i j.castSucc k) = L)).card

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

Paper theorem cited in the Lean source

A source comment for OAI.CoordinateSweeps.conditional_main names this theorem’s label in the paper’s TeX source. Read the source comment from line 451

/-- Manuscript03 thm:conditional / ac:main, with c₀=e₀=a/100,
a=1/100, natural logarithms, unnormalized integer Schatten moments,
all feasible disjoint paths (including none and all), and log(0)=−∞. -/

Theorem 3.1 of Conditional coordinate sweeps and analytic transfer

Theorem (Conditional sweep moment estimate)

There are a power of two R\ge2, a positive integer q, and z_*>0 such that every grid and every family \mathcal H of prescribed disjoint trajectories defined above satisfy, for every z\in[0,z_*] and every \lambda\vdash s-h,

\log T_\lambda^{\mathcal H}\ \le\ -c(s)F(\lambda)+e(s)h\log s-C(\mathcal H).

Here a zero moment gives -\infty. Use parameters

a=\frac1{100},\quad c_0=e_0=\frac a{100},\qquad c(s)=c_0+\frac1{\sqrt{\log s}},\qquad e(s)=e_0-\frac1{\sqrt{\log s}}.

The empty diagram has dimension one. If h=s, the remaining bijection is the unique empty bijection and its moment is one. The assertion also includes the one-dimensional trivial representation for every h.

Section 3, Prescribed trajectories and their cost Read it in the paper TeX source, line 137

Show the TeX

There are a power of two \(R\ge2\), a positive integer \(q\), and \(z_*>0\) such
that every grid and every family \(\mathcal H\) of prescribed disjoint
trajectories defined above satisfy, for every \(z\in[0,z_*]\) and every
\(\lambda\vdash s-h\),
\begin{equation}
 \log T_\lambda^{\mathcal H}\ \le\
 -c(s)F(\lambda)+e(s)h\log s-C(\mathcal H).
 
\end{equation}
Here a zero moment gives \(-\infty\). Use parameters
\[
 a=\frac1{100},\quad c_0=e_0=\frac a{100},\qquad
 c(s)=c_0+\frac1{\sqrt{\log s}},\qquad
 e(s)=e_0-\frac1{\sqrt{\log s}}.
\]
The empty diagram has dimension one. If \(h=s\), the remaining bijection is
the unique empty bijection and its moment is one. The assertion also includes
the one-dimensional trivial representation for every \(h\).

Other possible paper matches, closest wording first

Theorem 1.1 of Compatibility entropy and the spectrum of a Thorp sweep. Words of the Lean statement, its names split into words, that also appear in the theorem: natural, symmetric, functions, logarithms, weighted, take and six more. Found in the paper’s title, which its main theorem states: sweep.
Proposition 10.2 of Signed tensor densities and diagram budgets for the Thorp shuffle. Words of the Lean statement, its names split into words, that also appear in the theorem: 1/100, xi, using, grid, hook, first and twelve more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: powers.
Theorem 1.1 of Conditional permutations in a revealed switching environment. Words of the Lean statement, its names split into words, that also appear in the theorem: slot, free, conditional, full, distance, permutation and five more. Found in the paper’s title, which its main theorem states: switching. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: physical and shuffles.
Theorem 12.2 of Conditional information under deterministic coordinate sweeps. Words of the Lean statement, its names split into words, that also appear in the theorem: natural, logarithms, positive, input, independent, permutation and three more. Found in the paper’s title, which its main theorem states: conditional, coordinate and sweeps. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: physical and shuffles.
Proposition 6.3 of Conditional permutations in a revealed switching environment. Words of the Lean statement, its names split into words, that also appear in the theorem: without, base, layers, pairs, switches, group and nine more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: shuffle.
Proposition 7.1 of Compatibility entropy and the spectrum of a Thorp sweep. Words of the Lean statement, its names split into words, that also appear in the theorem: compatible, reference, split, weights, pairs, independent and six more.

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 618 s; peak memory 6.6 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 238’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 agree 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: ThorpResults.remaining_main (ThorpRemaining) is a conjunction whose last conjunct OptimalOrderMain := IsTheta atTop (mixingTime d) (log (2^d)) states exactly the Θ(log N) mixing for N = 2^d cards, with mixingTime d := sInf {t | distance d t ≤ 1/4}, distance d t := tv (law d t) (uniform d) the total-variation distance of the full permutation law (and InformationMain records that lawFrom from any starting deck has the same distance, plus distance d (2048 d) → 0); the 12 other challenge files are supporting Fourier/representation estimates (regular_trace_smoothing, binary_sweep_contraction_and_mixing, robust_permanent, ...). All of them state estimates for the same hand-built Thorp shuffle.
Definitions the group asks readers to compare with the paper: Custom Thorp shuffle and 'mixing time': step (d+1) c := (pairSwitch d c on the head bit, then rotate the coordinate order) on State d := Equiv.Perm (Fin d → Bool) with Coins (d+1) = (Fin d → Bool) → Bool iid fair coins, fairMass, tv, mixingTime d := sInf {t | distance d t ≤ 1/4}; this is the standard pair-swap-and-interleave Thorp shuffle and TV-from-uniform with ε = 1/4, but it is a hand-written model and the Θ-statement hides explicit constants (1600d, 2048d, ...). OccupiedOverlap shows an axiom_files hit (HookModel.lean) that is a doc-comment false positive.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“A separate conditional moment estimate covers allowed power-of-two grids and disjoint coordinate-respecting trajectories, with its stated feasibility premise. Physical-shuffle law identification is not included.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a supporting result”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

