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Lean statement

Equidistribution of prime-degree torus packets

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.DukePrimeDegree.prime_degree_packet_measure_equidistribution_unconditional OAI.DukePrimeDegree.prime_degree_packet_measures_tendsto_haar_unconditional

Lean statement file: DukePrimeDegree.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 015 link it.

See the candidate theoremWho ran the proof-verification programReport a comparison with the paper (GitHub issue)

The Lean statement

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

theorem prime_degree_packet_measure_equidistribution_unconditional    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]    [∀ i, NumberField.IsTotallyReal (K i)]    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :    ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧      WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧      IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by  sorrytheorem prime_degree_packet_measures_tendsto_haar_unconditional    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]    [∀ i, NumberField.IsTotallyReal (K i)]    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)    (m : ProbabilityMeasure (X (d+1)))    (hm : IsHaarProbability (m : Measure (X (d+1)))) :    WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i))      (m : Measure (X (d+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.PrimeDegreePackets.G line 11

    abbrev G (n : ℕ) := Matrix.SpecialLinearGroup (Fin n) ℝ

  • definition OAI.PrimeDegreePackets.integralGroup line 13

    def integralGroup (n : ℕ) : Subgroup (G n) :=  (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)).range

  • abbreviation OAI.PrimeDegreePackets.X line 16

    abbrev X (n : ℕ) := Quotient (QuotientGroup.rightRel (integralGroup n))

  • definition OAI.PrimeDegreePackets.rightAction line 21

    def rightAction {n : ℕ} (x : X n) (g : G n) : X n :=  Quotient.map (fun h => h * g) (by    intro a b hab    change (QuotientGroup.rightRel (integralGroup n)) a b at hab    change (QuotientGroup.rightRel (integralGroup n)) (a * g) (b * g)    rw [QuotientGroup.rightRel_apply] at hab ⊢    simpa only [mul_inv_rev, mul_assoc, mul_inv_cancel_left] using hab) x

  • definition OAI.PrimeDegreePackets.signMatrices line 38

    def signMatrices (n : ℕ) : Set (G n) :=  {g | (∀ i j, i ≠ j → g.1 i j = 0) ∧ ∀ i, g.1 i i = 1 ∨ g.1 i i = -1}

  • definition OAI.PrimeDegreePackets.rowLattice line 41

    def rowLattice {n : ℕ} (g : Matrix (Fin n) (Fin n) ℝ) : Submodule ℤ (Fin n → ℝ) :=  Submodule.span ℤ (Set.range fun i => g i)

  • structure OAI.PrimeDegreePackets.FullLattice line 44

    structure FullLattice (n : ℕ) (K : Type*) [Field K] [NumberField K] where  carrier : Submodule ℤ K  basis : Basis (Fin n) ℤ carrier  spans : Submodule.span ℚ (carrier : Set K) = ⊤

  • definition OAI.PrimeDegreePackets.multiplierOrder line 49

    def multiplierOrder {n : ℕ} {K : Type*} [Field K] [NumberField K] (M : FullLattice n K) : AddSubgroup K where  carrier := {a | ∀ x ∈ M.carrier, a * x ∈ M.carrier}  zero_mem' := by simp  add_mem' := by    intro a b ha hb x hx    simpa only [add_mul] using M.carrier.add_mem (ha x hx) (hb x hx)  neg_mem' := by    intro a ha x hx    simpa only [neg_mul] using M.carrier.neg_mem (ha x hx)

  • definition OAI.PrimeDegreePackets.integralElements line 59

    def integralElements (K : Type*) [Field K] : AddSubgroup K :=  (integralClosure ℤ K).toSubring.toAddSubgroup

  • definition OAI.PrimeDegreePackets.multiplierDiscriminant line 62

    def multiplierDiscriminant {n : ℕ} {K : Type*} [Field K] [NumberField K]    (M : FullLattice n K) : ℝ :=  |(NumberField.discr K : ℝ)| *    ((multiplierOrder M).relIndex (integralElements K) : ℝ) ^ 2

  • abbreviation OAI.PrimeDegreePackets.OrderedEmbeddings line 67

    abbrev OrderedEmbeddings (n : ℕ) (K : Type*) [Field K] := Fin n ≃ (K →+* ℝ)

  • definition OAI.PrimeDegreePackets.embeddingMatrix line 69

    def embeddingMatrix {n : ℕ} {K : Type*} [Field K] [NumberField K]    (M : FullLattice n K) (σ : OrderedEmbeddings n K) : Matrix (Fin n) (Fin n) ℝ :=  fun i j => σ j (M.basis i)

