Plectis Atlas
Search

↑ and ↓ move, Enter opens, Esc closes.

Lean statement

Two periodic hard-sphere Boltzmann solutions with the same data

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.BoltzmannNonuniqueness.nonuniqueness

Lean statement file: BoltzmannNonuniqueness.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 363 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 nonuniqueness :    ∃ f₀ : Density, BoundedVelocitySupport f₀ ∧      ∃ (F G : Evolution) (T : ℝ), AdmissibleGlobal f₀ F ∧ AdmissibleGlobal f₀ G ∧        StrongEarly T F ∧ StrongEarly T G ∧        ∃ t ∈ Set.Icc 0 T, ¬(F t =ᵐ[MeasureTheory.volume] G t) :=  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.BoltzmannNonuniqueness.Velocity line 19

    abbrev Velocity := EuclideanSpace ℝ (Fin 3)

  • abbreviation OAI.BoltzmannNonuniqueness.Position line 20

    abbrev Position := Fin 3 → AddCircle (1 : ℝ)

  • abbreviation OAI.BoltzmannNonuniqueness.Phase line 21

    abbrev Phase := Position × Velocity

  • abbreviation OAI.BoltzmannNonuniqueness.Density line 22

    abbrev Density := Phase → ℝ

  • abbreviation OAI.BoltzmannNonuniqueness.Sphere line 23

    abbrev Sphere := Metric.sphere (0 : Velocity) 1

  • definition OAI.BoltzmannNonuniqueness.sphereArea line 26

    def sphereArea : Measure Sphere := (volume : Measure Velocity).toSphere

  • definition OAI.BoltzmannNonuniqueness.normalComponent line 28

    def normalComponent (v w ω : Velocity) : ℝ := inner ℝ (v - w) ω

  • definition OAI.BoltzmannNonuniqueness.scatterLeft line 30

    def scatterLeft (v w ω : Velocity) : Velocity := v - normalComponent v w ω • ω

  • definition OAI.BoltzmannNonuniqueness.scatterRight line 32

    def scatterRight (v w ω : Velocity) : Velocity := w + normalComponent v w ω • ω

  • definition OAI.BoltzmannNonuniqueness.collisionKernel line 34

    def collisionKernel (v w ω : Velocity) : ℝ := |normalComponent v w ω|

  • definition OAI.BoltzmannNonuniqueness.gain line 36

    def gain (f : Density) (x : Position) (v : Velocity) : ℝ :=  ∫ p : Velocity × Sphere,    collisionKernel v p.1 p.2 * f (x, scatterLeft v p.1 p.2) *      f (x, scatterRight v p.1 p.2)    ∂((volume : Measure Velocity).prod sphereArea)

  • definition OAI.BoltzmannNonuniqueness.loss line 42

    def loss (f : Density) (x : Position) (v : Velocity) : ℝ :=  ∫ p : Velocity × Sphere,    collisionKernel v p.1 p.2 * f (x, v) * f (x, p.1)    ∂((volume : Measure Velocity).prod sphereArea)

  • definition OAI.BoltzmannNonuniqueness.collision line 47

    def collision (f : Density) (x : Position) (v : Velocity) : ℝ :=  gain f x v - loss f x vrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}

  • abbreviation OAI.BoltzmannNonuniqueness.Evolution line 52

    abbrev Evolution := ℝ → Density

  • abbreviation OAI.BoltzmannNonuniqueness.SpaceTimePhase line 53

    abbrev SpaceTimePhase := ℝ × Phase

  • abbreviation OAI.BoltzmannNonuniqueness.LiftedSpaceTimePhase line 54

    abbrev LiftedSpaceTimePhase := ℝ × (Velocity × Velocity)

  • definition OAI.BoltzmannNonuniqueness.torusProjection line 56

    def torusProjection (x : Velocity) : Position := fun i => (x i : AddCircle (1 : ℝ))

  • definition OAI.BoltzmannNonuniqueness.torusRepresentative line 60

    def torusRepresentative (x : Position) : Velocity :=  WithLp.toLp 2 (fun i => ((AddCircle.equivIco (1 : ℝ) 0) (x i) : ℝ))

  • definition OAI.BoltzmannNonuniqueness.maxwellian line 63

    def maxwellian (v : Velocity) : ℝ :=  (2 * Real.pi) ^ (-(3 / 2 : ℝ)) * Real.exp (-‖v‖ ^ 2 / 2)