314 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: 99 to its own declarations, 136 to Mathlib’s documentation. 45 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 OAI/-!Conditional moments for one ordered coordinate sweep on power-of-two rectangulargrids. The line law mixes uniform permutations with a binary coordinate sweep.The trace is unnormalized, and the trajectory cost uses falling factorials.-/noncomputable sectionopen scoped BigOperators Matrix.Norms.L2Operator ComplexOrder ComplexConjugate MatrixOrder ENNRealopen MeasureTheoryattribute [local instance] Classical.propDecidablenamespace CoordinateSweepsstructure FiniteLaw (Ω : Type*) where  mass : Ω → ℝ  nonneg : ∀ x, 0 ≤ mass x  total : ∑ᶠ x, mass x = 1namespace FiniteLawvariable {Ω X Y : Type*}instance : CoeFun (FiniteLaw Ω) (fun _ => Ω → ℝ) := ⟨FiniteLaw.mass⟩@[ext] theorem ext {μ ν : FiniteLaw Ω} (h : ∀ x, μ x = ν x) : μ = ν := by  cases μ  cases ν  congr  exact funext hvariable [Fintype Ω] [Fintype X] [Fintype Y]@[simp] theorem sum_mass (μ : FiniteLaw Ω) : ∑ x, μ x = 1 := by  simpa only [finsum_eq_sum_of_fintype] using μ.totaldef uniform (Ω : Type*) [Fintype Ω] [Nonempty Ω] : FiniteLaw Ω where  mass _ := (Fintype.card Ω : ℝ)⁻¹  nonneg _ := inv_nonneg.mpr (Nat.cast_nonneg _)  total := by    rw [finsum_eq_sum_of_fintype]    simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]    exact mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero)def map (μ : FiniteLaw Ω) (f : Ω → X) : FiniteLaw X where  mass y := ∑ x, if f x = y then μ x else 0  nonneg y := Finset.sum_nonneg (fun x _ => by split_ifs; exact μ.nonneg x; exact le_rfl)  total := by    rw [finsum_eq_sum_of_fintype]    classical    rw [Finset.sum_comm]    simp [μ.sum_mass]end FiniteLawabbrev Cube (d : ℕ) := Fin d → Booldef pairSwitch {L : Type*} (bits : L → Bool) : Equiv.Perm (Bool × L) where  toFun x := (Bool.xor x.1 (bits x.2), x.2)  invFun x := (Bool.xor x.1 (bits x.2), x.2)  left_inv point := by    rcases point with ⟨bit, label⟩    change ((bit ^^ bits label) ^^ bits label, label) = (bit, label)    cases bit <;> cases bits label <;> simp  right_inv point := by    rcases point with ⟨bit, label⟩    change ((bit ^^ bits label) ^^ bits label, label) = (bit, label)    cases bit <;> cases bits label <;> simpdef binaryLayer {d : ℕ} (j : Fin d) (bits : ({k : Fin d // k ≠ j} → Bool) → Bool) :    Equiv.Perm (Cube d) :=  (Equiv.funSplitAt j Bool).trans ((pairSwitch bits).trans (Equiv.funSplitAt j Bool).symm)/-- Coordinate zero acts first. -/def orderedProduct {G : Type*} [Monoid G] {d : ℕ} (g : Fin d → G) : G :=  (List.ofFn g).reverse.prodabbrev BinaryChoices (d : ℕ) := ∀ j : Fin d, ({k : Fin d // k ≠ j} → Bool) → Boolinstance (d : ℕ) : Nonempty (BinaryChoices d) := ⟨fun _ _ => false⟩def binarySweep (d : ℕ) (bits : BinaryChoices d) : Equiv.Perm (Cube d) :=  orderedProduct (fun j => binaryLayer j (bits j))def binaryLaw (d : ℕ) : FiniteLaw (Equiv.Perm (Cube d)) :=  (FiniteLaw.uniform (BinaryChoices d)).map (binarySweep d)structure Grid where  b : ℕ  positive : 0 < b  bits : Fin b → ℕnamespace Griddef size (G : Grid) : ℕ := ∏ j, 2 ^ G.bits jabbrev Slot (G : Grid) := ∀ j, Cube (G.bits j)abbrev Line (G : Grid) (j : Fin G.b) := ∀ k : {k : Fin G.b // k ≠ j}, Cube (G.bits k)abbrev Choices (G : Grid) := ∀ j, G.Line j → Equiv.Perm (Cube (G.bits j))def fiberPerm {A B : Type*} (p : B → Equiv.Perm A) : Equiv.Perm (A × B) where  toFun x := (p x.2 x.1, x.2)  invFun x := ((p x.2).symm x.1, x.2)  left_inv x := by simp  right_inv x := by simpdef stage (G : Grid) (ω : G.Choices) (j : Fin G.b) : Equiv.Perm G.Slot :=  (Equiv.piSplitAt j (fun k => Cube (G.bits k))).trans    ((fiberPerm (ω j)).trans (Equiv.piSplitAt j (fun k => Cube (G.bits k))).symm)def boundary (G : Grid) (ω : G.Choices) (t : ℕ) : Equiv.Perm G.Slot :=  ((List.ofFn (G.stage ω)).take t).reverse.proddef sweep (G : Grid) (ω : G.Choices) : Equiv.Perm G.Slot := G.boundary ω G.b/-- The side lengths are in `[R, R²]`, where the single global scale is `R = 2^r`. -/def Allowed (G : Grid) (r : ℕ) : Prop := ∀ j, r ≤ G.bits j ∧ G.bits j ≤ 2 * r/-- Full disjoint paths, including every boundary of the ordered sweep. -/structure Holes (G : Grid) (h : ℕ) where  path : Fin h → Fin (G.b + 1) → G.Slot  disjoint : ∀ t, Function.Injective (fun i => path i t)  coordinate_step : ∀ i (j : Fin G.b) (k : Fin G.b), k ≠ j →    path i j.succ k = path i j.castSucc knamespace Holesvariable {G : Grid} {h : ℕ}def Compatible (H : G.Holes h) (ω : G.Choices) : Prop :=  ∀ i (t : Fin (G.b + 1)), G.boundary ω t (H.path i 0) = H.path i tdef Feasible (H : G.Holes h) : Prop := ∃ ω, H.Compatible ωdef lineCount (H : G.Holes h) (j : Fin G.b) (L : G.Line j) : ℕ :=  (Finset.univ.filter (fun i : Fin h =>    (fun k : {k : Fin G.b // k ≠ j} => H.path i j.castSucc k) = L)).carddef cost (H : G.Holes h) : ℝ :=  ∑ j, ∑ L : G.Line j,    Real.log (((2 ^ G.bits j : ℕ) : ℝ) ^ H.lineCount j L /      ((2 ^ G.bits j).descFactorial (H.lineCount j L) : ℝ))/-- The symmetric group on the unmarked input sites. -/def stabilizer (H : G.Holes h) : Subgroup (Equiv.Perm G.Slot) where  carrier g := ∀ i, g (H.path i 0) = H.path i 0  one_mem' _ := rfl  mul_mem' := by    intro g g' hg hg' i    change g (g' (H.path i 0)) = H.path i 0    rw [hg' i, hg i]  inv_mem' hg i := (Equiv.symm_apply_eq _).mpr (hg i).symmdef reference (H : G.Holes h) (hf : H.Feasible) : G.Choices := hf.choosetheorem reference_compatible (H : G.Holes h) (hf : H.Feasible) :    H.Compatible (H.reference hf) := hf.choose_spec/-- A reference completion identifies the free input and output sites. -/def residual (H : G.Holes h) (hf : H.Feasible) (ω : {ω : G.Choices // H.Compatible ω}) :    H.stabilizer :=  ⟨(G.sweep (H.reference hf))⁻¹ * G.sweep ω.val, by    intro i    change (G.sweep (H.reference hf)).symm (G.sweep ω.val (H.path i 0)) = H.path i 0    apply (Equiv.symm_apply_eq _).mpr    exact (ω.property i (Fin.last _)).trans ((H.reference_compatible hf i (Fin.last _)).symm)⟩end Holes/-- Independent line permutations with law `(1-z) U + z B`. -/def choiceWeight (G : Grid) (z : ℝ) (ω : G.Choices) : ℝ :=  ∏ j, ∏ L : G.Line j,    ((1-z) * (Fintype.card (Equiv.Perm (Cube (G.bits j))) : ℝ)⁻¹ +      z * binaryLaw (G.bits j) (ω j L))namespace Holesvariable {G : Grid} {h : ℕ}def probability (H : G.Holes h) (z : ℝ) : ℝ :=  ∑ ω : {ω : G.Choices // H.Compatible ω}, G.choiceWeight z ω.valend Holesdef eventDistance (G : Grid) (E : Finset G.Choices) (z : ℝ) : ℝ :=  ∑ ω ∈ E, |G.choiceWeight z ω/(∑ v ∈ E, G.choiceWeight z v)-    G.choiceWeight 0 ω/(∑ v ∈ E, G.choiceWeight 0 v)|end Gridstructure UnitaryIrrep (Γ : Type*) [Group Γ] where  dimension : ℕ  positive : 0 < dimension  matrix : Γ →* Matrix (Fin dimension) (Fin dimension) ℂ  unitary : ∀ g, (matrix g).conjTranspose * matrix g = 1  irreducible : ∀ S : Submodule ℂ (Fin dimension → ℂ),    (∀ g v, v ∈ S → (matrix g).mulVec v ∈ S) → S = ⊥ ∨ S = ⊤/-- Unnormalized Schatten `2q` moment, without division by the dimension. -/def schattenMoment {D : ℕ} (q : ℕ) (K : Matrix (Fin D) (Fin D) ℂ) : ℝ :=  (Matrix.trace ((K.conjTranspose * K) ^ q)).renamespace Grid.Holesvariable {G : Grid} {h : ℕ}def conditionalAverage (H : G.Holes h) (hf : H.Feasible) (z : ℝ)    (ρ : UnitaryIrrep H.stabilizer) : Matrix (Fin ρ.dimension) (Fin ρ.dimension) ℂ :=  (H.probability z : ℂ)⁻¹ •    ∑ ω : {ω : G.Choices // H.Compatible ω},      (G.choiceWeight z ω.val : ℂ) • ρ.matrix (H.residual hf ω)end Grid.Holesdef a : ℝ := 1 / 100def c0 : ℝ := a / 100def e0 : ℝ := a / 100def c (s : ℕ) : ℝ := c0 + 1 / Real.sqrt (Real.log s)def e (s : ℕ) : ℝ := e0 - 1 / Real.sqrt (Real.log s)/-- The logarithm of a zero moment is minus infinity. -/def logMoment (t : ℝ) : EReal := if t = 0 then ⊥ else (Real.log t : EReal)def ConditionalMain : Prop :=  ∃ (r q : ℕ) (zStar : ℝ), 1 ≤ r ∧ 1 ≤ q ∧ 0 < zStar ∧ zStar ≤ 1 / ((2 : ℕ) : ℝ) ∧    ∀ (G : Grid), G.Allowed r → ∀ (h : ℕ) (H : G.Holes h) (hf : H.Feasible),    ∀ z ∈ Set.Icc 0 zStar, ∀ ρ : UnitaryIrrep H.stabilizer,      logMoment (schattenMoment q (H.conditionalAverage hf z ρ)) ≤        ((-c G.size * Real.log ρ.dimension + e G.size * h * Real.log G.size - H.cost : ℝ) : EReal)def MomentBound (G : Grid) (q : ℕ) (z : ℝ) : Prop :=  ∀ (h : ℕ) (H : G.Holes h) (hf : H.Feasible) (σ : UnitaryIrrep H.stabilizer),    schattenMoment q (H.conditionalAverage hf z σ) ≤      Real.exp (-c G.size*Real.log σ.dimension+e G.size*h*Real.log G.size-H.cost)def ScaleLarge (r : ℕ) : Prop :=  1 ≤ r ∧ (2/e0)^2 ≤ (r : ℝ)*Real.log 2 ∧ 4/e0 ≤ (r : ℝ)*Real.log 2def baseWidth (r : ℕ) : ℝ := ((2^(2*r)).factorial : ℕ)def baseZeta (r : ℕ) : ℝ := 1/(4*(baseWidth r)^2)namespace ScalarInductionstructure MainSizeReady (q : ℕ) (s : ℝ) : Prop where  one : 1 ≤ Real.log s  sparse : (100004:ℝ)*s^(1-a/4) ≤ s^(1-(1:ℝ)/1600)  hook : (((2*q+1:ℕ):ℝ)*4+82*c0)*2*s^((4:ℝ)/5+2*a)*Real.log (s+1) ≤ s^((9:ℝ)/10)  removed : s^((9:ℝ)/10)+s^(1-a/2)*Real.log s ≤ s^(1-a/4)/(20*Real.sqrt (Real.log s))end ScalarInductionnamespace SparseScalardef theta : ℝ := 1/1600def K0 : ℝ := 100000def B0 : ℕ := 12800def xi : ℝ := 1/2048000000def rho : ℝ := 1/32768000000def q : ℕ := 65536000000def uniformRate : ℝ := Real.log ((2 : ℕ) : ℝ) + (B0 : ℝ) *  (K0 + 1 + Real.log ((4 : ℕ) : ℝ) +    Real.log (((16 : ℕ) : ℝ) * Real.exp ((2 : ℕ) : ℝ)) +    Real.log (1 + ((16 : ℕ) : ℝ) * Real.exp ((2 : ℕ) : ℝ)))structure SizeReady (s : ℝ) : Prop where  two : 2 ≤ s  logpow : Real.log s ≤ s^(theta/4)  powtwo : 2 ≤ s^(theta/4)  powB : (B0+1:ℝ) ≤ s^(theta/2)  count : 4/s^(1/(4*(B0:ℝ))) ≤ theta/(64*B0)  absorb : Real.log (1+2*(B0:ℝ))+Real.log 2 ≤ xi*Real.log s  eta : s^(-xi) ≤ 1/8  prefactor : uniformRate ≤ xi*Real.log s/16  logtwo : Real.log 2 ≤ xi*Real.log s/8structure ScaleReady (r : ℕ) : Prop where  two : 2 ≤ r  logone : 1 ≤ (r:ℝ)*Real.log 2  absorb : 3*Real.log 2+2*(K0+1+Real.log 4) ≤ theta*((r:ℝ)*Real.log 2)/8  size : ∀ s : ℝ, (2:ℝ)^r ≤ s → SizeReady sdef slack (s : ℝ) : ℝ :=  if 1 < s then s ^ (-((2 : ℕ) : ℝ) * rho * s) * (s ^ rho - 1) else 1end SparseScalardef SparsePerturb (r : ℕ) (zStar : ℝ) : Prop :=  0 < zStar ∧ zStar ≤ baseZeta r ∧ ∀ G : Grid, G.b ≤ SparseScalar.B0 → G.Allowed r →    ∀ z ∈ Set.Icc 0 zStar, ∀ E : Finset G.Choices, E.Nonempty →      G.eventDistance E z < SparseScalar.slack G.size/-- The quantitative all-grid estimate under the explicit scale and perturbation bounds. -/theorem all_grid_moment {r : ℕ} (hr : ScaleLarge r) (hrS : SparseScalar.ScaleReady r)    (hS : ∀ G : Grid, G.Allowed r → ScalarInduction.MainSizeReady SparseScalar.q G.size)    {zStar : ℝ} (hζ : SparsePerturb r zStar) {z : ℝ} (hz : z∈Set.Icc 0 zStar) :    ∀ G : Grid, G.Allowed r → MomentBound G SparseScalar.q z := by  sorry/-- One global choice of scale, moment order, and positive perturbation interval. -/theorem conditional_main : ConditionalMain := by  sorryend CoordinateSweepsendend 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.

