Lean statement

Finite cylinder approximation for the selected ruled families

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.RuledApproximation.fullMain

Lean statement file: RuledCovering.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 100 link it.

See the closest candidates 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 FullMain : Prop :=  HyperbolicStatement ∧ PhysicalHyperbolicStatement ∧ LogarithmicStatement ∧  ProductCellStatement ∧ OrthogonalGridStatement ∧ NilpotentStatement ∧ SingularStatement⋮theorem fullMain : FullMain := 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.RuledApproximation.Plane line 10
```
abbrev Plane := EuclideanSpace ℝ (Fin 2)
```
abbreviation OAI.RuledApproximation.Space line 11
```
abbrev Space := EuclideanSpace ℝ (Fin 3)
```

definition OAI.RuledApproximation.vec2 line 13

def vec2 (x y : ℝ) : Plane := WithLp.toLp 2 ![x, y]

definition OAI.RuledApproximation.vec3 line 14

def vec3 (x y z : ℝ) : Space := WithLp.toLp 2 ![x, y, z]

definition OAI.RuledApproximation.axial line 15

def axial (s : ℝ) (p : Plane) : Space := vec3 s (p 0) (p 1)

definition OAI.RuledApproximation.IsParallelogram line 17

def IsParallelogram {E : Type*} [AddCommGroup E] [Module ℝ E] (P : Set E) : Prop :=  ∃ c u v : E, LinearIndependent ℝ ![u, v] ∧    P = {x | ∃ s ∈ Icc (0 : ℝ) 1, ∃ t ∈ Icc (0 : ℝ) 1,      x = c + s • u + t • v}

definition OAI.RuledApproximation.IsTriangle line 22

def IsTriangle {E : Type*} [AddCommGroup E] [Module ℝ E] (P : Set E) : Prop :=  ∃ c u v : E, LinearIndependent ℝ ![u, v] ∧    P = {x | ∃ s : ℝ, ∃ t : ℝ, 0 ≤ s ∧ 0 ≤ t ∧ s + t ≤ 1 ∧      x = c + s • u + t • v}

structure OAI.RuledApproximation.Cylinder line 27

structure Cylinder where  axis : Space  unit_axis : ‖axis‖ = 1  base : Set ((Submodule.span ℝ ({axis} : Set Space))ᗮ)

definition OAI.RuledApproximation.Cylinder.carrier line 34

def carrier (C : Cylinder) : Set Space :=  {x | ∃ b ∈ C.base, ∃ t : ℝ, x = (b : Space) + t • C.axis}

definition OAI.RuledApproximation.Cylinder.area line 37

def area (C : Cylinder) : ℝ := volume.real C.base

definition OAI.RuledApproximation.Cylinder.parallelogram line 39

def parallelogram (C : Cylinder) : Prop := IsParallelogram C.base

definition OAI.RuledApproximation.Cylinder.triangle line 41

def triangle (C : Cylinder) : Prop := IsTriangle C.base

definition OAI.RuledApproximation.Cylinder.openBounded line 43

def openBounded (C : Cylinder) : Prop :=  IsOpen C.base ∧ Bornology.IsBounded C.base ∧ MeasurableSet C.base

definition OAI.RuledApproximation.familySet line 47

def familySet {n : ℕ} (C : Fin n → Cylinder) : Set Space := ⋃ i, (C i).carrier

definition OAI.RuledApproximation.familyArea line 48

def familyArea {n : ℕ} (C : Fin n → Cylinder) : ℝ := ∑ i, (C i).area

definition OAI.RuledApproximation.HasFiniteCover line 50

def HasFiniteCover (shape : Cylinder → Prop) (S : Set Space) (A : ℝ) : Prop :=  ∃ n : ℕ, ∃ C : Fin n → Cylinder,    (∀ i, shape (C i)) ∧ S ⊆ familySet C ∧ familyArea C ≤ A

definition OAI.RuledApproximation.SmoothNear line 55

def SmoothNear (n : WithTop ℕ∞) (D : Set Plane) (v : Plane → Plane) : Prop :=  ∃ U : Set Plane, IsOpen U ∧ D ⊆ U ∧ ContDiffOn ℝ n v U