  • definition OAI.PrimeDegreePackets.normalizedLattice line 73

    def normalizedLattice {n : ℕ} {K : Type*} [Field K] [NumberField K]    (M : FullLattice n K) (σ : OrderedEmbeddings n K) : Submodule ℤ (Fin n → ℝ) :=  let B := embeddingMatrix M σ  rowLattice (|(Matrix.det B)| ^ (-(1 : ℝ) / n) • B)

  • definition OAI.PrimeDegreePackets.latticePoint line 78

    def latticePoint {n : ℕ} {K : Type*} [Field K] [NumberField K]    (M : FullLattice n K) (σ : OrderedEmbeddings n K) : X n :=  if h : ∃ g : G n, rowLattice g.1 = normalizedLattice M σ then    Quotient.mk _ h.choose  else Quotient.mk _ (1 : G n)

  • abbreviation OAI.PrimeDegreePackets.LogCoordinates line 84

    abbrev LogCoordinates (d : ℕ) := Fin d → ℝ

  • definition OAI.PrimeDegreePackets.diagonalFlow line 86

    def diagonalFlow {d : ℕ} (u : LogCoordinates d) : G (d + 1) :=  ⟨Matrix.diagonal (Fin.snoc (fun i => Real.exp (u i)) (Real.exp (-∑ i, u i))), by    rw [Matrix.det_diagonal, Fin.prod_univ_castSucc]    simp only [Fin.snoc_castSucc, Fin.snoc_last]    rw [← Real.exp_sum, ← Real.exp_add, add_neg_cancel, Real.exp_zero]⟩

  • definition OAI.PrimeDegreePackets.orbitMap line 110

    def orbitMap {d : ℕ} (x : X (d + 1)) (u : LogCoordinates d) : X (d + 1) :=  rightAction x (diagonalFlow u)

  • definition OAI.PrimeDegreePackets.periodGroup line 120

    def periodGroup {d : ℕ} (x : X (d + 1)) : AddSubgroup (LogCoordinates d) where
    Show the remaining 11 lines
      carrier := {u | orbitMap x u = x}  zero_mem' := orbitMap_zero x  add_mem' := by    intro u v hu hv    change orbitMap x (u + v) = x    rw [orbitMap_add, hu, hv]  neg_mem' := by    intro u hu    have h := congrArg (fun y => orbitMap y (-u)) hu    rw [← orbitMap_add, add_neg_cancel, orbitMap_zero] at h    exact h.symm

  • definition OAI.PrimeDegreePackets.diagonalOrbit line 133

    def diagonalOrbit {d : ℕ} (x : X (d + 1)) : Set (X (d + 1)) := Set.range (orbitMap x)

  • definition OAI.PrimeDegreePackets.orbitVolume line 135

    def orbitVolume {d : ℕ} (x : X (d + 1)) : ℝ≥0∞ :=  addCovolume (periodGroup x) (LogCoordinates d) volume

  • definition OAI.PrimeDegreePackets.orbitalHaar line 138

    def orbitalHaar {d : ℕ} (x : X (d + 1)) : Measure (X (d + 1)) :=  if h : HasAddFundamentalDomain (periodGroup x) (LogCoordinates d) volume then    Measure.map (orbitMap x) (volume.restrict h.ExistsIsAddFundamentalDomain.choose)  else 0

  • definition OAI.PrimeDegreePackets.orbitProbability line 143

    def orbitProbability {d : ℕ} (x : X (d + 1)) : Measure (X (d + 1)) :=  (orbitVolume x)⁻¹ • orbitalHaar x

  • abbreviation OAI.PrimeDegreePackets.LocalAlgebra line 146

    abbrev LocalAlgebra (p : ℕ) [Fact p.Prime] (K : Type*) [Field K] [NumberField K] :=  Padic p ⊗[ℚ] K

  • definition OAI.PrimeDegreePackets.completedLattice line 149

    def completedLattice {n : ℕ} {K : Type*} [Field K] [NumberField K]    (p : ℕ) [Fact p.Prime] (M : FullLattice n K) :    Submodule (PadicInt p) (LocalAlgebra p K) :=  Submodule.span (PadicInt p)    ((Algebra.TensorProduct.includeRight : K →ₐ[ℚ] LocalAlgebra p K) '' (M.carrier : Set K))

6 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: continuous, convergence, prime, function, weak, measures and probability. Found in the paper’s title, which its main theorem states: degree, equidistribution, local and packets. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: mass, escapes and sequence.