  • definition OAI.BoltzmannNonuniqueness.entropyMoment line 66

    def entropyMoment (f : Density) (z : Phase) : ℝ :=  f z * (1 + ‖z.2‖ ^ 2 + |Real.log (f z)|)

  • definition OAI.BoltzmannNonuniqueness.FiniteEntropyDatum line 70

    def FiniteEntropyDatum (f : Density) : Prop :=  (∀ᵐ z, 0 ≤ f z) ∧ AEStronglyMeasurable f ∧ Integrable (entropyMoment f)

  • definition OAI.BoltzmannNonuniqueness.BoundedVelocitySupport line 73

    def BoundedVelocitySupport (f : Density) : Prop :=  ∃ R : ℝ, 0 < R ∧ ∀ᵐ z, R < ‖z.2‖ → f z = 0

  • definition OAI.BoltzmannNonuniqueness.energy line 76

    def energy (f : Density) : ℝ := ∫ z, ‖z.2‖ ^ 2 * f z

  • definition OAI.BoltzmannNonuniqueness.momentum line 78

    def momentum (f : Density) : Velocity := ∫ z, f z • z.2

13 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: space, sphere, strongly, log, support, definition, datum, gains and 15 more. Found in the paper’s title, which its main theorem states: periodic and nonuniqueness. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: distinct, boltzmann, two, hard and solutions.

Statements in Nonuniqueness for the periodic hard-sphere Boltzmann equation come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Nonuniqueness for the periodic hard-sphere Boltzmann equation

Theorem

There is a nonnegative density F_0 on \T^3\times\R^3, with bounded velocity support and

\int_{\T^3\times\R^3} F_0(1+|v|^2+|\log F_0|)\dd x\dd v<\infty,

for which the hard-sphere Boltzmann equation has two distinct global admissible solutions, in the sense of Definition def:admissible, with initial datum F_0. On some common initial interval both solutions are strongly continuous in L^1, and their collision gains and losses are integrable in space-time-velocity.

Section 1, Introduction Read it in the paper TeX source, line 55
Show the TeX
There is a nonnegative density \(F_0\) on \(\T^3\times\R^3\), with bounded
velocity support and
\[
\int_{\T^3\times\R^3}
 F_0(1+|v|^2+|\log F_0|)\dd x\dd v<\infty,
\]
for which the hard-sphere Boltzmann equation has two distinct global
admissible solutions, in the sense of Definition~\ref{def:admissible},
with initial datum \(F_0\).
On some common initial interval both solutions are strongly continuous
in \(L^1\), and their collision gains and losses are integrable in
space-time-velocity.

Other possible paper matches, closest wording first

  • Proposition 7.1 of Nonuniqueness for the periodic hard-sphere Boltzmann equation. Words of the Lean statement, its names split into words, that also appear in the theorem: continuity, compact, representative, terms, extend, spacetime and 16 more.
  • Proposition 6.1 of Nonuniqueness for the periodic hard-sphere Boltzmann equation. Words of the Lean statement, its names split into words, that also appear in the theorem: inequalities, condition, including, definition, common, datum and 13 more.
  • Proposition 4.3 of Nonuniqueness with local conservation for the hard-sphere Boltzmann equation. Words of the Lean statement, its names split into words, that also appear in the theorem: dissipation, norm, energy, momentum, conserved, mass and 18 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: data and solution.
  • Theorem 1.2 of Nonuniqueness with local conservation for the hard-sphere Boltzmann equation. Words of the Lean statement, its names split into words, that also appear in the theorem: log, conservation, entropy, definition, datum, admissible and seven more. Found in the paper’s title, which its main theorem states: sphere and nonuniqueness. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two and solutions.
  • Proposition 4.4 of Nonuniqueness for the periodic hard-sphere Boltzmann equation. Words of the Lean statement, its names split into words, that also appear in the theorem: additional, spatial, restricting, probability, condition, mass and nine more.
  • Proposition 9.1 of Nonuniqueness with local conservation for the hard-sphere Boltzmann equation. Words of the Lean statement, its names split into words, that also appear in the theorem: continuity, compact, representative, terms, extend, spacetime and 16 more.

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 363’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: “weaker-statement”.