definition OAI.RuledApproximation.HyperbolicAt line 59

def HyperbolicAt (L : ℝ) (v : Plane → Plane) (p : Plane) : Prop :=  ∃ lam : ℝ, 0 < L * lam ∧ L * lam < 1 ∧    Module.End.HasEigenvalue (fderiv ℝ v p).toLinearMap lam ∧    Module.End.HasEigenvalue (fderiv ℝ v p).toLinearMap (-lam)

definition OAI.RuledApproximation.weight line 64

def weight (v : Plane) : ℝ := 1 / Real.sqrt (1 + ‖v‖ ^ 2)

definition OAI.RuledApproximation.physicalWeight line 65

def physicalWeight (h : ℝ) (w : Plane) : ℝ :=  1 / Real.sqrt (1 + w 0 ^ 2 + h ^ 2 * w 1 ^ 2)

definition OAI.RuledApproximation.ruled line 68

def ruled (D : Set Plane) (v : Plane → Plane) (L : ℝ) : Set Space :=  {z | ∃ p ∈ D, ∃ r : ℝ, |r| ≤ L ∧    z = vec3 (p 0 + r * v p 0) (p 1 + r * v p 1) r}

definition OAI.RuledApproximation.physicalRuled line 72

def physicalRuled (h : ℝ) (D : Set Plane) (v : Plane → Plane) (L : ℝ) : Set Space :=  {z | ∃ p ∈ D, ∃ s : ℝ, |s| ≤ L ∧    z = vec3 s (p 0 + s * v p 0) (h * (p 1 + s * v p 1))}

definition OAI.RuledApproximation.HyperbolicStatement line 76

def HyperbolicStatement : Prop :=  ∀ (D : Set Plane) (v : Plane → Plane) (L : ℝ),    IsCompact D → SmoothNear 1 D v → 0 < L →    (∀ p ∈ D, HyperbolicAt L v p) →    ∀ e : ℝ, 0 < e → HasFiniteCover Cylinder.parallelogram (ruled D v L)      ((∫ p in D, weight (v p)) + e)

definition OAI.RuledApproximation.PhysicalHyperbolicStatement line 83

def PhysicalHyperbolicStatement : Prop :=  ∀ (h L : ℝ) (D : Set Plane) (v : Plane → Plane),    0 < h → 0 < L → IsCompact D → SmoothNear 1 D v →    (∀ p ∈ D, HyperbolicAt L v p) →    ∀ (e : ℝ), 0 < e → ∀ F : Space ≃ᵢ Space,    HasFiniteCover Cylinder.parallelogram (F '' physicalRuled h D v L)      (h * (∫ p in D, physicalWeight h (v p)) + e)

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

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Statements in Finite cylinder approximation of ruled sets come first because the release’s name for this Lean statement shares words with that paper’s title.

Other possible paper matches, closest wording first

Corollary 4.4 of Finite cylinder approximation of ruled sets. Words of the Lean statement, its names split into words, that also appear in the theorem: coordinates, logarithmic, rectangle, det, cells, real and ten more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: approximation and perpendicular.
Theorem 4.1 of Finite cylinder approximation of ruled sets. Words of the Lean statement, its names split into words, that also appear in the theorem: product, eigenvalues, neighborhood, cells, real, measurable and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: approximation and perpendicular.
Theorem 5.1 of Slope-field perturbations of the two-cylinder covering. Words of the Lean statement, its names split into words, that also appear in the theorem: asymptotic, gamma, kappa, regular, edge, two and 13 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: needed, errors, positive, approximations and perpendicular.
Theorem 1.1 of Slope-field perturbations of the two-cylinder covering. Words of the Lean statement, its names split into words, that also appear in the theorem: regular, edge, measurable, parallelogram, bounded, tetrahedron and five more. Found in the paper’s title, which its main theorem states: field and two. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive and perpendicular.
Theorem 1.2 of Finite cylinder approximation of ruled sets. Words of the Lean statement, its names split into words, that also appear in the theorem: square, det, neighborhood, parallelograms, finite, area and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: condition, approximation and perpendicular.