Statements in Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type

Theorem

For every fixed prime n\ge5 and every sequence (K_i,M_i,\sigma_i) as above with D(M_i)\to\infty,

\mu_{K_i,M_i,\sigma_i}\longrightarrow m_n.

The convergence is weak convergence of probability measures. In particular, it holds against every continuous compactly supported function, and no mass escapes into the cusp.

Section 1, Introduction Read it in the paper TeX source, line 103
Show the TeX
For every fixed prime $n\ge5$ and every sequence
$(K_i,M_i,\sigma_i)$ as above with \mbox{$D(M_i)\to\infty$},
\[
 \mu_{K_i,M_i,\sigma_i}\longrightarrow m_n.
\]
The convergence is weak convergence of probability measures. In
particular, it holds against every continuous compactly supported
function, and no mass escapes into the cusp.

Other possible paper matches, closest wording first

  • Proposition A.1 of Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type. Words of the Lean statement, its names split into words, that also appear in the theorem: includes, rational, changing, basis, quotient, local and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: homothety, allowed and torus.
  • Theorem 1.1 of Equidistribution of Primitive Sextic Torus Packets. Words of the Lean statement, its names split into words, that also appear in the theorem: choose, degree, epsilon, totally, field, real and two more. Found in the paper’s title, which its main theorem states: equidistribution and packets. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: escape and mass.
  • Proposition 3.6 of Equidistribution of primitive quartic torus packets for arbitrary orders. Words of the Lean statement, its names split into words, that also appear in the theorem: tightness, ranges, tight, epsilon, totally, weak and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sequence.
  • Theorem 4.4 of Equidistribution of primitive quartic torus packets for arbitrary orders. Words of the Lean statement, its names split into words, that also appear in the theorem: action, matrices, diagonal, point, subgroup, group and two more.
  • Theorem 1.1 of Equidistribution of primitive quartic torus packets for arbitrary orders. Words of the Lean statement, its names split into words, that also appear in the theorem: totally, field, real, embeddings and order. Found in the paper’s title, which its main theorem states: equidistribution and packets. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: escape, mass and sequence.
  • Proposition 2.3 of Equidistribution of Primitive Sextic Torus Packets. Words of the Lean statement, its names split into words, that also appear in the theorem: family, tight, weak, bound, measure, packet and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: form.

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

One independent group reports that Comparator accepted a run for this Lean statement file: Carlos Toledo’s cert-machine. 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

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 015’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: Prime degree >= 5 only; the primitive quartic (Picard packets) and primitive sextic (maximal-order ideal-class packets) results of the summary are not stated.

    Definitions the group asks readers to compare with the paper: Whole packet-measure apparatus is custom (packetOrbits, orbitBasepoint := Classical.epsilon ..., orbitalHaar := if h : HasAddFundamentalDomain ... then ... else 0, latticePoint := if h : ∃ g, ... then ⟦h.choose⟧ else ⟦1⟧): junk else branches are unreachable for genuine torus packets, but the encoding of 'complete volume-weighted packet' is not checkable against a standard Mathlib notion

  • 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: The primitive quartic (arbitrary orders) and primitive sextic (maximal orders) equidistribution theorems are not stated; only prime degree ≥ 5.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves equidistribution of volume-weighted torus packets for totally real number fields of every fixed prime degree at least five. For any sequence of full lattices whose multiplier-order discriminants tend to infinity, the packet measures converge weakly to Haar probability measure and form a tight family, so no mass escapes. Arbitrary local homothety types are allowed, and the fields may vary along the sequence.”

