Lean statement

Global classical Vlasov–Maxwell solutions

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.RVM.global_classical_solution

Lean statement file: VlasovMaxwell.leanShow on the mapFamily 362. OpenAI lists this as a main result in its Lean catalogue.

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 global_classical_solution (d : Datum) (hd : Admissible d) :    ∃ s : Solution, Classical d s ∧ SmoothOnFiniteHorizons s ∧      ∀ s' : Solution, Classical d s' → SameNonnegativeTime s s' := 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.RVM.Vec line 10

abbrev Vec := EuclideanSpace ℝ (Fin 3)

abbreviation OAI.RVM.Phase line 11
```
abbrev Phase := Vec × Vec
```
abbreviation OAI.RVM.Particle line 12
```
abbrev Particle := ℝ × Phase → ℝ
```
abbreviation OAI.RVM.Field line 13
```
abbrev Field := ℝ × Vec → Vec
```

definition OAI.RVM.q line 15

def q (v : Vec) : ℝ := Real.sqrt (1 + ‖v‖ ^ 2)

definition OAI.RVM.velocity line 16

def velocity (v : Vec) : Vec := (q v)⁻¹ • v

definition OAI.RVM.basis line 18

def basis (i : Fin 3) : Vec := WithLp.toLp 2 (Pi.single i (1 : ℝ))

definition OAI.RVM.cross line 20

def cross (a b : Vec) : Vec := WithLp.toLp 2  ![a 1 * b 2 - a 2 * b 1, a 2 * b 0 - a 0 * b 2,    a 0 * b 1 - a 1 * b 0]

definition OAI.RVM.spatialPartial line 24

def spatialPartial (g : Vec → ℝ) (x : Vec) (i : Fin 3) : ℝ :=  fderiv ℝ g x (basis i)

definition OAI.RVM.divergence line 26

def divergence (F : Vec → Vec) (x : Vec) : ℝ :=  ∑ i : Fin 3, spatialPartial (fun y => F y i) x i

definition OAI.RVM.curl line 29

def curl (F : Vec → Vec) (x : Vec) : Vec := WithLp.toLp 2  ![spatialPartial (fun y => F y 2) x 1 - spatialPartial (fun y => F y 1) x 2,    spatialPartial (fun y => F y 0) x 2 - spatialPartial (fun y => F y 2) x 0,    spatialPartial (fun y => F y 1) x 0 - spatialPartial (fun y => F y 0) x 1]

definition OAI.RVM.rho line 34

def rho (g : Phase → ℝ) (x : Vec) : ℝ := ∫ v : Vec, g (x, v)

definition OAI.RVM.current line 35

def current (g : Phase → ℝ) (x : Vec) : Vec :=  ∫ v : Vec, g (x, v) • velocity v

structure OAI.RVM.Datum line 38

structure Datum where  f₀ : Phase → ℝ  E₀ : Vec → Vec  B₀ : Vec → Vec

definition OAI.RVM.BoundedSmooth line 43

def BoundedSmooth (F : Vec → Vec) : Prop :=  ContDiff ℝ ∞ F ∧ ∀ n : ℕ, ∃ C : ℝ, ∀ x : Vec,    ‖iteratedFDeriv ℝ n F x‖ ≤ C

definition OAI.RVM.Admissible line 47

def Admissible (d : Datum) : Prop :=  ContDiff ℝ ∞ d.f₀ ∧ HasCompactSupport d.f₀ ∧ (∀ z, 0 ≤ d.f₀ z) ∧  BoundedSmooth d.E₀ ∧ BoundedSmooth d.B₀ ∧  MemLp d.E₀ 2 volume ∧ MemLp d.B₀ 2 volume ∧  (∀ x, divergence d.E₀ x = rho d.f₀ x) ∧  (∀ x, divergence d.B₀ x = 0)

structure OAI.RVM.Solution line 54

structure Solution where  f : Particle  E : Field  B : Field

definition OAI.RVM.particleHalfSpace line 59

def particleHalfSpace : Set (ℝ × Phase) := {z | 0 ≤ z.1}

definition OAI.RVM.fieldHalfSpace line 60

def fieldHalfSpace : Set (ℝ × Vec) := {z | 0 ≤ z.1}

definition OAI.RVM.particleSlab line 61

def particleSlab (T : ℝ) : Set (ℝ × Phase) := {z | z.1 ∈ Icc 0 T}

definition OAI.RVM.fieldSlab line 62

def fieldSlab (T : ℝ) : Set (ℝ × Vec) := {z | z.1 ∈ Icc 0 T}

definition OAI.RVM.TimeContinuousL2 line 64

def TimeContinuousL2 (F : Field) : Prop :=  ∃ F₂ : ℝ → Lp Vec 2 (volume : Measure Vec),    ContinuousOn F₂ (Ici 0) ∧    ∀ t, 0 ≤ t → (F₂ t : Vec → Vec) =ᵐ[volume] (fun x => F (t, x))