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

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 471 s; peak memory 6.4 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 100’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: None for the headline; the aggregate file additionally states ruled-set and radial-sweep approximation results.
Definitions the group asks readers to compare with the paper: CylinderCovering.lean contains ten run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {} lines (environment-resetting metaprogramming inside a challenge file; a memory/performance hack that does not change statements)
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 formalization approximates compact ruled families of line segments by finite cylinder covers with perpendicular-base area at most the integral projection cost plus any prescribed positive error. The selected results cover the stated $C^1$ hyperbolic differential condition, square-zero and nilpotent cases, physical rescalings, logarithmic and product-cell estimates, and singular cases. Compact label sets need not have regular boundary.”
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

222 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: 171 to its own declarations, 116 to Mathlib’s documentation. 47 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 OAInoncomputable sectionopen Set MeasureTheory Filteropen scoped Topology ContDiffnamespace RuledApproximationabbrev Plane := EuclideanSpace ℝ (Fin 2)abbrev Space := EuclideanSpace ℝ (Fin 3)def vec2 (x y : ℝ) : Plane := WithLp.toLp 2 ![x, y]def vec3 (x y z : ℝ) : Space := WithLp.toLp 2 ![x, y, z]def axial (s : ℝ) (p : Plane) : Space := vec3 s (p 0) (p 1)def IsParallelogram {E : Type*} [AddCommGroup E] [Module ℝ E] (P : Set E) : Prop :=  ∃ c u v : E, LinearIndependent ℝ ![u, v] ∧    P = {x | ∃ s ∈ Icc (0 : ℝ) 1, ∃ t ∈ Icc (0 : ℝ) 1,      x = c + s • u + t • v}def IsTriangle {E : Type*} [AddCommGroup E] [Module ℝ E] (P : Set E) : Prop :=  ∃ c u v : E, LinearIndependent ℝ ![u, v] ∧    P = {x | ∃ s : ℝ, ∃ t : ℝ, 0 ≤ s ∧ 0 ≤ t ∧ s + t ≤ 1 ∧      x = c + s • u + t • v}structure Cylinder where  axis : Space  unit_axis : ‖axis‖ = 1  base : Set ((Submodule.span ℝ ({axis} : Set Space))ᗮ)namespace Cylinderdef carrier (C : Cylinder) : Set Space :=  {x | ∃ b ∈ C.base, ∃ t : ℝ, x = (b : Space) + t • C.axis}def area (C : Cylinder) : ℝ := volume.real C.basedef parallelogram (C : Cylinder) : Prop := IsParallelogram C.basedef triangle (C : Cylinder) : Prop := IsTriangle