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

231 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: 139 to its own declarations, 172 to Mathlib’s documentation. 40 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAIsectionnoncomputable sectionopen scoped Matrix Topology BigOperators Classical TensorProduct ENNRealopen MeasureTheory Filter Set Modulenamespace PrimeDegreePacketsabbrev G (n : ℕ) := Matrix.SpecialLinearGroup (Fin n) ℝdef integralGroup (n : ℕ) : Subgroup (G n) :=  (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)).rangeabbrev X (n : ℕ) := Quotient (QuotientGroup.rightRel (integralGroup n))instance xMeasurableSpace (n : ℕ) : MeasurableSpace (X n) := borel (X n)instance xBorelSpace (n : ℕ) : BorelSpace (X n) := ⟨rfl⟩def rightAction {n : ℕ} (x : X n) (g : G n) : X n :=  Quotient.map (fun h => h * g) (by    intro a b hab    change (QuotientGroup.rightRel (integralGroup n)) a b at hab    change (QuotientGroup.rightRel (integralGroup n)) (a * g) (b * g)    rw [QuotientGroup.rightRel_apply] at hab ⊢    simpa only [mul_inv_rev, mul_assoc, mul_inv_cancel_left] using hab) x@[simp] theorem rightAction_one {n : ℕ} (x : X n) : rightAction x 1 = x := by  induction x using Quotient.inductionOn with  | h a => exact congrArg (Quotient.mk _) (mul_one a)@[simp] theorem rightAction_mul {n : ℕ} (x : X n) (g h : G n) :    rightAction (rightAction x g) h = rightAction x (g * h) := by  induction x using Quotient.inductionOn with  | h a => exact congrArg (Quotient.mk _) (mul_assoc a g h)def signMatrices (n : ℕ) : Set (G n) :=  {g | (∀ i j, i ≠ j → g.1 i j = 0) ∧ ∀ i, g.1 i i = 1 ∨ g.1 i i = -1}def rowLattice {n : ℕ} (g : Matrix (Fin n) (Fin n) ℝ) : Submodule ℤ (Fin n → ℝ) :=  Submodule.span ℤ (Set.range fun i => g i)structure FullLattice (n : ℕ) (K : Type*) [Field K] [NumberField K] where  carrier : Submodule ℤ K  basis : Basis (Fin n) ℤ carrier  spans : Submodule.span ℚ (carrier : Set K) = ⊤def multiplierOrder {n : ℕ} {K : Type*} [Field K] [NumberField K] (M : FullLattice n K) : AddSubgroup K where  carrier := {a | ∀ x ∈ M.carrier, a * x ∈ M.carrier}  zero_mem' := by simp  add_mem' := by    intro a b ha hb x hx    simpa only [add_mul] using M.carrier.add_mem (ha x hx) (hb x hx)  neg_mem' := by    intro a ha x hx    simpa only [neg_mul] using M.carrier.neg_mem (ha x hx)def integralElements (K : Type*) [Field K] : AddSubgroup K :=  (integralClosure ℤ K).toSubring.toAddSubgroupdef multiplierDiscriminant {n : ℕ} {K : Type*} [Field K] [NumberField K]    (M : FullLattice n K) : ℝ :=  |(NumberField.discr K : ℝ)| *    ((multiplierOrder M).relIndex (integralElements K) : ℝ) ^ 2abbrev OrderedEmbeddings (n : ℕ) (K : Type*) [Field K] := Fin n ≃ (K →+* ℝ)def embeddingMatrix {n : ℕ} {K : Type*} [Field K] [NumberField K]    (M : FullLattice n K) (σ : OrderedEmbeddings n K) : Matrix (Fin n) (Fin n) ℝ :=  fun i j => σ j (M.basis i)def normalizedLattice {n : ℕ} {K : Type*} [Field K] [NumberField K]    (M : FullLattice n K) (σ : OrderedEmbeddings n K) : Submodule ℤ (Fin n → ℝ) :=  let B := embeddingMatrix M σ  rowLattice (|(Matrix.det B)| ^ (-(1 : ℝ) / n) • B)def latticePoint {n : ℕ} {K : Type*} [Field K] [NumberField K]    (M : FullLattice n K) (σ : OrderedEmbeddings n K) : X n :=  if h : ∃ g : G n, rowLattice g.1 = normalizedLattice M σ then    Quotient.mk _ h.choose  else Quotient.mk _ (1 : G n)abbrev LogCoordinates (d : ℕ) := Fin d → ℝdef diagonalFlow {d : ℕ} (u : LogCoordinates d) : G (d + 1) :=  ⟨Matrix.diagonal (Fin.snoc (fun i => Real.exp (u i)) (Real.exp (-∑ i, u i))), by    rw [Matrix.det_diagonal, Fin.prod_univ_castSucc]    simp only [Fin.snoc_castSucc, Fin.snoc_last]    rw [← Real.exp_sum, ← Real.exp_add, add_neg_cancel, Real.exp_zero]⟩@[simp] theorem diagonalFlow_zero (d : ℕ) : diagonalFlow (0 : LogCoordinates d) = 1 := by  apply Subtype.ext  change Matrix.diagonal _ = (1 : Matrix (Fin (d + 1)) (Fin (d + 1)) ℝ)  simp  ext i  refine Fin.lastCases ?_ (fun j => ?_) i <;> simp@[simp] theorem diagonalFlow_add {d : ℕ} (u v : LogCoordinates d) :    diagonalFlow (u + v) = diagonalFlow u * diagonalFlow v := by  apply Subtype.ext  change Matrix.diagonal _ = Matrix.diagonal _ * Matrix.diagonal _  rw [Matrix.diagonal_mul_diagonal]  congr 1  funext i  refine Fin.lastCases ?_ (fun j => ?