    The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record

    Their note: nonuniqueness gives bounded-velocity-support data f₀ and two AdmissibleGlobal renormalized solutions F, G (all renormalizations β, gain/loss integrability, weak continuity, local mass conservation as a weak identity, global momentum conservation, global energy inequality ≤ and entropy-dissipation inequality) that are strongly L¹-continuous with integrable gain/loss on a common [0,T] only and differ at some t ≤ T. The summary's exact local conservation of momentum and kinetic energy, and strong L¹ continuity for all times, are not stated (docs: 'later local-conservation refinements are outside it').

    Definitions the group asks readers to compare with the paper: Hand-built kinetic setup: gain/loss as Bochner integrals over ℝ³×S² with sphereArea := volume.toSphere (the collision_fibers hypotheses prevent silent zero values of non-integrable integrals), extendedEntropyProduction (lower semicontinuous extension of (A-B)log(A/B)), entropyDissipation, relativeEntropy against a normalized Maxwellian; a statement-level run_cmd Lean.modifyEnv ... auxLemmasExt meta command

  • Carlos Toledo’s cert-machine

    Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.

    The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record

    Their note: The October refinement in the headline is not stated: exact LOCAL conservation of momentum and kinetic energy, and strong L1 continuity for all times. The Lean states the September result: local mass, total momentum, the energy and entropy inequalities, and strong L1 continuity on [0,T].

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions on $\mathbb T^3\times\mathbb R^3$ with the same nonnegative initial density. The data have bounded velocity support and finite mass, energy, and absolute entropy. Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities.”