Optimal-order mixing of the Thorp shuffle

Theorem (Optimal-order mixing)

As d\to\infty,

\|q_d^{*(1600d)}-U_n\|_{\mathrm{TV}}\longrightarrow0.

For every integer t\ge0,

\|q_d^{*t}-U_n\|_{\mathrm{TV}} \ge 1-\frac{2^{tn/2}}{n!}.

Consequently the mixing time at distance 1/4 satisfies

t_{\mathrm{mix}}(d)\ge \left\lceil\frac2n\log_2\frac{3n!}{4}\right\rceil =2d-O(1), \qquad t_{\mathrm{mix}}(d)=\Theta(d).

Section 1, Introduction Read it in the paper TeX source, line 53

Show the TeX

As $d\to\infty$,
\[
 \|q_d^{*(1600d)}-U_n\|_{\mathrm{TV}}\longrightarrow0.
\]
For every integer $t\ge0$,
\[
 \|q_d^{*t}-U_n\|_{\mathrm{TV}}
 \ge 1-\frac{2^{tn/2}}{n!}.
\]
Consequently the mixing time at distance $1/4$ satisfies
\[
 t_{\mathrm{mix}}(d)\ge
 \left\lceil\frac2n\log_2\frac{3n!}{4}\right\rceil
 =2d-O(1),
 \qquad t_{\mathrm{mix}}(d)=\Theta(d).
\]

From partial permutation information to Fourier bounds

Theorem (Complementary cosets control Fourier matrices)

Partition X into b nonempty sets M_1,\ldots,M_b. Put H_i=H_{M_i}. Suppose f is a nonnegative measure on G of mass at most one and, for every g\in G and i,

f(gH_i)\le \frac{B}{[G:H_i]}.

For every u>0 and every irreducible \rho of degree D,

\|\widehat f(\rho)\|_{\HS}^2 \le bBC_uD^{-1+(u+2)/b}.

There is also an independent orbit-span proof of the bound

\|\widehat f(\rho)\|_{\op}^2 \le bBC_uD^{-1+(u+2)/b}.

No restriction is imposed on block sizes or on the multiplicities in \rho|_{H_1\times\cdots\times H_b}.

Section 1, What partial observations can determine Read it in the paper TeX source, line 18

Show the TeX

Partition $X$ into $b$ nonempty sets $M_1,\ldots,M_b$. Put $H_i=H_{M_i}$. Suppose $f$ is a nonnegative measure on $G$ of mass at most one and, for every $g\in G$ and $i$,
\begin{equation}
 f(gH_i)\le \frac{B}{[G:H_i]}.
\end{equation}
For every $u>0$ and every irreducible $\rho$ of degree $D$,
\begin{equation}
 \|\widehat f(\rho)\|_{\HS}^2
 \le bBC_uD^{-1+(u+2)/b}.
\end{equation}
There is also an independent orbit-span proof of the bound
\begin{equation}
 \|\widehat f(\rho)\|_{\op}^2
 \le bBC_uD^{-1+(u+2)/b}.
\end{equation}
No restriction is imposed on block sizes or on the multiplicities in $\rho|_{H_1\times\cdots\times H_b}$.

Conditional permutations in a revealed switching environment

Theorem (Conditional permutation and full-deck mixing)

Let n=2^d, and start the shuffle from any deterministic deck. After J(s_0+1) sweeps, where s_0=400 and J=40000, the conditional free-slot law defined above satisfies

\sup_{\text{initial layouts}}\E_{\mathcal E} \|\nu_{\mathcal E}-U_{n/2}\|_{\TV}\longrightarrow0 \qquad(d\longrightarrow\infty).

Consequently the full-deck law approaches uniform, uniformly in the starting deck, after 16\,040\,400d physical shuffles. For every integer t\ge0, after t physical shuffles the full-deck distance from uniform is at least 1-2^{tn/2}/n!. Thus the full-deck mixing time is of order d.

Section 1, Introduction Read it in the paper TeX source, line 11

Show the TeX

Let $n=2^d$, and start the shuffle from any deterministic deck.
After $J(s_0+1)$ sweeps, where $s_0=400$ and $J=40000$, the conditional free-slot law defined above satisfies
\[
 \sup_{\text{initial layouts}}\E_{\mathcal E}
       \|\nu_{\mathcal E}-U_{n/2}\|_{\TV}\longrightarrow0
       \qquad(d\longrightarrow\infty).
\]
Consequently the full-deck law approaches uniform, uniformly in the starting deck, after $16\,040\,400d$ physical shuffles. For every integer $t\ge0$, after $t$ physical shuffles the full-deck distance from uniform is at least
$1-2^{tn/2}/n!$. Thus the full-deck mixing time is of order $d$.

