Lean statement

Negatively pinched Kähler threefold without bounded coordinates

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

Theorems selected for verification: OAI.PinchedHartogs.main_theorem OAI.PinchedHartogs.controlled_potential OAI.PinchedHartogs.metric_transfer

Lean statement file: PinchedKahler.leanShow on the mapFamily 359. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

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

def MainTheorem : Prop :=  ∃ (M : Set Ambient) (g : Ambient → CMatrix) (A B : ℝ),    IsOpen M ∧ M.Nonempty ∧ ContractibleSpace M ∧ IsKahlerMetric M g ∧      GeodesicallyComplete M g ∧ IsNegativelyPinched M g A B ∧      NoBoundedCoordinates M ∧ ¬ BiholomorphicToBoundedDomain M⋮def ControlledPotential : Prop :=  ∃ φ : Base → ℝ, ∃ C : ℝ, ContDiffOn ℝ ∞ φ ball ∧    ChartControl φ 1 C ∧ ChartJets φ C ∧ noMinorant φ⋮def MetricTransfer : Prop :=  ∀ c C : ℝ, 0 < c → c ≤ C → ∃ lam0 : ℝ, 0 < lam0 ∧    ∀ φ : Base → ℝ, ContDiffOn ℝ ∞ φ ball → ChartControl φ c C → ChartJets φ C →      ∀ lam : ℝ, lam0 ≤ lam → ContractibleSpace (hartogs φ) ∧        IsKahlerMetric (hartogs φ) (hartogsMetric φ lam) ∧        GeodesicallyComplete (hartogs φ) (hartogsMetric φ lam) ∧        ∃ A B : ℝ, IsNegativelyPinched (hartogs φ) (hartogsMetric φ lam) A B⋮theorem main_theorem : MainTheorem := by  sorry⋮theorem controlled_potential : ControlledPotential := by  sorry⋮theorem metric_transfer : MetricTransfer := 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.PinchedHartogs.Base line 12

abbrev Base := EuclideanSpace ℂ (Fin 2)

abbreviation OAI.PinchedHartogs.Ambient line 13
```
abbrev Ambient := Base × ℂ
```
abbreviation OAI.PinchedHartogs.Target line 14
```
abbrev Target := Fin 3 → ℂ
```

definition OAI.PinchedHartogs.ball line 16

def ball : Set Base := {z | ‖z‖ < 1}

definition OAI.PinchedHartogs.hartogs line 18

def hartogs (φ : Base → ℝ) : Set Ambient :=  {p | p.1 ∈ ball ∧ Real.exp (φ p.1) * ‖p.2‖ ^ 2 < 1}

definition OAI.PinchedHartogs.noMinorant line 21

def noMinorant (φ : Base → ℝ) : Prop :=  ¬ ∃ (H : Base → ℂ) (C : ℝ), AnalyticOnNhd ℂ H ball ∧    ∀ z ∈ ball, (H z).re ≤ C + 4 * Real.log (1 / (1 - ‖z‖)) + φ z

definition OAI.PinchedHartogs.coordVector line 25

def coordVector (j : Fin 3) : Ambient :=  ![(EuclideanSpace.single 0 1, 0), (EuclideanSpace.single 1 1, 0), (0, 1)] j

definition OAI.PinchedHartogs.jacobian line 28

def jacobian (F : Ambient → Target) (p : Ambient) : Matrix (Fin 3) (Fin 3) ℂ :=  fun i j ↦ (fderiv ℂ F p (coordVector j)) i

abbreviation OAI.PinchedHartogs.CMatrix line 33

abbrev CMatrix := Matrix (Fin 3) (Fin 3) ℂ

abbreviation OAI.PinchedHartogs.CurvatureTensor line 34

abbrev CurvatureTensor := Fin 3 → Fin 3 → Fin 3 → Fin 3 → ℂ

definition OAI.PinchedHartogs.coordinates line 36

def coordinates (u : Ambient) : Fin 3 → ℂ := ![u.1 0, u.1 1, u.2]

definition OAI.PinchedHartogs.dz line 38

def dz (f : Ambient → ℂ) (p : Ambient) (j : Fin 3) : ℂ :=  (fderiv ℝ f p (coordVector j) - Complex.I * fderiv ℝ f p (Complex.I • coordVector j)) / 2

definition OAI.PinchedHartogs.dbar line 41

def dbar (f : Ambient → ℂ) (p : Ambient) (j : Fin 3) : ℂ :=  (fderiv ℝ f p (coordVector j) + Complex.I * fderiv ℝ f p (Complex.I • coordVector j)) / 2