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

215 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: 164 to its own declarations, 82 to Mathlib’s documentation. 39 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 OAI/-!# Nonuniqueness for the periodic hard-sphere Boltzmann equationTheorem 1.1, with the full entropy-class admissibility of Definition 2.1 andstrong initial L¹ continuity with integrable collision gain and loss.-/noncomputable sectionopen MeasureTheory Set Filteropen scoped RealInnerProductSpace ENNReal Topologynamespace BoltzmannNonuniquenessabbrev Velocity := EuclideanSpace ℝ (Fin 3)abbrev Position := Fin 3 → AddCircle (1 : ℝ)abbrev Phase := Position × Velocityabbrev Density := Phase → ℝabbrev Sphere := Metric.sphere (0 : Velocity) 1/-- Ordinary surface area, rather than normalized probability on the sphere. -/def sphereArea : Measure Sphere := (volume : Measure Velocity).toSpheredef normalComponent (v w ω : Velocity) : ℝ := inner ℝ (v - w) ωdef scatterLeft (v w ω : Velocity) : Velocity := v - normalComponent v w ω • ωdef scatterRight (v w ω : Velocity) : Velocity := w + normalComponent v w ω • ωdef collisionKernel (v w ω : Velocity) : ℝ := |normalComponent v w ω|def gain (f : Density) (x : Position) (v : Velocity) : ℝ :=  ∫ p : Velocity × Sphere,    collisionKernel v p.1 p.2 * f (x, scatterLeft v p.1 p.2) *      f (x, scatterRight v p.1 p.2)    ∂((volume : Measure Velocity).prod sphereArea)def loss (f : Density) (x : Position) (v : Velocity) : ℝ :=  ∫ p : Velocity × Sphere,    collisionKernel v p.1 p.2 * f (x, v) * f (x, p.1)    ∂((volume : Measure Velocity).prod sphereArea)def collision (f : Density) (x : Position) (v : Velocity) : ℝ :=  gain f x v - loss f x vrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}abbrev Evolution := ℝ → Densityabbrev SpaceTimePhase := ℝ × Phaseabbrev LiftedSpaceTimePhase := ℝ × (Velocity × Velocity)def torusProjection (x : Velocity) : Position := fun i => (x i : AddCircle (1 : ℝ))/-- A representative in the fundamental half-open unit cube, used only to evaluate periodicderivatives. Periodicity makes those derivatives independent of this choice. -/def torusRepresentative (x : Position) : Velocity :=  WithLp.toLp 2 (fun i => ((AddCircle.equivIco (1 : ℝ) 0) (x i) : ℝ))def maxwellian (v : Velocity) : ℝ :=  (2 * Real.pi) ^ (-(3 / 2 : ℝ)) * Real.exp (-‖v‖ ^ 2 / 2)def entropyMoment (f : Density) (z : Phase) : ℝ :=  f z * (1 + ‖z.2‖ ^ 2 + |Real.log (f z)|)/-- The convention `0 |log 0| = 0` is automatic in this real-valued moment. -/def FiniteEntropyDatum (f : Density) : Prop :=  (∀ᵐ z, 0 ≤ f z) ∧ AEStronglyMeasurable f ∧ Integrable (entropyMoment f)def BoundedVelocitySupport (f : Density) : Prop :=  ∃ R : ℝ, 0 < R ∧ ∀ᵐ z, R < ‖z.2‖ → f z = 0def energy (f : Density) : ℝ := ∫ z, ‖z.2‖ ^ 2 * f zdef momentum (f : Density) : Velocity := ∫ z, f z • z.2def relativeEntropy (f : Density) : ℝ :=  ∫ z, f z * Real.log (f z / maxwellian z.2) - f z + maxwellian z.2/-- Lower semicontinuous extension of `(A-B) log(A/B)` on the nonnegative quadrant.The off-quadrant value is immaterial for nonnegative densities. -/def extendedEntropyProduction (A B : ℝ) : ℝ≥0∞ :=  if A = 0 ∧ B = 0 then 0  else if 0 < A ∧ 0 < B then ENNReal.ofReal ((A - B) * Real.log (A / B))  else ⊤def entropyDissipation (f : Density) : ℝ≥0∞ :=  (4 : ℝ≥0∞)⁻¹ * ∫⁻ p : Position × (Velocity × (Velocity × Sphere)),    ENNReal.ofReal (collisionKernel p.2.1 p.2.2.1 p.2.2.2) *      extendedEntropyProduction        (f (p.1, scatterLeft p.2.1 p.2.2.1 p.2.2.2) *          f (p.1, scatterRight p.2.1 p.2.2.1 p.2.2.2))        (f (p.1, p.2.1) * f (p.1, p.2.2.1))    ∂((volume : Measure Position).prod ((volume : Measure Velocity).prod      ((volume : Measure Velocity).prod sphereArea)))/-- The renormalizations of Definition 2.1, with the right derivative at zero. -/structure Renormalization where  toFun : ℝ → ℝ  regular : ContDiffOn ℝ 1 toFun (Ici 0)  at_zero : toFun 0 = 0  derivative_bound : ∃ C : ℝ, 0 ≤ C ∧ ∀ z ∈ Ici (0 : ℝ),    |derivWithin toFun (Ici 0) z| ≤ C / (1 + z)/-- A smooth test, periodic in the spatial variable and compact in time and velocity. -/structure KineticTest where  toFun : ℝ → Position → Velocity → ℝ  regular : ContDiff ℝ (⊤ : ℕ∞) (fun p : LiftedSpaceTimePhase =>    toFun p.1 (torusProjection p.2.1) p.2.2)  compact_time_velocity : ∃ R : ℝ, 0 < R ∧ ∀ t x v,    R < |t| ∨ R < ‖v‖ → toFun t x v = 0def KineticTest.transportDerivative (φ : KineticTest) (t : ℝ) (z : Phase) : ℝ :=  fderiv ℝ (fun p : LiftedSpaceTimePhase => φ.toFun p.1 (torusProjection p.2.1) p.2.2)    (t, torusRepresentative z.1, z.2) (1, z.2, 0)/-- Smooth