Routing densities and representation contraction for Thorp sweeps

Theorem (Routing density and representation contraction)

There are absolute constants \eta,c,C>0 such that, with a=1/1000,

\log\E_{y\sim u_{N,k}}R_x(y)^{1+a} \le Ck(k/N)^\eta \qquad(1\le k\le N),

uniformly over every starting injection x. For every sufficiently large dyadic N and every nontrivial irreducible representation,

T_N(\lambda)=0\quad\hbox{or}\quad \|T_N(\lambda)\|_{\op}\le D_\lambda^{-c}.

Consequently an absolute number of forward sweeps has worst-start total-variation distance from uniform tending to zero as d\to\infty.

Section 1, Introduction Read it in the paper TeX source, line 30

Show the TeX

There are absolute constants $\eta,c,C>0$ such that, with $a=1/1000$,
\[
 \log\E_{y\sim u_{N,k}}R_x(y)^{1+a}
 \le Ck(k/N)^\eta
 \qquad(1\le k\le N),
\]
uniformly over every starting injection $x$. For every sufficiently
large dyadic $N$ and every nontrivial irreducible representation,
\[
 T_N(\lambda)=0\quad\hbox{or}\quad
 \|T_N(\lambda)\|_{\op}\le D_\lambda^{-c}.
\]
Consequently an absolute number of forward sweeps has worst-start
total-variation distance from uniform tending to zero as $d\to\infty$.

Row–column symmetry and contraction of coordinate sweeps

Theorem (Weighted sweep moments)

There are positive weights W_\lambda, an absolute finite p_*, and real exponents 2\le p_d\le p_* such that, for every d\ge1 and every \lambda\vdash2^d,

D_\lambda^{3/4}\le W_\lambda\le D_\lambda, \qquad W_\lambda\|K_d(\lambda)\|_{p_d}^{p_d}\le1.

Consequently \|K_d(\lambda)\|_{\rm op}\le D_\lambda^{-3/(4p_*)}.

Section 1, Introduction Read it in the paper TeX source, line 60

Show the TeX

There are positive weights $W_\lambda$, an absolute finite $p_*$, and real
exponents $2\le p_d\le p_*$ such that, for every $d\ge1$ and every
$\lambda\vdash2^d$,
\begin{equation}
 D_\lambda^{3/4}\le W_\lambda\le D_\lambda,
 \qquad W_\lambda\|K_d(\lambda)\|_{p_d}^{p_d}\le1.
\end{equation}
Consequently $\|K_d(\lambda)\|_{\rm op}\le D_\lambda^{-3/(4p_*)}$.

Corollary (Optimal order of mixing)

There is an absolute constant C such that, for every d\ge1,

\left\lceil\frac2N\log_2\frac{3N!}{4}\right\rceil \le t_{\rm mix}(d)\le Cd, \qquad N=2^d.

In particular t_{\rm mix}(d)=\Theta(d), and its lower bound is 2d-O(1).

Section 1, Introduction Read it in the paper TeX source, line 93

Show the TeX

There is an absolute constant $C$ such that, for every $d\ge1$,
\[
 \left\lceil\frac2N\log_2\frac{3N!}{4}\right\rceil
 \le t_{\rm mix}(d)\le Cd,
 \qquad N=2^d.
\]
In particular $t_{\rm mix}(d)=\Theta(d)$, and its lower bound is
$2d-O(1)$.

Random-subspace tests and trace smoothing for coordinate sweeps

Theorem (Regular-trace smoothing)

There is an absolute P\ge2 such that for every dyadic n\ge2,

\operatorname{Tr}_{\rm reg}|Q_n|^P\le1+n^{-10}.

If v is an integer with 2v\ge P, then v independent sweeps have total-variation distance at most \frac12n^{-5} from uniform, from every deterministic initial deck. Here \|\mu-\nu\|_{\rm TV}=\tfrac12\sum_g|\mu(g)-\nu(g)|.

Section 1, Introduction Read it in the paper TeX source, line 24

Show the TeX

There is an absolute $P\ge2$ such that for every dyadic $n\ge2$,
\begin{equation}
 \operatorname{Tr}_{\rm reg}|Q_n|^P\le1+n^{-10}.
\end{equation}
If $v$ is an integer with $2v\ge P$, then $v$ independent sweeps have total-variation distance at most $\frac12n^{-5}$ from uniform, from every deterministic initial deck. Here $\|\mu-\nu\|_{\rm TV}=\tfrac12\sum_g|\mu(g)-\nu(g)|$.

Compatibility entropy and the spectrum of a Thorp sweep

Theorem (Weighted compatibility)

There are absolute constants L,C_0>0 such that the following holds. Put m=\sqrt n, assume m/2\le A,D\le2m, and take m sufficiently large that \theta=1-L/\log m>0. For all nonnegative functions w_i:S_D\to[0,\infty) and v_k:S_A\to[0,\infty),

\frac{n!}{|R||C|}\, \mathbb E_U\!\left[\mathbf1_{\mathcal I} \prod_{i=1}^A w_i(r_i)\prod_{k=1}^D v_k(c_k)\right] \le e^{C_0n^{.54}} \prod_{i=1}^A(\mathbb E w_i^{1/\theta})^\theta \prod_{k=1}^D(\mathbb E v_k^{1/\theta})^\theta .