_) i  · simp [Real.exp_add, Finset.sum_add_distrib, neg_add_rev, mul_comm]  · simp [Real.exp_add]def orbitMap {d : ℕ} (x : X (d + 1)) (u : LogCoordinates d) : X (d + 1) :=  rightAction x (diagonalFlow u)@[simp] theorem orbitMap_zero {d : ℕ} (x : X (d + 1)) : orbitMap x 0 = x := by  simp [orbitMap]@[simp] theorem orbitMap_add {d : ℕ} (x : X (d + 1)) (u v : LogCoordinates d) :    orbitMap x (u + v) = orbitMap (orbitMap x u) v := by  simp [orbitMap]def periodGroup {d : ℕ} (x : X (d + 1)) : AddSubgroup (LogCoordinates d) where  carrier := {u | orbitMap x u = x}  zero_mem' := orbitMap_zero x  add_mem' := by    intro u v hu hv    change orbitMap x (u + v) = x    rw [orbitMap_add, hu, hv]  neg_mem' := by    intro u hu    have h := congrArg (fun y => orbitMap y (-u)) hu    rw [← orbitMap_add, add_neg_cancel, orbitMap_zero] at h    exact h.symmdef diagonalOrbit {d : ℕ} (x : X (d + 1)) : Set (X (d + 1)) := Set.range (orbitMap x)def orbitVolume {d : ℕ} (x : X (d + 1)) : ℝ≥0∞ :=  addCovolume (periodGroup x) (LogCoordinates d) volumedef orbitalHaar {d : ℕ} (x : X (d + 1)) : Measure (X (d + 1)) :=  if h : HasAddFundamentalDomain (periodGroup x) (LogCoordinates d) volume then    Measure.map (orbitMap x) (volume.restrict h.ExistsIsAddFundamentalDomain.choose)  else 0def orbitProbability {d : ℕ} (x : X (d + 1)) : Measure (X (d + 1)) :=  (orbitVolume x)⁻¹ • orbitalHaar xabbrev LocalAlgebra (p : ℕ) [Fact p.Prime] (K : Type*) [Field K] [NumberField K] :=  Padic p ⊗[ℚ] Kdef completedLattice {n : ℕ} {K : Type*} [Field K] [NumberField K]    (p : ℕ) [Fact p.Prime] (M : FullLattice n K) :    Submodule (PadicInt p) (LocalAlgebra p K) :=  Submodule.span (PadicInt p)    ((Algebra.TensorProduct.includeRight : K →ₐ[ℚ] LocalAlgebra p K) '' (M.carrier : Set K))def LocallyHomothetic {n : ℕ} {K : Type*} [Field K] [NumberField K]    (M M' : FullLattice n K) : Prop :=  ∀ (p : ℕ) (hp : p.Prime), letI : Fact p.Prime := ⟨hp⟩    ∃ c : (LocalAlgebra p K)ˣ,      (completedLattice p M' : Set (LocalAlgebra p K)) =        (fun z => (c : LocalAlgebra p K) * z) '' (completedLattice p M : Set (LocalAlgebra p K))def packetOrbits {d : ℕ} {K : Type*} [Field K] [NumberField K]    (M : FullLattice (d + 1) K) (σ : OrderedEmbeddings (d + 1) K) :    Set (Set (X (d + 1))) :=  {O | ∃ M' : FullLattice (d + 1) K, LocallyHomothetic M M' ∧    ∃ w ∈ signMatrices (d + 1), O = diagonalOrbit (rightAction (latticePoint M' σ) w)}def orbitBasepoint {d : ℕ} {K : Type*} [Field K] [NumberField K]    {M : FullLattice (d + 1) K} {σ : OrderedEmbeddings (d + 1) K}    (O : packetOrbits M σ) : X (d + 1) :=  Classical.epsilon (fun x => diagonalOrbit x = O.1)def packetMeasure {d : ℕ} {K : Type*} [Field K] [NumberField K]    (M : FullLattice (d + 1) K) (σ : OrderedEmbeddings (d + 1) K) : Measure (X (d + 1)) :=  (∑' O : packetOrbits M σ, orbitVolume (orbitBasepoint O))⁻¹ •    Measure.sum (fun O : packetOrbits M σ =>      orbitVolume (orbitBasepoint O) • orbitProbability (orbitBasepoint O))def IsHaarProbability {n : ℕ} (μ : Measure (X n)) : Prop :=  IsProbabilityMeasure μ ∧ ∀ g : G n, Measure.map (fun x => rightAction x g) μ = μdef WeakProbabilityConvergence {n : ℕ} (μ : ℕ → Measure (X n)) (m : Measure (X n)) : Prop :=  (∀ i, IsProbabilityMeasure (μ i)) ∧ IsProbabilityMeasure m ∧    ∀ f : BoundedContinuousFunction (X n) ℝ,    Tendsto (fun i => ∫ x, f x ∂μ i) atTop (𝓝 (∫ x, f x ∂m))end PrimeDegreePacketsendendopen MeasureTheory Filteropen scoped Topologynamespace DukePrimeDegreeopen PrimeDegreePacketsuniverse u/-- Theorem 1.1 as weak convergence of the actual complete packet measures,including tightness of the entire family. -/theorem prime_degree_packet_measure_equidistribution_unconditional    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]    [∀ i, NumberField.IsTotallyReal (K i)]    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :    ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧      WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧      IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by  sorry/-- The complete packet measures converge weakly to any specified Haarprobability, in the formulation by bounded continuous test functions. -/theorem prime_degree_packet_measures_tendsto_haar_unconditional    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]    [∀ i, NumberField.IsTotallyReal (K i)]    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)    (m : ProbabilityMeasure (X (d+1)))    (hm : IsHaarProbability (m : Measure (X (d+1)))) :    WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i))      (m : Measure (X (d+1))) := by  sorryend DukePrimeDegreeend OAI
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Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type