definition OAI.RVM.Equations line 69

def Equations (s : Solution) : Prop := ∀ t, 0 ≤ t →

Show the remaining 11 lines

  (∀ x v,    derivWithin (fun t' => s.f (t', x, v)) (Ici 0) t +    (∑ i : Fin 3, velocity v i * spatialPartial (fun y => s.f (t, y, v)) x i) +    (∑ i : Fin 3, (s.E (t, x) + cross (velocity v) (s.B (t, x))) i *      spatialPartial (fun v' => s.f (t, x, v')) v i) = 0) ∧  (∀ x, derivWithin (fun t' => s.E (t', x)) (Ici 0) t -    curl (fun y => s.B (t, y)) x = -current (fun z => s.f (t, z)) x) ∧  (∀ x, derivWithin (fun t' => s.B (t', x)) (Ici 0) t +    curl (fun y => s.E (t, y)) x = 0) ∧  (∀ x, divergence (fun y => s.E (t, y)) x = rho (fun z => s.f (t, z)) x) ∧  (∀ x, divergence (fun y => s.B (t, y)) x = 0)

definition OAI.RVM.CompactOnFiniteHorizons line 82

def CompactOnFiniteHorizons (s : Solution) : Prop :=  ∀ T : ℝ, 0 ≤ T → ∃ K : Set Phase, IsCompact K ∧    ∀ t ∈ Icc 0 T, tsupport (fun z => s.f (t, z)) ⊆ K

3 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: phase, divergence, global, space, support, time, finite, admissible and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constraints and unique.

Theorem 1.1 of Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system

Theorem

Every smooth admissible datum (eq:data) generates a unique global classical solution of (eq:rvm). For every finite T, this solution satisfies

f\in C^\infty([0,T]\times\R^3_x\times\R^3_v),\qquad E,B\in C^\infty([0,T]\times\R^3_x;\R^3),

and f has compact phase-space support on [0,T]. The two divergence constraints hold for all time.

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

Show the TeX

Every smooth admissible datum \eqref{eq:data} generates a unique global
classical solution of \eqref{eq:rvm}. For every finite $T$, this solution
satisfies
\[
 f\in C^\infty([0,T]\times\R^3_x\times\R^3_v),\qquad
 E,B\in C^\infty([0,T]\times\R^3_x;\R^3),
\]
and $f$ has compact phase-space support on $[0,T]$.
The two divergence constraints hold for all time.

Other possible paper matches, closest wording first

Proposition 2.2 of Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system. Words of the Lean statement, its names split into words, that also appear in the theorem: time, finite, admissible, smooth, solution, datum and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unique.
Proposition 2.1 of Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system. Words of the Lean statement, its names split into words, that also appear in the theorem: supported, finite, admissible, smooth, datum, solution and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one and interval.
Proposition 6.3 of Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system. Words of the Lean statement, its names split into words, that also appear in the theorem: spatial, horizon, bound, pi, time, compact and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: derivative, one, interval and initial.
Proposition 3.3 of Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system. Words of the Lean statement, its names split into words, that also appear in the theorem: field, partial, pi and solutions. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: initial.
Proposition 5.1 of Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system. Words of the Lean statement, its names split into words, that also appear in the theorem: bounds and sqrt.
Proposition 4.3 of Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system. Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: energy.

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 362’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: global_classical_solution: for every admissible datum (f₀ ≥ 0 smooth compactly supported in ℝ³×ℝ³; E₀, B₀ C^∞ with bounded derivatives of all orders, in L², div E₀ = ρ[f₀], div B₀ = 0) there is a solution of the full relativistic system (v̂ = v/√(1+|v|²), force E + v̂×B, ∂ₜE - curl B = -j, ∂ₜB + curl E = 0, Gauss laws) that is C^∞ on every [0,T], and every C¹ solution in the class (nonnegative, L²-continuous fields, compact phase support on finite horizons) coincides with it for t ≥ 0; no smallness, symmetry or neutrality assumption.
Definitions the group asks readers to compare with the paper: def Classical (d) (s) inside namespace RVM shadows the Classical namespace name (harmless); the Vlasov, Maxwell and Gauss equations are written out by hand with derivWithin (Ici 0) at t = 0
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

“The formalized result gives global existence and uniqueness for the three-dimensional one-species relativistic Vlasov–Maxwell system. The initial particle density is nonnegative, smooth, and compactly supported; the initial fields have bounded derivatives of all orders, finite energy, and satisfy both Gauss constraints. The solution is smooth on every finite time interval, has compact particle phase support there, and is unique among classical solutions. No smallness, symmetry, or neutrality assumption is imposed.”
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

112 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: 100 to its own declarations, 50 to Mathlib’s documentation. 29 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 global_classical_solution exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches 27 of the file’s 28 other declarations; with the theorem itself they span 79 lines. 6 of them are named in the statement itself.