C.basedef openBounded (C : Cylinder) : Prop :=  IsOpen C.base ∧ Bornology.IsBounded C.base ∧ MeasurableSet C.baseend Cylinderdef familySet {n : ℕ} (C : Fin n → Cylinder) : Set Space := ⋃ i, (C i).carrierdef familyArea {n : ℕ} (C : Fin n → Cylinder) : ℝ := ∑ i, (C i).areadef HasFiniteCover (shape : Cylinder → Prop) (S : Set Space) (A : ℝ) : Prop :=  ∃ n : ℕ, ∃ C : Fin n → Cylinder,    (∀ i, shape (C i)) ∧ S ⊆ familySet C ∧ familyArea C ≤ A/-- No boundary hypothesis is included in this neighborhood regularity predicate. -/def SmoothNear (n : WithTop ℕ∞) (D : Set Plane) (v : Plane → Plane) : Prop :=  ∃ U : Set Plane, IsOpen U ∧ D ⊆ U ∧ ContDiffOn ℝ n v U/-- In dimension two these two distinct nonzero eigenvalues are all the eigenvalues. -/def HyperbolicAt (L : ℝ) (v : Plane → Plane) (p : Plane) : Prop :=  ∃ lam : ℝ, 0 < L * lam ∧ L * lam < 1 ∧    Module.End.HasEigenvalue (fderiv ℝ v p).toLinearMap lam ∧    Module.End.HasEigenvalue (fderiv ℝ v p).toLinearMap (-lam)def weight (v : Plane) : ℝ := 1 / Real.sqrt (1 + ‖v‖ ^ 2)def physicalWeight (h : ℝ) (w : Plane) : ℝ :=  1 / Real.sqrt (1 + w 0 ^ 2 + h ^ 2 * w 1 ^ 2)def ruled (D : Set Plane) (v : Plane → Plane) (L : ℝ) : Set Space :=  {z | ∃ p ∈ D, ∃ r : ℝ, |r| ≤ L ∧    z = vec3 (p 0 + r * v p 0) (p 1 + r * v p 1) r}def physicalRuled (h : ℝ) (D : Set Plane) (v : Plane → Plane) (L : ℝ) : Set Space :=  {z | ∃ p ∈ D, ∃ s : ℝ, |s| ≤ L ∧    z = vec3 s (p 0 + s * v p 0) (h * (p 1 + s * v p 1))}def HyperbolicStatement : Prop :=  ∀ (D : Set Plane) (v : Plane → Plane) (L : ℝ),    IsCompact D → SmoothNear 1 D v → 0 < L →    (∀ p ∈ D, HyperbolicAt L v p) →    ∀ e : ℝ, 0 < e → HasFiniteCover Cylinder.parallelogram (ruled D v L)      ((∫ p in D, weight (v p)) + e)def PhysicalHyperbolicStatement : Prop :=  ∀ (h L : ℝ) (D : Set Plane) (v : Plane → Plane),    0 < h → 0 < L → IsCompact D → SmoothNear 1 D v →    (∀ p ∈ D, HyperbolicAt L v p) →    ∀ (e : ℝ), 0 < e → ∀ F : Space ≃ᵢ Space,    HasFiniteCover Cylinder.parallelogram (F '' physicalRuled h D v L)      (h * (∫ p in D, physicalWeight h (v p)) + e)def OpenRectangleC2 (D : Set Plane) (v : Plane → Plane) : Prop :=  ∃ a b : Plane, (∀ i, a i < b i) ∧    D ⊆ {p | ∀ i, p i ∈ Ioo (a i) (b i)} ∧    ContDiffOn ℝ 2 v {p | ∀ i, p i ∈ Ioo (a i) (b i)}def LogarithmicStatement : Prop :=  ∀ (X k : ℝ) (D : Set Plane) (v : Plane → Plane),    0 < X → 0 < k → IsCompact D → volume (frontier D) = 0 →    OpenRectangleC2 D v →    (∀ p ∈ D, LinearMap.trace ℝ Plane (fderiv ℝ v p).toLinearMap = 0 ∧      LinearMap.det (fderiv ℝ v p).toLinearMap < 0 ∧      k * X * Real.sqrt (-LinearMap.det (fderiv ℝ v p).toLinearMap) < 1) →    ∀ e : ℝ, 0 < e → HasFiniteCover Cylinder.openBounded      (physicalRuled 1 D (fun p => k • v p) X)      ((∫ p in D, 1 / Real.sqrt (1 + k ^ 2 * ‖v p‖ ^ 2)) + e)def ProductCellStatement : Prop :=  ∀ (h M : ℝ) (D : Set Plane) (v : Plane → Plane),    0 < h → 0 < M → IsCompact D → volume (frontier D) = 0 →    SmoothNear 1 D v → (∀ p ∈ D, HyperbolicAt M v p) →    ∀ e : ℝ, 0 < e → HasFiniteCover Cylinder.openBounded (physicalRuled h D v M)      (h * (∫ p in D, physicalWeight h (v p)) + e)def OrthogonalGridStatement : Prop :=  ∀ (D : Set Plane) (v : Plane → Plane) (kappa : ℝ),    IsCompact D → volume (frontier D) = 0 → SmoothNear 1 D v → kappa < 1 →    (∀ p ∈ D, LinearMap.trace ℝ Plane (fderiv ℝ v p).toLinearMap = 0 ∧      LinearMap.det (fderiv ℝ v p).toLinearMap < 0 ∧ ‖fderiv ℝ v p‖ ≤ kappa) →    ∀ e : ℝ, 0 < e → HasFiniteCover Cylinder.parallelogram      (physicalRuled (Real.sqrt 2) D v 1)      (Real.sqrt 2 * (∫ p in D,        1 / Real.sqrt (1 + v p 0 ^ 2 + 2 * v p 1 ^ 2)) + e)def unitSquare : Set Plane := {x | ∀ i, |x i| ≤ (1 : ℝ) / 2}def IsSquareTile (T : Set Plane) : Prop :=  ∃ c : Plane, ∃ R : Plane ≃ₗᵢ[ℝ] Plane, ∃ a : ℝ, 0 < a ∧    T = (fun x => c + a • R x) '' unitSquaredef auxiliaryCylinder (T : Set Plane) (g : Plane) : Set Space :=  {z | ∃ s : ℝ, ∃ b ∈ T, z = axial s (b + s • g)}def firstPhysicalMap (h : ℝ) (z : Space) : Space := vec3 (z 0) (z 1) (h * z 2)def secondPhysicalMap (h a : ℝ) (z : Space) : Space :=  vec3 (z 1) (z 0) (h * (a - z 2))def NilpotentStatement : Prop :=  ∀ (D : Set Plane) (G : Plane → Plane), IsCompact D → SmoothNear ∞ D G →    (∀ b ∈ D, (fderiv ℝ G b).comp (fderiv ℝ G b) = 0) →    ∀ h eps : ℝ, 0 < h → 0 < eps →    ∃ n : ℕ, ∃ T : Fin n → Set Plane, ∃ g : Fin n → Plane,      (∀ i, IsSquareTile (T i)) ∧      physicalRuled 1 D G 1 ⊆ ⋃ i, auxiliaryCylinder (T i) (g i) ∧      (∑ i, volume.real (T i) * physicalWeight h (g i)) ≤        (∫ b in D, physicalWeight h (G b)) + eps ∧      ∀ a : ℝ, ∃ C₁ C₂ : Fin n → Cylinder,        (∀ i, (C₁ i).parallelogram ∧ (C₂ i).parallelogram ∧          (C₁ i).carrier = firstPhysicalMap h '' auxiliaryCylinder (T i) (g i) ∧          (C₂ i).carrier = secondPhysicalMap h a '' auxiliaryCylinder (T i) (g i) ∧          (C₁ i).area = h * volume.real (T i) * physicalWeight h (g i) ∧          (C₂ i).area = h * volume.real (T i) * physicalWeight h (g i))/-- The regular edge-2 tetrahedron in physical coordinates. -/def tetrahedron : Set Space :=  {z | ∃ t : ℝ, 0 ≤ t ∧ t ≤ 1 ∧ |z 0| ≤ 1 - t ∧ |z 1| ≤ t ∧    z 2 = Real.sqrt 2 * t}def singularField (p : ℝ) (b : Plane) : Plane :=  vec2 (((1 / 2 : ℝ) / 2) * (b 1 / (1 / 2)) ^ (p - 1) *    (p + (2 - p) * (b 0 / b 1) ^ 2))    ((b 1 / (1 / 2)) ^ (p - 2) * b 0)def ceilingPolynomial (p b : ℝ) : ℝ :=  b ^ 2 * ((1 / 2 : ℝ) * p / 2 + (2 - p) / (2 * (1 / 2)) * b ^ 2)def singularDomain (p gamma c : ℝ) : Set Plane :=  {b | c ^ gamma ≤ b 1 ∧ b 1 ≤ 2 ∧ |b 0| ≤ b 1 ∧    b 1 ≤ (1 / 2 : ℝ) + c ^ 2 * ceilingPolynomial p (b 0) + c ^ (5 / 2 : ℝ)}def singularLower (p gamma c : ℝ) : Set Space :=  physicalRuled (Real.sqrt 2) (singularDomain p gamma c)    (fun b => c • singularField p b) 1def singularUpper (p gamma c : ℝ) : Set Space :=  {z | ∃ b ∈ singularDomain p gamma c, ∃ y : ℝ, |y| ≤ 1 ∧    z = vec3 (b 0 - c * y * singularField p b 0) y      (Real.sqrt 2 * (1 - (b 1 - c * y * singularField p b 1)))}def lowerTip (d : ℝ) : Set Space :=  {z | ∃ t : ℝ, 0 ≤ t ∧ t ≤ 2 * d ∧ |z 1| ≤ t ∧ z 2 = Real.sqrt 2 * t}def upperTip (d : ℝ) : Set Space :=  {z | ∃ s : ℝ, 0 ≤ s ∧ s ≤ 2 * d ∧ |z 0| ≤ s ∧ z 2 = Real.sqrt 2 * (1 - s)}def singularCost (p gamma c : ℝ) : ℝ :=  Real.sqrt 2 * ∫ b in singularDomain p gamma c,    physicalWeight (Real.sqrt 2) (c • singularField p b)def singularCoefficient (p : ℝ) : ℝ :=  (1 / 2 : ℝ) ^ 4 * (2 * p ^ 2 + 10 * p - 13) / (30 * p)def SingularStatement : Prop :=  ∀ p gamma : ℝ, 1 < p → p < (Real.sqrt 51 - 5) / 2 →    1 < gamma → gamma < 1 / (2 - p) →    singularCoefficient p < 0 ∧    Asymptotics.IsLittleO (𝓝[>] (0 : ℝ))      (fun c => 2 * singularCost p gamma c + 8 * Real.sqrt 2 * (c ^ gamma) ^ 2 -        (Real.sqrt 2 / 2 + 2 * Real.sqrt 2 * singularCoefficient p * c ^ 2))      (fun c => c ^ 2) ∧    ∀ᶠ c : ℝ in 𝓝[>] (0 : ℝ),      ∃ n m : ℕ, ∃ C : Fin n → Cylinder, ∃ D : Fin m → Cylinder,      ∃ B₁ B₂ : Cylinder,        (∀ i, (C i).parallelogram) ∧ (∀ j, (D j).parallelogram) ∧        B₁.triangle ∧ B₂.triangle ∧        singularLower p gamma c ⊆ familySet C ∧        singularUpper p gamma c ⊆ familySet D ∧        B₁.carrier = lowerTip (c ^ gamma) ∧ B₂.carrier = upperTip (c ^ gamma) ∧        tetrahedron ⊆ familySet C ∪ familySet D ∪ B₁.carrier ∪ B₂.carrier ∧        familyArea C + familyArea D + B₁.area + B₂.area < Real.sqrt 2 / 2def FullMain : Prop :=  HyperbolicStatement ∧ PhysicalHyperbolicStatement ∧ LogarithmicStatement ∧  ProductCellStatement ∧ OrthogonalGridStatement ∧ NilpotentStatement ∧ SingularStatementtheorem fullMain : FullMain := by  sorryend RuledApproximationendend 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.

