Cylinder-covering counterexamples and companion approximations (Lean)

The Lean statement

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

theorem fourPaperMain :    AngularExplicitStatement ∧    RuledApproximation.HyperbolicStatement ∧    RuledApproximation.NilpotentC1Statement ∧    RegularParallelogramCounterexampleStatement ∧    RegularOpenCounterexampleStatement ∧    RadialSweep.AlignedStatement ∧    RadialSweep.ActualSweepStatement := 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.TriangularCovering.Space line 12
```
abbrev Space := EuclideanSpace ℝ (Fin 3)
```
definition OAI.TriangularCovering.H line 14
```
def H : ℝ := Real.sqrt 2
```

definition OAI.TriangularCovering.point line 16

def point (x y t : ℝ) : Space := !₂[x, y, H * t]

definition OAI.TriangularCovering.K line 18

def K : Set Space :=  {z | ∃ x y t : ℝ, 0 ≤ t ∧ t ≤ 1 ∧ |x| ≤ 1 - t ∧ |y| ≤ t ∧ z = point x y t}

definition OAI.TriangularCovering.projection line 21

def projection (u z : Space) : Space := z - (inner ℝ u z) • u

definition OAI.TriangularCovering.area line 23

def area (B : Set Space) : ℝ := (Measure.euclideanHausdorffMeasure 2 B).toReal

definition OAI.TriangularCovering.shadowArea line 25

def shadowArea (u : Space) : ℝ := area (projection u '' K)

definition OAI.TriangularCovering.A_min line 27

def A_min : ℝ := sInf {a : ℝ | ∃ u : Space, ‖u‖ = 1 ∧ a = shadowArea u}

structure OAI.TriangularCovering.TriCylinder line 29

structure TriCylinder where

Show the remaining 10 lines

  axis : Space  base : Set Space  unit_axis : ‖axis‖ = 1  perpendicular : ∀ z ∈ base, inner ℝ axis z = 0  triangular : ∃ v : Fin 3 → Space, AffineIndependent ℝ v ∧    base = convexHull ℝ (Set.range v)  compact : IsCompact base  convex : Convex ℝ base  measurable : MeasurableSet base  finite_area : Measure.euclideanHausdorffMeasure 2 base ≠ ⊤

definition OAI.TriangularCovering.TriCylinder.carrier line 41

def TriCylinder.carrier (c : TriCylinder) : Set Space :=  {z | ∃ b ∈ c.base, ∃ r : ℝ, z = b + r • c.axis}

definition OAI.TriangularCovering.Covers line 44

def Covers {m : ℕ} (c : Fin m → TriCylinder) : Prop :=  ∀ z ∈ K, ∃ i : Fin m, z ∈ (c i).carrier

definition OAI.TriangularCovering.totalArea line 47

def totalArea {m : ℕ} (c : Fin m → TriCylinder) : ℝ :=  ∑ i : Fin m, area (c i).base

definition OAI.TriangularCovering.relativeCost line 50

def relativeCost {m : ℕ} (c : Fin m → TriCylinder) : ℝ :=  ∑ i : Fin m, area (c i).base / shadowArea (c i).axis

definition OAI.TriangularCovering.n line 53

def n (ε : ℝ) : ℕ := ⌈2 / ε ^ 2⌉₊

definition OAI.TriangularCovering.η line 67
```
def η : ℝ := 1 / 1000
```

definition OAI.TriangularCovering.alpha line 69

def alpha (a b : ℝ) : ℝ := (1 + a * b) / 4

definition OAI.TriangularCovering.beta line 71

def beta (a b : ℝ) : ℝ := (a + b) / 4

definition OAI.TriangularCovering.d line 73

def d (q : ℝ) : ℝ := q ^ 2 * (1 + q ^ 2) / 16

definition OAI.TriangularCovering.cutoffCoeff line 75

def cutoffCoeff (a b : ℝ) : ℝ := η + max (d a) (d b)

definition OAI.TriangularCovering.step line 77
```
def step (N : ℕ) : ℝ := 2 / N
```

definition OAI.TriangularCovering.mesh line 79

def mesh (N i : ℕ) : ℝ := -1 + i * step N

definition OAI.TriangularCovering.cutoff line 81

def cutoff (ε a b : ℝ) : ℝ := 1 / 2 + ε ^ 2 * cutoffCoeff a b

definition OAI.TriangularCovering.speedSq line 83

def speedSq (a b : ℝ) : ℝ := (alpha a b)^2 + 2*(beta a b)^2

definition OAI.TriangularCovering.normalizedTerm line 85

def normalizedTerm (ε a b : ℝ) : ℝ :=  (b-a) * (cutoff ε a b)^2 / Real.sqrt (1+ε^2*speedSq a b)

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

Possible paper matches for the parts of the Lean statement

Atlas selected possible matches by comparing words in each part of the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

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

The Lean statement joins several assertions with “and”. Atlas selected possible matches to two paper results from their shared wording, listed below. This does not establish that their mathematical meanings agree.