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

abbrev Vecline 10library
abbrev Phaseline 11library
abbrev Particleline 12library
abbrev Fieldline 13library
def qline 15library
def velocityline 16library
def basisline 18library
def crossline 20library
def spatialPartialline 24library
def divergenceline 26library
def curlline 29library
def rholine 34library
def currentline 35library
structure Datumline 38in the statementlibrary
def BoundedSmoothline 43library
def Admissibleline 47in the statementlibrary
structure Solutionline 54in the statementlibrary
def particleHalfSpaceline 59library
def fieldHalfSpaceline 60library
def particleSlabline 61library
def fieldSlabline 62library
def TimeContinuousL2line 64library
def Equationsline 69library
def CompactOnFiniteHorizonsline 82library
def Classicalline 86in the statementlibrary
def SmoothOnFiniteHorizonsline 95in the statementlibrary
def SameNonnegativeTimeline 99in the statementlibrary
theorem global_classical_solutionline 103selected theoremlibrary

1 declaration the statement does not reach by name

These definitions and lemmas were not found by following names in the statement. Lean may still use them through rules such as instances or macros.

def dot line 19

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheory Setopen scoped ContDiffnamespace RVMabbrev Vec := EuclideanSpace ℝ (Fin 3)abbrev Phase := Vec × Vecabbrev Particle := ℝ × Phase → ℝabbrev Field := ℝ × Vec → Vecdef q (v : Vec) : ℝ := Real.sqrt (1 + ‖v‖ ^ 2)def velocity (v : Vec) : Vec := (q v)⁻¹ • vdef basis (i : Fin 3) : Vec := WithLp.toLp 2 (Pi.single i (1 : ℝ))def dot (a b : Vec) : ℝ := ∑ i : Fin 3, a i * b idef cross (a b : Vec) : Vec := WithLp.toLp 2  ![a 1 * b 2 - a 2 * b 1, a 2 * b 0 - a 0 * b 2,    a 0 * b 1 - a 1 * b 0]def spatialPartial (g : Vec → ℝ) (x : Vec) (i : Fin 3) : ℝ :=  fderiv ℝ g x (basis i)def divergence (F : Vec → Vec) (x : Vec) : ℝ :=  ∑ i : Fin 3, spatialPartial (fun y => F y i) x idef curl (F : Vec → Vec) (x : Vec) : Vec := WithLp.toLp 2  ![spatialPartial (fun y => F y 2) x 1 - spatialPartial (fun y => F y 1) x 2,    spatialPartial (fun y => F y 0) x 2 - spatialPartial (fun y => F y 2) x 0,    spatialPartial (fun y => F y 1) x 0 - spatialPartial (fun y => F y 0) x 1]def rho (g : Phase → ℝ) (x : Vec) : ℝ := ∫ v : Vec, g (x, v)def current (g : Phase → ℝ) (x : Vec) : Vec :=  ∫ v : Vec, g (x, v) • velocity vstructure Datum where  f₀ : Phase → ℝ  E₀ : Vec → Vec  B₀ : Vec → Vecdef BoundedSmooth (F : Vec → Vec) : Prop :=  ContDiff ℝ ∞ F ∧ ∀ n : ℕ, ∃ C : ℝ, ∀ x : Vec,    ‖iteratedFDeriv ℝ n F x‖ ≤ Cdef Admissible (d : Datum) : Prop :=  ContDiff ℝ ∞ d.f₀ ∧ HasCompactSupport d.f₀ ∧ (∀ z, 0 ≤ d.f₀ z) ∧  BoundedSmooth d.E₀ ∧ BoundedSmooth d.B₀ ∧  MemLp d.E₀ 2 volume ∧ MemLp d.B₀ 2 volume ∧  (∀ x, divergence d.E₀ x = rho d.f₀ x) ∧  (∀ x, divergence d.B₀ x = 0)structure Solution where  f : Particle  E : Field  B : Fielddef particleHalfSpace : Set (ℝ × Phase) := {z | 0 ≤ z.1}def fieldHalfSpace : Set (ℝ × Vec) := {z | 0 ≤ z.1}def particleSlab (T : ℝ) : Set (ℝ × Phase) := {z | z.1 ∈ Icc 0 T}def fieldSlab (T : ℝ) : Set (ℝ × Vec) := {z | z.1 ∈ Icc 0 T}def TimeContinuousL2 (F : Field) : Prop :=  ∃ F₂ : ℝ → Lp Vec 2 (volume : Measure Vec),    ContinuousOn F₂ (Ici 0) ∧    ∀ t, 0 ≤ t → (F₂ t : Vec → Vec) =ᵐ[volume] (fun x => F (t, x))def Equations (s : Solution) : Prop := ∀ t, 0 ≤ t →  (∀ x v,    derivWithin (fun t' => s.f (t', x, v)) (Ici 0) t +    (∑ i : Fin 3, velocity v i * spatialPartial (fun y => s.f (t, y, v)) x i) +    (∑ i : Fin 3, (s.E (t, x) + cross (velocity v) (s.B (t, x))) i *      spatialPartial (fun v' => s.f (t, x, v')) v i) = 0) ∧  (∀ x, derivWithin (fun t' => s.E (t', x)) (Ici 0) t -    curl (fun y => s.B (t, y)) x = -current (fun z => s.f (t, z)) x) ∧  (∀ x, derivWithin (fun t' => s.B (t', x)) (Ici 0) t +    curl (fun y => s.E (t, y)) x = 0) ∧  (∀ x, divergence (fun y => s.E (t, y)) x = rho (fun z => s.f (t, z)) x) ∧  (∀ x, divergence (fun y => s.B (t, y)) x = 0)def CompactOnFiniteHorizons (s : Solution) : Prop :=  ∀ T : ℝ, 0 ≤ T → ∃ K : Set Phase, IsCompact K ∧    ∀ t ∈ Icc 0 T, tsupport (fun z => s.f (t, z)) ⊆ Kdef Classical (d : Datum) (s : Solution) : Prop :=  ContDiffOn ℝ 1 s.f particleHalfSpace ∧  ContDiffOn ℝ 1 s.E fieldHalfSpace ∧ ContDiffOn ℝ 1 s.B fieldHalfSpace ∧  (∀ t, 0 ≤ t → ∀ z, 0 ≤ s.f (t, z)) ∧  TimeContinuousL2 s.E ∧ TimeContinuousL2 s.B ∧  CompactOnFiniteHorizons s ∧ Equations s ∧  (∀ z, s.f (0, z) = d.f₀ z) ∧  (∀ x, s.E (0, x) = d.E₀ x) ∧ (∀ x, s.B (0, x) = d.B₀ x)def SmoothOnFiniteHorizons (s : Solution) : Prop := ∀ T : ℝ, 0 ≤ T →  ContDiffOn ℝ ∞ s.f (particleSlab T) ∧  ContDiffOn ℝ ∞ s.E (fieldSlab T) ∧ ContDiffOn ℝ ∞ s.B (fieldSlab T)def SameNonnegativeTime (s s' : Solution) : Prop := ∀ t, 0 ≤ t →  (∀ z, s.f (t, z) = s'.f (t, z)) ∧  (∀ x, s.E (t, x) = s'.E (t, x)) ∧ (∀ x, s.B (t, x) = s'.B (t, x))theorem global_classical_solution (d : Datum) (hd : Admissible d) :    ∃ s : Solution, Classical d s ∧ SmoothOnFiniteHorizons s ∧      ∀ s' : Solution, Classical d s' → SameNonnegativeTime s s' := by  sorryend RVMendend 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.

Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system

Theorem

Every smooth admissible datum (eq:data) generates a unique global classical solution of (eq:rvm). For every finite T, this solution satisfies

f\in C^\infty([0,T]\times\R^3_x\times\R^3_v),\qquad E,B\in C^\infty([0,T]\times\R^3_x;\R^3),

and f has compact phase-space support on [0,T]. The two divergence constraints hold for all time.

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

Show the TeX

Every smooth admissible datum \eqref{eq:data} generates a unique global
classical solution of \eqref{eq:rvm}. For every finite $T$, this solution
satisfies
\[
 f\in C^\infty([0,T]\times\R^3_x\times\R^3_v),\qquad
 E,B\in C^\infty([0,T]\times\R^3_x;\R^3),
\]
and $f$ has compact phase-space support on $[0,T]$.
The two divergence constraints hold for all time.

Configuration and file facts

File: lean/ComparatorChallenges/VlasovMaxwell.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/VlasovMaxwell/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: 28 local vocabulary declarations
Lines of Lean, without comments or blank lines: 80
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