local mass tests have no velocity cutoff, since their velocity dependence isintroduced only by the transport derivative. -/structure MassTest where  toFun : ℝ → Position → ℝ  regular : ContDiff ℝ (⊤ : ℕ∞) (fun p : ℝ × Velocity => toFun p.1 (torusProjection p.2))  compact_time : ∃ R : ℝ, 0 < R ∧ ∀ t x, R < |t| → toFun t x = 0def MassTest.transportDerivative (φ : MassTest) (t : ℝ) (z : Phase) : ℝ :=  fderiv ℝ (fun p : ℝ × Velocity => φ.toFun p.1 (torusProjection p.2))    (t, torusRepresentative z.1) (1, z.2)def relativeDensity (F : Evolution) (t : ℝ) (z : Phase) : ℝ := F t z / maxwellian z.2def renormalizedIntegrand (F : Evolution) (β : Renormalization) (φ : KineticTest)    (p : SpaceTimePhase) : ℝ :=  maxwellian p.2.2 * β.toFun (relativeDensity F p.1 p.2) *      φ.transportDerivative p.1 p.2 +    derivWithin β.toFun (Ici 0) (relativeDensity F p.1 p.2) *      collision (F p.1) p.2.1 p.2.2 * φ.toFun p.1 p.2.1 p.2.2def nonnegativeTimes : Set SpaceTimePhase := Ici (0 : ℝ) ×ˢ univdef boundedTimes (T : ℝ) : Set SpaceTimePhase := Icc 0 T ×ˢ univ/-- Full global admissibility, including both separately renormalized collision terms,local mass, total momentum, and the every-time energy and entropy inequalities. -/structure AdmissibleGlobal (f₀ : Density) (F : Evolution) : Prop where  datum : FiniteEntropyDatum f₀  spacetime_measurable : AEStronglyMeasurable (fun p : SpaceTimePhase => F p.1 p.2)    (volume.restrict nonnegativeTimes)  finite_each_time : ∀ t ∈ Ici (0 : ℝ), FiniteEntropyDatum (F t)  initial : F 0 =ᵐ[volume] f₀  weak_continuity : ∀ ψ : Phase → ℝ, Measurable ψ →    (∃ C : ℝ, ∀ z, |ψ z| ≤ C) →    ContinuousOn (fun t => ∫ z, F t z * ψ z) (Ici 0)  uniform_moment : ∀ T : ℝ, 0 ≤ T → ∃ C : ℝ, ∀ t ∈ Icc 0 T,    (∫ z, entropyMoment (F t) z) ≤ C  -- These finiteness conditions prevent the real Bochner integrals defining `gain` and  -- `loss` from silently taking the value zero at divergent collision fibers.  collision_fibers : ∀ T : ℝ, 0 ≤ T → ∀ᵐ p ∂volume.restrict (boundedTimes T),    Integrable (fun q : Velocity × Sphere =>      collisionKernel p.2.2 q.1 q.2 *        F p.1 (p.2.1, scatterLeft p.2.2 q.1 q.2) *        F p.1 (p.2.1, scatterRight p.2.2 q.1 q.2))      ((volume : Measure Velocity).prod sphereArea) ∧    Integrable (fun q : Velocity × Sphere =>      collisionKernel p.2.2 q.1 q.2 * F p.1 p.2 * F p.1 (p.2.1, q.1))      ((volume : Measure Velocity).prod sphereArea)  gain_integrable : ∀ T : ℝ, 0 ≤ T → IntegrableOn    (fun p : SpaceTimePhase => gain (F p.1) p.2.1 p.2.2 /      (1 + relativeDensity F p.1 p.2)) (boundedTimes T)  loss_integrable : ∀ T : ℝ, 0 ≤ T → IntegrableOn    (fun p : SpaceTimePhase => loss (F p.1) p.2.1 p.2.2 /      (1 + relativeDensity F p.1 p.2)) (boundedTimes T)  renormalized_integrable : ∀ β φ, IntegrableOn (renormalizedIntegrand F β φ)    nonnegativeTimes  renormalized_equation : ∀ β : Renormalization, ∀ φ : KineticTest,    (∫ p in nonnegativeTimes, renormalizedIntegrand F β φ p) +      (∫ z, maxwellian z.2 * β.toFun (f₀ z / maxwellian z.2) *        φ.toFun 0 z.1 z.2) = 0  local_mass_integrable : ∀ φ : MassTest, IntegrableOn    (fun p : SpaceTimePhase => F p.1 p.2 * φ.transportDerivative p.1 p.2)    nonnegativeTimes  local_mass : ∀ φ : MassTest,    (∫ p in nonnegativeTimes, F p.1 p.2 * φ.transportDerivative p.1 p.2) +      (∫ z, f₀ z * φ.toFun 0 z.1) = 0  momentum_conservation : ∀ t ∈ Ici (0 : ℝ), momentum (F t) = momentum f₀  energy_inequality : ∀ t ∈ Ici (0 : ℝ), energy (F t) ≤ energy f₀  entropy_inequality : ∀ t ∈ Ici (0 : ℝ),    ENNReal.ofReal (relativeEntropy (F t)) +      (∫⁻ s in Icc (0 : ℝ) t, entropyDissipation (F s)) ≤        ENNReal.ofReal (relativeEntropy f₀)/-- Additional regularity asserted on the common initial interval. The `eLpNorm` statementis strong `L¹` continuity and cannot be made vacuous by a nonintegrable Bochner integral. -/structure StrongEarly (T : ℝ) (F : Evolution) : Prop where  positive_time : 0 < T  strong_continuity : ∀ t ∈ Icc 0 T,    Tendsto (fun s => eLpNorm (fun z => F s z - F t z) 1 volume)      (𝓝[Icc 0 T] t) (𝓝 0)  gain_integrable : IntegrableOn (fun p : SpaceTimePhase => gain (F p.1) p.2.1 p.2.2)    (boundedTimes T)  loss_integrable : IntegrableOn (fun p : SpaceTimePhase => loss (F p.1) p.2.1 p.2.2)    (boundedTimes T)theorem nonuniqueness :    ∃ f₀ : Density, BoundedVelocitySupport f₀ ∧      ∃ (F G : Evolution) (T : ℝ), AdmissibleGlobal f₀ F ∧ AdmissibleGlobal f₀ G ∧        StrongEarly T F ∧ StrongEarly T G ∧        ∃ t ∈ Set.Icc 0 T, ¬(F t =ᵐ[MeasureTheory.volume] G t) :=  sorryend BoltzmannNonuniquenessendend 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.