Theorem 1.1 of Finite triangular approximation of radial sweeps

Words of the Lean statement, its names split into words, that also appear in the theorem: frozen, data, mesh, projection, direction, actual, continuous, max and 14 more. Found in the paper’s title, which its main theorem states: approximation. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nondegenerate.

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 1.2 of Finite triangular approximation of radial sweeps

Words of the Lean statement, its names split into words, that also appear in the theorem: unit, length, regular, edge, perp, min, triangular, tetrahedron and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nondegenerate.

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

Show the TeX

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

Other possible paper matches, closest wording first

Proposition B.1 of Finite triangular approximation of radial sweeps. Words of the Lean statement, its names split into words, that also appear in the theorem: four, tilt, costs, normalized, cap, sweeps and seven more.
Corollary 3.2 of Finite angular cylinder covers below the half-area bound. Words of the Lean statement, its names split into words, that also appear in the theorem: axis, triangles, directions, integer, max, alpha and nine more.
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: kappa, errors, regular, edge, two, families and 16 more.
Proposition 4.4 of Finite triangular approximation of radial sweeps. Words of the Lean statement, its names split into words, that also appear in the theorem: actual, normalized, cost, sqrt, min, triangular and ten more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nondegenerate.
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: length, regular, edge, measurable, perp, parallelogram and nine more. Found in the paper’s title, which its main theorem states: two.
Theorem 5.1 of Finite angular cylinder covers below the half-area bound. Words of the Lean statement, its names split into words, that also appear in the theorem: range, independent, triangle, error, continuous, caps and 13 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

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 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 linked aggregate includes the accompanying explicit angular covers and finite approximation theorems for ruled sets and radial sweeps. The affine extension of the directionwise conclusion to every nondegenerate tetrahedron is outside the selected regular-tetrahedron statements.”
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