definition OAI.PinchedHartogs.hermitianValue line 44

def hermitianValue (h : CMatrix) (u v : Ambient) : ℂ :=  ∑ i, ∑ j, h i j * coordinates u i * star (coordinates v j)

definition OAI.PinchedHartogs.realArea line 47

def realArea (h : CMatrix) (u v : Ambient) : ℝ :=  (hermitianValue h u u).re * (hermitianValue h v v).re - (hermitianValue h u v).re ^ 2

definition OAI.PinchedHartogs.tensorValue line 50

def tensorValue (R : CurvatureTensor) (u v w y : Ambient) : ℂ :=  ∑ a, ∑ b, ∑ c, ∑ d,    R a b c d * coordinates u a * star (coordinates v b) * coordinates w c * star (coordinates y d)

definition OAI.PinchedHartogs.curvature line 54

def curvature (g : Ambient → CMatrix) (p : Ambient) : CurvatureTensor := fun a b c d ↦  -dbar (fun q ↦ dz (fun r ↦ g r c d) q a) p b +    ∑ e, ∑ k, (g p)⁻¹ e k * dz (fun q ↦ g q c e) p a * dbar (fun q ↦ g q k d) p b

definition OAI.PinchedHartogs.sectional line 58

def sectional (g : Ambient → CMatrix) (p u v : Ambient) : ℝ :=  ((tensorValue (curvature g p) u u v v).re - (tensorValue (curvature g p) u v u v).re) /    (2 * realArea (g p) u v)

definition OAI.PinchedHartogs.IsKahlerMetric line 62

def IsKahlerMetric (M : Set Ambient) (g : Ambient → CMatrix) : Prop :=  ContDiffOn ℝ ∞ g M ∧  (∀ p ∈ M, ∀ i j, g p i j = star (g p j i)) ∧  (∀ p ∈ M, ∀ u : Ambient, u ≠ 0 → 0 < (hermitianValue (g p) u u).re) ∧  (∀ p ∈ M, ∀ i j k, dz (fun q ↦ g q k j) p i = dz (fun q ↦ g q i j) p k)

definition OAI.PinchedHartogs.christoffel line 68

def christoffel (g : Ambient → CMatrix) (p : Ambient) (k a c : Fin 3) : ℂ :=  ∑ e, (g p)⁻¹ e k * dz (fun q ↦ g q c e) p a

definition OAI.PinchedHartogs.GeodesicallyComplete line 71

def GeodesicallyComplete (M : Set Ambient) (g : Ambient → CMatrix) : Prop :=  ∀ p ∈ M, ∀ v : Ambient, ∃ γ : ℝ → Ambient,    ContDiff ℝ ∞ γ ∧ (∀ t, γ t ∈ M) ∧ γ 0 = p ∧ deriv γ 0 = v ∧      ∀ t k, coordinates (deriv (deriv γ) t) k +        ∑ a, ∑ c, christoffel g (γ t) k a c * coordinates (deriv γ t) a *          coordinates (deriv γ t) c = 0

definition OAI.PinchedHartogs.IsNegativelyPinched line 78

def IsNegativelyPinched (M : Set Ambient) (g : Ambient → CMatrix) (A B : ℝ) : Prop :=  0 < A ∧ A ≤ B ∧ ∀ p ∈ M, ∀ u v : Ambient, LinearIndependent ℝ ![u, v] →    -B ≤ sectional g p u v ∧ sectional g p u v ≤ -A

definition OAI.PinchedHartogs.NoBoundedCoordinates line 82

def NoBoundedCoordinates (M : Set Ambient) : Prop :=  ∀ F : Ambient → Target, AnalyticOnNhd ℂ F M →    (∃ B : ℝ, ∀ p ∈ M, ∀ i, ‖F p i‖ ≤ B) →      ¬ ∀ p ∈ M, (jacobian F p).det ≠ 0

definition OAI.PinchedHartogs.BiholomorphicToBoundedDomain line 87

def BiholomorphicToBoundedDomain (M : Set Ambient) : Prop :=  ∃ N : Set Ambient, IsOpen N ∧ Bornology.IsBounded N ∧    ∃ F G : Ambient → Ambient, AnalyticOnNhd ℂ F M ∧ AnalyticOnNhd ℂ G N ∧      MapsTo F M N ∧ MapsTo G N M ∧ (∀ p ∈ M, G (F p) = p) ∧ (∀ q ∈ N, F (G q) = q)

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

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: biholomorphic, bounded, complete, constants, contractible, domain, holomorphic, metric and one more.