Finite angular cylinder covers below the half-area bound

Theorem

The tetrahedron (eq:tetrahedron) has A_{\min}(K)=\sqrt2. For every 0<\eps\le1/2000, it admits a covering by m=2\lceil2/\eps^2\rceil cylinders with compact triangular perpendicular bases such that

\frac1{\sqrt2}\sum_{i=1}^m|B_i| =\frac12-\frac{13}{6000}\eps^2+O(\eps^4).

The remainder has absolute value at most 2\eps^4, and the total base area is strictly less than A_{\min}(K)/2.

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

Show the TeX

The tetrahedron \eqref{eq:tetrahedron} has $A_{\min}(K)=\sqrt2$.
For every $0<\eps\le1/2000$, it admits a covering by
$m=2\lceil2/\eps^2\rceil$ cylinders with compact triangular perpendicular
bases such that
\begin{equation}
 \frac1{\sqrt2}\sum_{i=1}^m|B_i|
 =\frac12-\frac{13}{6000}\eps^2+O(\eps^4).
\end{equation}
The remainder has absolute value at most $2\eps^4$, and the total base
area is strictly less than $A_{\min}(K)/2$.

Corollary

Every nondegenerate tetrahedron T\subset\R^3 admits a finite cylinder covering with compact triangular perpendicular bases for which

\sum_{i=1}^m\frac{|B_i|}{|\pi_{u_i^\perp}T|}<\frac12.

Thus the 1-Codimensional Cylinder Covering Conjecture is false.

Section 1, Introduction Read it in the paper TeX source, line 50

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Every nondegenerate tetrahedron $T\subset\R^3$ admits a finite cylinder
covering with compact triangular perpendicular bases for which
\[
 \sum_{i=1}^m\frac{|B_i|}{|\pi_{u_i^\perp}T|}<\frac12.
\]
Thus the 1-Codimensional Cylinder Covering Conjecture is false.

Finite cylinder approximation of ruled sets

Proposition (Perpendicular base area)

Let h>0, let T\subset\R^2 be bounded and measurable, and let g\in\R^2. The image F_h(\Cyl(T,g)) is a cylinder whose perpendicular base has area h|T|J_h(g). A compact nondegenerate parallelogram tile has a compact nondegenerate parallelogram base. An open tile gives an open base. The formula is unchanged after a Euclidean isometry.

Section 1, The finite approximation problem Read it in the paper TeX source, line 50

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Let $h>0$, let $T\subset\R^2$ be bounded and measurable, and let
$g\in\R^2$. The image $F_h(\Cyl(T,g))$ is a cylinder whose perpendicular
base has area $h|T|J_h(g)$. A compact nondegenerate parallelogram tile
has a compact nondegenerate parallelogram base. An open tile gives an
open base. The formula is unchanged after a Euclidean isometry.

Theorem (Finite approximation by square tiles)

Let D\subset\R^2 be compact, let V be C^1 on an open neighborhood of D, and let L,h>0. Suppose that

\tr DV(p)=0,\qquad -L^{-2}<\det DV(p)\le0 \quad(p\in D).

For every \varepsilon>0, there are finitely many compact nondegenerate square tiles T_i\subset\R^2, with possibly different orientations, and velocities g_i\in\R^2 such that

E(D,V,L)\subset\bigcup_i\Cyl(T_i,g_i),\qquad \sum_i |T_i|J_h(g_i) \le\int_D J_h(V(p))\,dp+\varepsilon.

Under F_h, the perpendicular bases are compact nondegenerate parallelograms and have total area h\sum_i|T_i|J_h(g_i). No condition on the area of \partial D or smallness of \|DV\| is required.

Section 1, The finite approximation problem Read it in the paper TeX source, line 74

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Let $D\subset\R^2$ be compact, let $V$ be $C^1$ on an open
neighborhood of $D$, and let $L,h>0$. Suppose that
\begin{equation}
 \tr DV(p)=0,\qquad -L^{-2}<\det DV(p)\le0
 \quad(p\in D).
\end{equation}
For every $\varepsilon>0$, there are finitely many compact
nondegenerate square tiles $T_i\subset\R^2$, with possibly different
orientations, and velocities $g_i\in\R^2$ such that
\begin{equation}
 E(D,V,L)\subset\bigcup_i\Cyl(T_i,g_i),\qquad
 \sum_i |T_i|J_h(g_i)
 \le\int_D J_h(V(p))\,dp+\varepsilon.
\end{equation}
Under $F_h$, the perpendicular bases are compact nondegenerate
parallelograms and have total area $h\sum_i|T_i|J_h(g_i)$.
No condition on the area of $\partial D$ or smallness of $\|DV\|$
is required.

Slope-field perturbations of the two-cylinder covering

Theorem

Every regular tetrahedron K\subset\R^3 of positive edge length admits a finite cylinder cover with compact parallelogram perpendicular bases such that

\sum_i|B_i|<\tfrac12 A_{\min}(K), \qquad \sum_i\frac{|B_i|}{|\pi_{u_i^\perp}K|}<\tfrac12.

It also admits finite covers with bounded open measurable bases satisfying both inequalities.

Section 1, Introduction Read it in the paper TeX source, line 51

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Every regular tetrahedron $K\subset\R^3$ of positive edge length admits a finite
cylinder cover with compact parallelogram perpendicular bases such that
\[
 \sum_i|B_i|<\tfrac12 A_{\min}(K),
 \qquad
 \sum_i\frac{|B_i|}{|\pi_{u_i^\perp}K|}<\tfrac12.
\]
It also admits finite covers with bounded open measurable bases
satisfying both inequalities.