573 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: 356 to its own declarations, 210 to Mathlib’s documentation. 109 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 OAIsectionopen Set MeasureTheoryopen scoped InnerProductSpace BigOperators Intervalnoncomputable sectionnamespace TriangularCoveringabbrev Space := EuclideanSpace ℝ (Fin 3)def H : ℝ := Real.sqrt 2def point (x y t : ℝ) : Space := !₂[x, y, H * t]def K : Set Space :=  {z | ∃ x y t : ℝ, 0 ≤ t ∧ t ≤ 1 ∧ |x| ≤ 1 - t ∧ |y| ≤ t ∧ z = point x y t}def projection (u z : Space) : Space := z - (inner ℝ u z) • udef area (B : Set Space) : ℝ := (Measure.euclideanHausdorffMeasure 2 B).toRealdef shadowArea (u : Space) : ℝ := area (projection u '' K)def A_min : ℝ := sInf {a : ℝ | ∃ u : Space, ‖u‖ = 1 ∧ a = shadowArea u}structure TriCylinder where  axis : Space  base : Set Space  unit_axis : ‖axis‖ = 1  perpendicular : ∀ z ∈ base, inner ℝ axis z = 0  triangular : ∃ v : Fin 3 → Space, AffineIndependent ℝ v ∧    base = convexHull ℝ (Set.range v)  compact : IsCompact base  convex : Convex ℝ base  measurable : MeasurableSet base  finite_area : Measure.euclideanHausdorffMeasure 2 base ≠ ⊤def TriCylinder.carrier (c : TriCylinder) : Set Space :=  {z | ∃ b ∈ c.base, ∃ r : ℝ, z = b + r • c.axis}def Covers {m : ℕ} (c : Fin m → TriCylinder) : Prop :=  ∀ z ∈ K, ∃ i : Fin m, z ∈ (c i).carrierdef totalArea {m : ℕ} (c : Fin m → TriCylinder) : ℝ :=  ∑ i : Fin m, area (c i).basedef relativeCost {m : ℕ} (c : Fin m → TriCylinder) : ℝ :=  ∑ i : Fin m, area (c i).base / shadowArea (c i).axisdef n (ε : ℝ) : ℕ := ⌈2 / ε ^ 2⌉₊end TriangularCoveringendendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}sectionopen Set MeasureTheoryopen scoped InnerProductSpace BigOperators Intervalnoncomputable sectionnamespace TriangularCoveringdef η : ℝ := 1 / 1000def alpha (a b : ℝ) : ℝ := (1 + a * b) / 4def beta (a b : ℝ) : ℝ := (a + b) / 4def d (q : ℝ) : ℝ := q ^ 2 * (1 + q ^ 2) / 16def cutoffCoeff (a b : ℝ) : ℝ := η + max (d a) (d b)def step (N : ℕ) : ℝ := 2 / Ndef mesh (N i : ℕ) : ℝ := -1 + i * step Ndef cutoff (ε a b : ℝ) : ℝ := 1 / 2 + ε ^ 2 * cutoffCoeff a bdef speedSq (a b : ℝ) : ℝ := (alpha a b)^2 + 2*(beta a b)^2def normalizedTerm (ε a b : ℝ) : ℝ :=  (b-a) * (cutoff ε a b)^2 / Real.sqrt (1+ε^2*speedSq a b)end TriangularCoveringendendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}sectionopen Set MeasureTheoryopen scoped InnerProductSpace BigOperators Intervalnoncomputable sectionnamespace TriangularCoveringdef unit (v : Space) : Space := ‖v‖⁻¹ • vend TriangularCoveringendendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}sectionopen Set MeasureTheoryopen scoped InnerProductSpace BigOperators Intervalnoncomputable sectionnamespace TriangularCoveringdef tiltV (ε A B : ℝ) : Space := !₂[1, ε*A, H*ε*B]def tiltW (ε A B : ℝ) : Space := !₂[-ε*A, 1, H*ε*B]def lowerVertices (a b T : ℝ) : Fin 3 → Space :=  ![point 0 0 0, point 0 (a*T) T, point 0 (b*T) T]def upperVertices (a b T : ℝ) : Fin 3 → Space :=  ![point 0 0 1, point (a*T) 0 (1-T), point (b*T) 0 (1-T)]end TriangularCoveringendendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}sectionopen Set MeasureTheoryopen scoped InnerProductSpace BigOperators Intervalnoncomputable sectionnamespace TriangularCovering.Planarabbrev Coord := Fin 2 → ℝdef standard : Set Coord := {x | 0 ≤ x 0 ∧ 0 ≤ x 1 ∧ x 0 + x 1 ≤ 1}def vertices : Fin 3 → Coord := ![0, ![1, 0], ![0, 1] ]end TriangularCovering.Planarnamespace TriangularCoveringsection Trianglesvariable {E F : Type*} [AddCommGroup E] [Module ℝ E] [AddCommGroup F] [Module ℝ F]def triangle (a b c : E) : Set E := convexHull ℝ (range (![a, b, c] : Fin 3 → E))end Trianglesend TriangularCoveringendendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}sectionopen Set MeasureTheoryopen scoped InnerProductSpacenoncomputable