Theorem

For every fixed prime n\ge5 and every sequence (K_i,M_i,\sigma_i) as above with D(M_i)\to\infty,

\mu_{K_i,M_i,\sigma_i}\longrightarrow m_n.

The convergence is weak convergence of probability measures. In particular, it holds against every continuous compactly supported function, and no mass escapes into the cusp.

Section 1, Introduction Read it in the paper TeX source, line 103
Show the TeX
For every fixed prime $n\ge5$ and every sequence
$(K_i,M_i,\sigma_i)$ as above with \mbox{$D(M_i)\to\infty$},
\[
 \mu_{K_i,M_i,\sigma_i}\longrightarrow m_n.
\]
The convergence is weak convergence of probability measures. In
particular, it holds against every continuous compactly supported
function, and no mass escapes into the cusp.
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 015 link it.
Solution module
OAI.NumberTheory.DukePrimeDegree.MainUnconditional
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
32 local vocabulary declarations
Lines of Lean, without comments or blank lines
164
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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Plectis Atlas

Equidistribution of prime-degree torus packets

Family 015: Torus-packet equidistribution in prime, quartic, and sextic degrees

Candidate pairing

Theorem

For every fixed prime n\ge5 and every sequence (K_i,M_i,\sigma_i) as above with D(M_i)\to\infty,

\mu_{K_i,M_i,\sigma_i}\longrightarrow m_n.

The convergence is weak convergence of probability measures. In particular, it holds against every continuous compactly supported function, and no mass escapes into the cusp.

Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type

TeX label thm:main, build/paper.tex, line 103

The Lean statement says

theorem prime_degree_packet_measure_equidistribution_unconditional    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]    [∀ i, NumberField.IsTotallyReal (K i)]    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))    (hn : ∀ i, Module.finrank ℚ (K i)=d+1)    (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :    ∃ ν : Measure (X (d+1)), IsHaarProbability ν ∧      WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧      IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by  sorrytheorem prime_degree_packet_measures_tendsto_haar_unconditional    (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1)    (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]    [∀ i, NumberField.IsTotallyReal (K i)]    (M : ∀ i, PrimeDegreePackets.FullLattice (d+1) (K i))    (σ : ∀ i, OrderedEmbeddings (d+1) (K i))

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

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

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