Nonuniqueness for the periodic hard-sphere Boltzmann equation

Theorem

There is a nonnegative density F_0 on \T^3\times\R^3, with bounded velocity support and

\int_{\T^3\times\R^3} F_0(1+|v|^2+|\log F_0|)\dd x\dd v<\infty,

for which the hard-sphere Boltzmann equation has two distinct global admissible solutions, in the sense of Definition def:admissible, with initial datum F_0. On some common initial interval both solutions are strongly continuous in L^1, and their collision gains and losses are integrable in space-time-velocity.

Section 1, Introduction Read it in the paper TeX source, line 55
Show the TeX
There is a nonnegative density \(F_0\) on \(\T^3\times\R^3\), with bounded
velocity support and
\[
\int_{\T^3\times\R^3}
 F_0(1+|v|^2+|\log F_0|)\dd x\dd v<\infty,
\]
for which the hard-sphere Boltzmann equation has two distinct global
admissible solutions, in the sense of Definition~\ref{def:admissible},
with initial datum \(F_0\).
On some common initial interval both solutions are strongly continuous
in \(L^1\), and their collision gains and losses are integrable in
space-time-velocity.
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 363 link it.
Solution module
OAI.MathematicalPhysics.Boltzmann.Main
Allowed axioms
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
38 local vocabulary declarations
Lines of Lean, without comments or blank lines
143
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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

Plectis Atlas

Two periodic hard-sphere Boltzmann solutions with the same data

Family 363: Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

Candidate pairing

Theorem

There is a nonnegative density F_0 on \T^3\times\R^3, with bounded velocity support and

\int_{\T^3\times\R^3} F_0(1+|v|^2+|\log F_0|)\dd x\dd v<\infty,

for which the hard-sphere Boltzmann equation has two distinct global admissible solutions, in the sense of Definition def:admissible, with initial datum F_0. On some common initial interval both solutions are strongly continuous in L^1, and their collision gains and losses are integrable in space-time-velocity.

Nonuniqueness for the periodic hard-sphere Boltzmann equation

TeX label thm:main, build/sections/01-introduction.tex, line 55

The Lean statement says

theorem nonuniqueness :    ∃ f₀ : Density, BoundedVelocitySupport f₀ ∧      ∃ (F G : Evolution) (T : ℝ), AdmissibleGlobal f₀ F ∧ AdmissibleGlobal f₀ G ∧        StrongEarly T F ∧ StrongEarly T G ∧        ∃ t ∈ Set.Icc 0 T, ¬(F t =ᵐ[MeasureTheory.volume] G t) :=  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.

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

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

Recorded comparisons with the paper

Atlas records no contributed comparison of this statement with the paper.

Contributed Comparator run reports

Atlas records no contributed report of running Comparator for this statement file.

Questions to resolve when comparing this statement with the paper

  • Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.

Compare this statement with the paperShare a Comparator run reportSuggest a change