The expectations on the right are uniform on the indicated symmetric groups, and logarithms are natural.

Section 1, Introduction Read it in the paper TeX source, line 24

Show the TeX

There are absolute constants $L,C_0>0$ such that the following holds. Put $m=\sqrt n$, assume $m/2\le A,D\le2m$, and take $m$ sufficiently large that $\theta=1-L/\log m>0$. For all nonnegative functions $w_i:S_D\to[0,\infty)$ and $v_k:S_A\to[0,\infty)$,
\begin{equation}
 \frac{n!}{|R||C|}\,
 \mathbb E_U\!\left[\mathbf1_{\mathcal I}
       \prod_{i=1}^A w_i(r_i)\prod_{k=1}^D v_k(c_k)\right]
 \le e^{C_0n^{.54}}
       \prod_{i=1}^A(\mathbb E w_i^{1/\theta})^\theta
       \prod_{k=1}^D(\mathbb E v_k^{1/\theta})^\theta .
\end{equation}
The expectations on the right are uniform on the indicated symmetric groups, and logarithms are natural.

Theorem (A bounded regular moment)

There is an absolute finite p_*>0 such that, for every dyadic n,

Z_n(p_*)\le1+\frac1{16}.

For every integer M\ge p_*, M independent forward sweeps have worst-start total-variation distance at most 1/8 from uniform.

Section 1, Introduction Read it in the paper TeX source, line 45

Show the TeX

There is an absolute finite $p_*>0$ such that, for every dyadic $n$,
\begin{equation}
 Z_n(p_*)\le1+\frac1{16}.
\end{equation}
For every integer $M\ge p_*$, $M$ independent forward sweeps have worst-start total-variation distance at most $1/8$ from uniform.

Corollary (Singular ranks and full-deck mixing)

For every nontrivial \lambda\vdash n and 1\le r\le D_\lambda,

s_r(T_{n,\lambda})\le(16D_\lambda r)^{-1/(2p_*)}.

If s_j(T_n) denotes the full regular singular list, including all multiplicities and zeros, then

s_j(T_n)\le j^{-1/(2p_*)}\qquad(1\le j\le n!).

The constant singular value is the first value. The physical Thorp chain has worst-start 1/4 mixing time \Theta(d); more precisely its lower bound is 2d-O(1). For every fixed integer M>p_*, the worst-start total-variation distance after Md physical shuffles tends to zero as d\to\infty.

Section 1, Introduction Read it in the paper TeX source, line 53

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For every nontrivial $\lambda\vdash n$ and $1\le r\le D_\lambda$,
\[
 s_r(T_{n,\lambda})\le(16D_\lambda r)^{-1/(2p_*)}.
\]
If $s_j(T_n)$ denotes the full regular singular list, including all multiplicities and zeros, then
\[
 s_j(T_n)\le j^{-1/(2p_*)}\qquad(1\le j\le n!).
\]
The constant singular value is the first value. The physical Thorp chain has worst-start $1/4$ mixing time $\Theta(d)$; more precisely its lower bound is $2d-O(1)$. For every fixed integer $M>p_*$, the worst-start total-variation distance after $Md$ physical shuffles tends to zero as $d\to\infty$.

Signed tensor densities and diagram budgets for the Thorp shuffle

Theorem (A moment bound for every signed occurrence)

For each sufficiently small fixed \bar\eta>0, choose a sufficiently small \bar\kappa>0 with \bar\kappa<\bar\eta/256. There is an integer r\ge1, depending only on these fixed constants, such that for every d\ge1, every \lambda\vdash n=2^d, and every occurrence (ten:spin:eq:1),

\log\!\left(D_\lambda\operatorname{Tr}_{V_\lambda}(B_n^*B_n)^r\right) \le\eta_d F(\alpha,\beta)+g_n(l),

where

\eta_d=\bar\eta\left(1-\frac1{2\sqrt d}\right),\qquad \kappa_d=\bar\kappa\left(1-\frac1{2\sqrt d}\right),\qquad g_n(l)=\max\{0,\kappa_d l\log n-l\log(n/l)\}.

Set g_n(0)=0 and interpret the logarithm of a zero trace as -\infty.

Section 1, Introduction Read it in the paper TeX source, line 48

Show the TeX

For each sufficiently small fixed $\bar\eta>0$, choose a sufficiently
small $\bar\kappa>0$ with $\bar\kappa<\bar\eta/256$. There is an
integer $r\ge1$, depending only on these fixed constants, such that for
every $d\ge1$, every $\lambda\vdash n=2^d$, and every
occurrence~\eqref{ten:spin:eq:1},
\begin{equation}
 \log\!\left(D_\lambda\operatorname{Tr}_{V_\lambda}(B_n^*B_n)^r\right)
 \le\eta_d F(\alpha,\beta)+g_n(l),
\end{equation}
where
\[
 \eta_d=\bar\eta\left(1-\frac1{2\sqrt d}\right),\qquad
 \kappa_d=\bar\kappa\left(1-\frac1{2\sqrt d}\right),\qquad
 g_n(l)=\max\{0,\kappa_d l\log n-l\log(n/l)\}.
\]
Set $g_n(0)=0$ and interpret the logarithm of a zero trace as $-\infty$.