Corollary

Every nondegenerate tetrahedron T\subset\R^3 admits a finite cylinder cover \mathcal C with compact parallelogram perpendicular bases and \mathcal R(\mathcal C;T)<1/2. The same conclusion holds with bounded open measurable bases.

Section 1, Introduction Read it in the paper TeX source, line 67

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Every nondegenerate tetrahedron $T\subset\R^3$ admits a finite cylinder
cover $\mathcal C$ with compact parallelogram perpendicular bases and
$\mathcal R(\mathcal C;T)<1/2$. The same conclusion holds with bounded
open measurable bases.

Finite triangular approximation of radial sweeps

Theorem

For the data in (eq:radial-sweep), suppose that

w'(t)=\lambda(t)(a+tb)\qquad(t\in J)

for a continuous real function \lambda. For any finite partition \alpha=t_0<t_1<\cdots<t_N=\beta, put I_i=[t_{i-1},t_i] and choose arbitrary tags c_i\in I_i. There are N cylinders covering \mathcal S, with axes parallel to e+w(c_i) and nondegenerate compact triangular perpendicular bases. These covers can be chosen so that their total base area tends to \mathcal A as the mesh \delta=\max_i|I_i| tends to zero, uniformly over the partitions and tags.

Alternatively, take each base to be the perpendicular projection of the actual compact sweep with t\in I_i, in the same frozen direction e+w(c_i). These N cylinders cover \mathcal S, and their total base area also tends to \mathcal A, uniformly over the partitions and tags.

Section 1, Radial sweeps and finite cylinders Read it in the paper TeX source, line 64

Show the TeX

For the data in \eqref{eq:radial-sweep}, suppose that
\[
 w'(t)=\lambda(t)(a+tb)\qquad(t\in J)
\]
for a continuous real function $\lambda$.
For any finite partition
\(\alpha=t_0<t_1<\cdots<t_N=\beta\), put \(I_i=[t_{i-1},t_i]\)
and choose arbitrary tags \(c_i\in I_i\). There are \(N\) cylinders
covering \(\mathcal S\), with axes parallel to \(e+w(c_i)\) and
nondegenerate compact triangular perpendicular bases. These covers
can be chosen so that their total base area tends to \(\mathcal A\)
as the mesh \(\delta=\max_i|I_i|\) tends to zero, uniformly over the
partitions and tags.

Alternatively, take each base to be the perpendicular projection of
the actual compact sweep with \(t\in I_i\), in the same frozen
direction \(e+w(c_i)\). These \(N\) cylinders cover \(\mathcal S\),
and their total base area also tends to \(\mathcal A\), uniformly
over the partitions and tags.

Corollary

Every regular tetrahedron K\subset\R^3 of positive edge length has a finite cylinder cover B_i+\R u_i, with unit axes u_i and nondegenerate compact triangular bases B_i\subset u_i^\perp, such that

\sum_i\area(B_i)<\frac{A_{\min}(K)}2, \qquad \sum_i\frac{\area(B_i)}{\area(\pi_{u_i^\perp}K)}<\frac12.

Section 1, Radial sweeps and finite cylinders Read it in the paper TeX source, line 95

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Every regular tetrahedron $K\subset\R^3$ of positive edge length has
a finite cylinder cover $B_i+\R u_i$, with unit axes $u_i$ and
nondegenerate compact triangular bases $B_i\subset u_i^\perp$, such that
\[
 \sum_i\area(B_i)<\frac{A_{\min}(K)}2,
 \qquad
 \sum_i\frac{\area(B_i)}{\area(\pi_{u_i^\perp}K)}<\frac12.
\]

Configuration and file facts

File: lean/ComparatorChallenges/RuledCovering.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 100 link it.
Solution module: OAI.Geometry.RuledCovering.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: 46 local vocabulary declarations
Lines of Lean, without comments or blank lines: 161
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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