Statements in A negatively pinched Kähler threefold without bounded holomorphic coordinates come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of A negatively pinched Kähler threefold without bounded holomorphic coordinates

Theorem

There exist a contractible domain M\subset\C^3, a smooth complete Kähler metric g on M, and finite constants 0<A\le B such that

-B\le K_g(\sigma)\le -A<0

for every real two-plane \sigma\subset T_pM and every p\in M. No bounded holomorphic map F:M\to\C^3 has a nowhere-vanishing Jacobian determinant. In particular, M is not biholomorphic to a bounded domain in \C^3.

Section 1, Introduction Read it in the paper TeX source, line 39

Show the TeX

There exist a contractible domain $M\subset\C^3$, a smooth complete
K\"ahler metric $g$ on $M$, and finite constants $0<A\le B$ such that
\[
 -B\le K_g(\sigma)\le -A<0
\]
for every real two-plane $\sigma\subset T_pM$ and every $p\in M$.
No bounded holomorphic map $F:M\to\C^3$ has a nowhere-vanishing Jacobian
determinant. In particular, $M$ is not biholomorphic to a bounded domain
in $\C^3$.

Other possible paper matches, closest wording first

Proposition 2.2 of A negatively pinched Kähler threefold without bounded holomorphic coordinates. Words of the Lean statement, its names split into words, that also appear in the theorem: geodesically, hartogs, space, transfer, tensor, contractible and nine more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two and constants.
Proposition 2.1 of A negatively pinched Kähler threefold without bounded holomorphic coordinates. Words of the Lean statement, its names split into words, that also appear in the theorem: controlled, base, control, re, log, potential and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant.
Proposition 8.1 of A negatively pinched Kähler threefold without bounded holomorphic coordinates. Words of the Lean statement, its names split into words, that also appear in the theorem: euclidean, chart, tensor, sqrt, hermitian, centered and two more.
Theorem 4.2 of A negatively pinched Kähler threefold without bounded holomorphic coordinates. Words of the Lean statement, its names split into words, that also appear in the theorem: vector and sqrt. Found in the paper’s title, which its main theorem states: negatively, pinched, coordinates and bounded. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: separated, holomorphic, one and constant.
Proposition 6.4 of One-sided negative sectional curvature and the holomorphic Liouville property. Words of the Lean statement, its names split into words, that also appear in the theorem: matrix, psi, hermitian, log, complex, jet and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: negative, two and constant.
Theorem 1.1 of One-sided negative sectional curvature and the holomorphic Liouville property. Words of the Lean statement, its names split into words, that also appear in the theorem: domain, complete, complex, sectional, curvatures, metric and one 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 359’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: main_theorem gives a nonempty open contractible M ⊆ ℂ³ with a smooth Kähler metric that is geodesically complete (all geodesics extend for all time) and negatively pinched (-B ≤ sectional ≤ -A < 0), with no bounded holomorphic map with everywhere-nonvanishing Jacobian and no biholomorphism onto a bounded domain; controlled_potential and metric_transfer support it. The summary's second claim (a higher-dimensional example with sectional curvature ≤ -1, only constant bounded holomorphic functions, curvature not bounded below) is not stated.
Definitions the group asks readers to compare with the paper: Kähler geometry is hand-coded on ℂ²×ℂ: dz/dbar from real derivatives, curvature g (-∂̄_b∂_a g_{c d̄} + g^{-1}∂g ∂̄g), sectional := (R(u,ū,v,v̄) - R(u,v̄,u... ))/(2·realArea), IsKahlerMetric (symmetry, positivity, ∂_i g_{kj̄} = ∂_k g_{ij̄}), GeodesicallyComplete via smooth solutions of the geodesic ODE; NoBoundedCoordinates only excludes bounded holomorphic F with nonvanishing Jacobian (stronger than excluding injective coordinate charts)
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 higher-dimensional one-sided example (the companion 'Liouville' paper) is not stated.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result constructs a nonempty contractible complex threefold with a complete Kähler metric whose real sectional curvatures lie between two fixed negative constants. Nevertheless, it has no system of bounded holomorphic coordinates and is not biholomorphic to a bounded domain. The metric is obtained from an explicitly controlled potential. This is the negatively pinched construction; the different companion with only one-sided curvature control is separate.”
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

159 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: 165 to its own declarations, 90 to Mathlib’s documentation. 40 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main_theorem exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 37 other declarations of the file; with the theorem itself they span 105 lines. 3 of them are named in the statement itself.