sectionnamespace TriangularCoveringdef lowerAngularIndex {N : ℕ} (j : Fin N) : Fin (2*N) :=  Fin.cast (two_mul N).symm (Fin.castAdd N j)def upperAngularIndex {N : ℕ} (j : Fin N) : Fin (2*N) :=  Fin.cast (two_mul N).symm (Fin.natAdd N j)def lowerAngularAxis (ε : ℝ) (j : ℕ) : Space :=  let a := mesh (n ε) j  let b := mesh (n ε) (j+1)  unit (tiltV ε (alpha a b) (beta a b))def upperAngularAxis (ε : ℝ) (j : ℕ) : Space :=  let a := mesh (n ε) j  let b := mesh (n ε) (j+1)  unit (tiltW ε (alpha a b) (beta a b))def lowerAngularBase (ε : ℝ) (j : ℕ) : Set Space :=  let a := mesh (n ε) j  let b := mesh (n ε) (j+1)  let v := lowerVertices a b (cutoff ε a b)  projection (lowerAngularAxis ε j) '' triangle (v 0) (v 1) (v 2)def upperAngularBase (ε : ℝ) (j : ℕ) : Set Space :=  let a := mesh (n ε) j  let b := mesh (n ε) (j+1)  let v := upperVertices a b (cutoff ε a b)  projection (upperAngularAxis ε j) '' triangle (v 0) (v 1) (v 2)def ExplicitAngularFamily (ε : ℝ) (c : Fin (2*n ε) → TriCylinder) : Prop :=  ∀ j : Fin (n ε),    (c (lowerAngularIndex j)).axis = lowerAngularAxis ε j ∧    (c (lowerAngularIndex j)).base = lowerAngularBase ε j ∧    (c (upperAngularIndex j)).axis = upperAngularAxis ε j ∧    (c (upperAngularIndex j)).base = upperAngularBase ε jdef AngularAreaStatement : Prop :=  ∀ ε : ℝ, 0 < ε → ε ≤ 1 →    ∃ c : Fin (2*n ε) → TriCylinder,      ExplicitAngularFamily ε c ∧      totalArea c/H = ∑ j : Fin (n ε),        normalizedTerm ε (mesh (n ε) j) (mesh (n ε) (j+1)) ∧      |totalArea c/H - (1/2-(13/6000:ℝ)*ε^2)| ≤ 2*ε^4def ExplicitAngularStatement : Prop :=  A_min = H ∧  ∀ ε : ℝ, 0 < ε → ε ≤ 1/2000 →    ∃ c : Fin (2*n ε) → TriCylinder,      ExplicitAngularFamily ε c ∧ Covers c ∧      |totalArea c/H - (1/2-(13/6000:ℝ)*ε^2)| ≤ 2*ε^4 ∧      totalArea c < A_min/2 ∧ relativeCost c < 1/2end TriangularCoveringendendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}sectionnoncomputable sectionopen Set MeasureTheory Filter Metricopen scoped Topology ContDiff Pointwisenamespace 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}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 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 ≤ Adef SmoothNear (n : WithTop ℕ∞) (D : Set Plane) (v : Plane → Plane) : Prop :=  ∃ U : Set Plane, IsOpen U ∧ D ⊆ U ∧ ContDiffOn ℝ n v Udef 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 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))end RuledApproximationendendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}sectionnoncomputable sectionopen Set MeasureTheoryopen scoped InnerProductSpacenamespace CylinderCoveringabbrev Space := EuclideanSpace ℝ (Fin 3)def IsRegularTetrahedron (K : Set Space) : Prop :=  ∃ v : Fin 4 → Space, AffineIndependent ℝ v ∧    ∃ edge : ℝ, 0 < edge ∧      (∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = edge) ∧      K = convexHull ℝ (range v)def shadowArea (K : Set Space) (u : Space) : ℝ :=  TriangularCovering.area (TriangularCovering.projection u '' K)def minimumShadowArea (K : Set Space) : ℝ :=  sInf {a : ℝ | ∃ u : Space, ‖u‖ = 1 ∧ a = shadowArea K u}def triangularCovers {n : ℕ} (K : Set Space)    (C : Fin n → TriangularCovering.TriCylinder) : Prop :=  ∀ x ∈ K, ∃ i : Fin n, x ∈ (C i).carrierdef triangularRelativeCost {n : ℕ} (K : Set Space)    (C : Fin n → TriangularCovering.TriCylinder) : ℝ :=  ∑ i, TriangularCovering.area (C i).base / shadowArea K (C i).axisdef relativeCost {n : ℕ} (K : Set Space)    (C : Fin n → RuledApproximation.Cylinder) : ℝ :=  ∑ i, (C i).area / shadowArea K (C i).axisdef AngularExplicitStatement : Prop :=  TriangularCovering.AngularAreaStatement ∧ TriangularCovering.ExplicitAngularStatementdef RegularTriangularCounterexampleStatement : Prop :=  ∀ K : Set Space, IsRegularTetrahedron K →    ∃ n : ℕ, ∃ C : Fin n → TriangularCovering.TriCylinder,      