Corollary (Fixed-sweep convergence of the full deck)

There is an absolute positive integer L such that

\mathbb E_{U_{S_n}}|f_{d,L}-1|^2\longrightarrow0 \qquad(d\longrightarrow\infty).

The convergence is uniform over starting orders. Moreover t_{\mathrm{mix}}(d)=\Theta(d).

Section 1, Introduction Read it in the paper TeX source, line 82

Show the TeX

There is an absolute positive integer $L$ such that
\[
 \mathbb E_{U_{S_n}}|f_{d,L}-1|^2\longrightarrow0
 \qquad(d\longrightarrow\infty).
\]
The convergence is uniform over starting orders. Moreover
$t_{\mathrm{mix}}(d)=\Theta(d)$.

Conditional coordinate sweeps and analytic transfer

Theorem (Binary sweep contraction)

There are absolute constants g>0 and d_0 such that, for d\ge d_0 and every \lambda\vdash 2^d,

\|K_\lambda\|_{\op}\le D_\lambda^{-g}.

The sign representation is annihilated. There is an absolute positive integer w such that the law after w independent sweeps converges to uniform in total variation as d\to\infty, uniformly over deterministic initial decks.

Section 1, Introduction Read it in the paper TeX source, line 54

Show the TeX

There are absolute constants \(g>0\) and \(d_0\) such that, for \(d\ge d_0\)
and every \(\lambda\vdash 2^d\),
\[
 \|K_\lambda\|_{\op}\le D_\lambda^{-g}.
\]
The sign representation is annihilated. There is an absolute positive integer \(w\)
such that the law after \(w\) independent sweeps converges to uniform in
total variation as \(d\to\infty\), uniformly over deterministic initial
decks.

Proposition (Support obstruction)

For every integer t\ge0,

\|q_d^{*t}-U_N\|_{\TV}\ge1-\frac{2^{tN/2}}{N!}.

Consequently,

t_{\rm mix}(d)\ge \left\lceil\frac2N\log_2\!\left(\frac{3N!}{4}\right)\right\rceil =2d-O(1).

Section 1, Introduction Read it in the paper TeX source, line 68

Show the TeX

For every integer \(t\ge0\),
\[
 \|q_d^{*t}-U_N\|_{\TV}\ge1-\frac{2^{tN/2}}{N!}.
\]
Consequently,
\[
 t_{\rm mix}(d)\ge
 \left\lceil\frac2N\log_2\!\left(\frac{3N!}{4}\right)\right\rceil
 =2d-O(1).
\]

A strict four-row permanent inequality and permutation moments

Theorem (A robust four-row inequality)

There are absolute constants p_0\in(4/3,2) and \varepsilon>0 with the following property. Let \nu be a probability measure on S_4 such that

\|\nu-u\|_{\TV}<\varepsilon, \qquad \nu\{\pi:\pi(i)=j\}=\frac14\quad(1\le i,j\le4).

Then every four nonnegative functions on \{1,2,3,4\} satisfy

\E_\nu\prod_{i=1}^4f_i(\pi(i))\le\prod_{i=1}^4\|f_i\|_{p_0}.

Section 1, Introduction Read it in the paper TeX source, line 14

Show the TeX

There are absolute constants $p_0\in(4/3,2)$ and $\varepsilon>0$ with the following property. Let $\nu$ be a probability measure on $S_4$ such that
\[
 \|\nu-u\|_{\TV}<\varepsilon,
 \qquad
 \nu\{\pi:\pi(i)=j\}=\frac14\quad(1\le i,j\le4).
\]
Then every four nonnegative functions on $\{1,2,3,4\}$ satisfy
\begin{equation}
 \E_\nu\prod_{i=1}^4f_i(\pi(i))\le\prod_{i=1}^4\|f_i\|_{p_0}.
\end{equation}

Theorem (A fixed moment with a degree saving)

There exist an even integer r\ge2 and \eta>0, independent of N, such that for every dyadic N and every \lambda\vdash N,

D_\lambda\Tr B_N(\lambda)^r\le D_\lambda^{1-\eta}.

Consequently there is an absolute integer q for which the total-variation distance of q independent forward sweeps from uniform tends to zero as N\to\infty, uniformly over the starting deck.

Section 1, Introduction Read it in the paper TeX source, line 46

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There exist an even integer $r\ge2$ and $\eta>0$, independent of $N$, such that for every dyadic $N$ and every $\lambda\vdash N$,
\begin{equation}
 D_\lambda\Tr B_N(\lambda)^r\le D_\lambda^{1-\eta}.
\end{equation}
Consequently there is an absolute integer $q$ for which the total-variation distance of $q$ independent forward sweeps from uniform tends to zero as $N\to\infty$, uniformly over the starting deck.

Configuration and file facts

File: lean/ComparatorChallenges/CoordinateSweeps.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Probability/CoordinateSweeps/Main.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: 59 local vocabulary declarations
Lines of Lean, without comments or blank lines: 190
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.

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