triangularCovers K C ∧      TriangularCovering.totalArea C < minimumShadowArea K / 2 ∧      triangularRelativeCost K C < 1 / 2def RegularParallelogramCounterexampleStatement : Prop :=  ∀ K : Set Space, IsRegularTetrahedron K →    ∃ n : ℕ, ∃ C : Fin n → RuledApproximation.Cylinder,      (∀ i, (C i).parallelogram) ∧      K ⊆ RuledApproximation.familySet C ∧      RuledApproximation.familyArea C < minimumShadowArea K / 2 ∧      relativeCost K C < 1 / 2def RegularOpenCounterexampleStatement : Prop :=  ∀ K : Set Space, IsRegularTetrahedron K →    ∃ n : ℕ, ∃ C : Fin n → RuledApproximation.Cylinder,      (∀ i, (C i).openBounded) ∧      K ⊆ RuledApproximation.familySet C ∧      RuledApproximation.familyArea C < minimumShadowArea K / 2 ∧      relativeCost K C < 1 / 2end CylinderCoveringendendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}sectionnoncomputable sectionopen Set MeasureTheory Filteropen scoped Topology ContDiffnamespace RuledApproximationdef NilpotentC1Statement : Prop :=  ∀ (D : Set Plane) (G : Plane → Plane), IsCompact D → SmoothNear 1 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))end RuledApproximationendendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}sectionnoncomputable sectionnamespace CylinderCovering.RadialRiemannopen Setdef step (a b : ℝ) (N : ℕ) : ℝ := (b - a) / Ndef point (a b : ℝ) (N i : ℕ) : ℝ := a + i * step a b Ndef cell (a b : ℝ) (N : ℕ) (i : Fin N) : Set ℝ :=  Icc (point a b N i) (point a b N (i + 1))def IsTagged (a b : ℝ) (N : ℕ) (tag : Fin N → ℝ) : Prop :=  ∀ i, tag i ∈ cell a b N idef capMax (R : ℝ → ℝ) (a b : ℝ) (N : ℕ) (i : Fin N) : ℝ :=  sSup (R '' cell a b N i)end CylinderCovering.RadialRiemannendendrun_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}sectionopen Set MeasureTheory Filteropen scoped InnerProductSpace Topologynoncomputable sectionnamespace CylinderCovering.RadialSweepabbrev Space := TriangularCovering.Spacestructure Data where  e : Space  a : Space  b : Space  p₀ : Space  α : ℝ  β : ℝ  L : ℝ  R : ℝ → ℝ  lam : ℝ → ℝ  w : ℝ → Space  interval_nonempty : α < β  length_pos : 0 < L  normal_unit : ‖e‖ = 1  a_perp : inner ℝ e a = 0  b_perp : inner ℝ e b = 0  origin_perp : inner ℝ e p₀ = 0  independent : LinearIndependent ℝ ![a, b]  cap_continuous : ContinuousOn R (Icc α β)  cap_nonneg : ∀ t ∈ Icc α β, 0 ≤ R t  scalar_continuous : ContinuousOn lam (Icc α β)  velocity_continuous : ContinuousOn w (Icc α β)  velocity_perp : ∀ t ∈ Icc α β, inner ℝ e (w t) = 0  aligned : ∀ t ∈ Icc α β,    HasDerivWithinAt w (lam t • (a + t • b)) (Icc α β) tdef sweep (D : Data) : Set Space :=  {z | ∃ t ∈ Icc D.α D.β, ∃ ρ ∈ Icc (0 : ℝ) (D.R t),    ∃ s : ℝ, |s| ≤ D.L ∧      z = D.p₀ + ρ • (D.a + t • D.b) + s • (D.e + D.w t)}def kappa (D : Data) : ℝ :=  Real.sqrt (inner ℝ D.a D.a * inner ℝ D.b D.b - (inner ℝ D.a D.b) ^ 2)def direction (D : Data) (t : ℝ) : Space :=  ‖D.e + D.w t‖⁻¹ • (D.e + D.w t)def weightedArea (D : Data) : ℝ :=  kappa D * ∫ t in D.α..D.β,    D.R t ^ 2 / (2 * Real.sqrt (1 + ‖D.w t‖ ^ 2))def scalarBound (D : Data) : ℝ := sSup ((fun t => |D.lam t|) '' Icc D.α D.β)def radialError (D : Data) : ℝ := D.L * scalarBound Ddef collar (D : Data) : ℝ := 3 * radialError D + 1def midpoint (D : Data) (N : ℕ) (i : Fin N) : ℝ :=  (RadialRiemann.point D.α D.β N i + RadialRiemann.point D.α D.β N (i + 1)) / 2def leftTip (D : Data) (N : ℕ) : ℝ := -collar D * RadialRiemann.step D.α D.β Ndef rightEdge (D : Data) (N : ℕ) (i : Fin N) : ℝ :=  RadialRiemann.capMax D.R D.α D.β N i + radialError D * RadialRiemann.step D.α D.β Ndef coordinateMap (D : Data) (p : ℝ × ℝ) : Space := D.p₀ + p.1 • D.a + p.2 • D.bdef frozenVertices (D : Data) (N : ℕ) (i : Fin N) : Fin 3 → Space :=  let l := leftTip D N  let r := rightEdge D N i  let m := midpoint D N i  let d := RadialRiemann.step D.α D.β N  ![coordinateMap D (l, m * l),    coordinateMap D (r, m * r - d / 2 * (r - l)),    coordinateMap D (r, m * r + d / 2 * (r - l))]def frozenTriangle (D : Data) (N : ℕ) (i : Fin N) : Set Space :=  convexHull ℝ (Set.range (frozenVertices D N i))def frozenBase (D : Data) (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : Set Space :=  TriangularCovering.projection (direction D (tag i)) '' frozenTriangle D N idef frozenArea (D : Data) (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : ℝ :=  kappa D * RadialRiemann.step D.α D.β N *    (RadialRiemann.capMax D.R D.α D.β N i +      (radialError D + collar D) * RadialRiemann.step D.α D.β N) ^ 2 /    (2 * Real.sqrt (1 + ‖D.w (tag i)‖ ^ 2))def localSweep (D : Data) (N : ℕ) (i : Fin N) : Set Space :=  {z | ∃ t ∈ RadialRiemann.cell D.α D.β N i,    ∃ ρ ∈ Icc (0 : ℝ) (D.R t), ∃ s : ℝ, |s| ≤ D.L ∧      z = D.p₀ + ρ • (D.a + t • D.b) + s • (D.e + D.w t)}def actualBase (D : Data) (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : Set Space :=  TriangularCovering.projection (direction D (tag i)) '' localSweep D N idef cylinder (u : Space) (B : Set Space) : Set Space :=  {z | ∃ b ∈ B, ∃ s : ℝ, z = b + s • u}def actualCost (D : Data) (N : ℕ) (tag : Fin N → ℝ) : ℝ :=  ∑ i : Fin N, TriangularCovering.area (actualBase D N tag i)def AlignedStatement : Prop :=  ∀ D : Data, ∀ tag : (N : ℕ) → Fin N → ℝ,    (∀ N, RadialRiemann.IsTagged D.α D.β N (tag N)) →    ∃ C : (N : ℕ) → Fin N → TriangularCovering.TriCylinder,      (∀ N, 0 < N →        (∀ i, (C N i).axis = direction D (tag N i) ∧          (C N i).base = frozenBase D N (tag N) i ∧          TriangularCovering.area (C N i).base = frozenArea D N (tag N) i) ∧        (∀ z ∈ sweep D, ∃ i : Fin N, z ∈ (C N i).carrier)) ∧      Tendsto (fun N => TriangularCovering.totalArea (C N)) atTop (𝓝 (weightedArea D))def ActualSweepStatement : Prop :=  ∀ D : Data, ∀ tag : (N : ℕ) → Fin N → ℝ,    (∀ N, RadialRiemann.IsTagged D.α D.β N (tag N)) →    (∀ N, 0 < N →      (∀ i : Fin N, ‖direction D (tag N i)‖ = 1 ∧        IsCompact (actualBase D N (tag N) i) ∧        MeasurableSet (actualBase D N (tag N) i) ∧        Measure.euclideanHausdorffMeasure 2 (actualBase D N (tag N) i) ≠ ⊤ ∧        (∀ z ∈ actualBase D N (tag N) i, inner ℝ (direction D (tag N i)) z = 0) ∧        actualBase D N (tag N) i ⊆ frozenBase D N (tag N) i) ∧      (∀ z ∈ sweep D, ∃ i : Fin N,        z ∈ cylinder (direction D (tag N i)) (actualBase D N (tag N) i))) ∧    (∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop, actualCost D N (tag N) ≤ weightedArea D + ε) ∧    Filter.limsup (fun N => actualCost D N (tag N)) atTop ≤ weightedArea Dend CylinderCovering.RadialSweependendnamespace CylinderCoveringtheorem fourPaperMain :    AngularExplicitStatement ∧    RuledApproximation.HyperbolicStatement ∧    RuledApproximation.NilpotentC1Statement ∧    RegularParallelogramCounterexampleStatement ∧    RegularOpenCounterexampleStatement ∧    RadialSweep.AlignedStatement ∧    RadialSweep.ActualSweepStatement := by  sorryend CylinderCoveringend 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

Show the TeX

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

Show the TeX

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

Show the TeX

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

Show the TeX

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

Show the TeX

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

Show the TeX

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/CylinderCovering.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.CylinderCovering.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: 108 local vocabulary declarations
Lines of Lean, without comments or blank lines: 327
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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