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

abbrev Baseline 12library
abbrev Ambientline 13library
abbrev Targetline 14library
def ballline 16library
def hartogsline 18library
def noMinorantline 21library
def coordVectorline 25library
def jacobianline 28library
abbrev CMatrixline 33library
abbrev CurvatureTensorline 34library
def coordinatesline 36library
def dzline 38library
def dbarline 41library
def hermitianValueline 44library
def realArealine 47library
def tensorValueline 50library
def curvatureline 54library
def sectionalline 58library
def IsKahlerMetricline 62library
def christoffelline 68library
def GeodesicallyCompleteline 71library
def IsNegativelyPinchedline 78library
def NoBoundedCoordinatesline 82library
def BiholomorphicToBoundedDomainline 87library
def MainTheoremline 92in the statementlibrary
def psiline 98library
def metricPotentialline 99library
def complexHessianline 102library
def hartogsMetricline 105library
def dzBaseline 108library
def dbarBaseline 112library
def baseHessianline 116library
def centeredChartline 119library
def ChartControlline 123library
def ChartJetsline 128library
def ControlledPotentialline 134in the statementlibrary
def MetricTransferline 138in the statementlibrary
theorem main_theoremline 146selected theoremlibrary
theorem controlled_potentialline 149selected theoremlibrary
theorem metric_transferline 152selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen Set Filter Topologyopen scoped ContDiffnamespace PinchedHartogsabbrev Base := EuclideanSpace ℂ (Fin 2)abbrev Ambient := Base × ℂabbrev Target := Fin 3 → ℂdef ball : Set Base := {z | ‖z‖ < 1}def hartogs (φ : Base → ℝ) : Set Ambient :=  {p | p.1 ∈ ball ∧ Real.exp (φ p.1) * ‖p.2‖ ^ 2 < 1}def noMinorant (φ : Base → ℝ) : Prop :=  ¬ ∃ (H : Base → ℂ) (C : ℝ), AnalyticOnNhd ℂ H ball ∧    ∀ z ∈ ball, (H z).re ≤ C + 4 * Real.log (1 / (1 - ‖z‖)) + φ zdef coordVector (j : Fin 3) : Ambient :=  ![(EuclideanSpace.single 0 1, 0), (EuclideanSpace.single 1 1, 0), (0, 1)] jdef jacobian (F : Ambient → Target) (p : Ambient) : Matrix (Fin 3) (Fin 3) ℂ :=  fun i j ↦ (fderiv ℂ F p (coordVector j)) iopen scoped Matrix.Norms.Elementwiseabbrev CMatrix := Matrix (Fin 3) (Fin 3) ℂabbrev CurvatureTensor := Fin 3 → Fin 3 → Fin 3 → Fin 3 → ℂdef coordinates (u : Ambient) : Fin 3 → ℂ := ![u.1 0, u.1 1, u.2]def dz (f : Ambient → ℂ) (p : Ambient) (j : Fin 3) : ℂ :=  (fderiv ℝ f p (coordVector j) - Complex.I * fderiv ℝ f p (Complex.I • coordVector j)) / 2def dbar (f : Ambient → ℂ) (p : Ambient) (j : Fin 3) : ℂ :=  (fderiv ℝ f p (coordVector j) + Complex.I * fderiv ℝ f p (Complex.I • coordVector j)) / 2def hermitianValue (h : CMatrix) (u v : Ambient) : ℂ :=  ∑ i, ∑ j, h i j * coordinates u i * star (coordinates v j)def realArea (h : CMatrix) (u v : Ambient) : ℝ :=  (hermitianValue h u u).re * (hermitianValue h v v).re - (hermitianValue h u v).re ^ 2def tensorValue (R : CurvatureTensor) (u v w y : Ambient) : ℂ :=  ∑ a, ∑ b, ∑ c, ∑ d,    R a b c d * coordinates u a * star (coordinates v b) * coordinates w c * star (coordinates y d)def curvature (g : Ambient → CMatrix) (p : Ambient) : CurvatureTensor := fun a b c d ↦  -dbar (fun q ↦ dz (fun r ↦ g r c d) q a) p b +    ∑ e, ∑ k, (g p)⁻¹ e k * dz (fun q ↦ g q c e) p a * dbar (fun q ↦ g q k d) p bdef sectional (g : Ambient → CMatrix) (p u v : Ambient) : ℝ :=  ((tensorValue (curvature g p) u u v v).re - (tensorValue (curvature g p) u v u v).re) /    (2 * realArea (g p) u v)def IsKahlerMetric (M : Set Ambient) (g : Ambient → CMatrix) : Prop :=  ContDiffOn ℝ ∞ g M ∧  (∀ p ∈ M, ∀ i j, g p i j = star (g p j i)) ∧  (∀ p ∈ M, ∀ u : Ambient, u ≠ 0 → 0 < (hermitianValue (g p) u u).re) ∧  (∀ p ∈ M, ∀ i j k, dz (fun q ↦ g q k j) p i = dz (fun q ↦ g q i j) p k)def christoffel (g : Ambient → CMatrix) (p : Ambient) (k a c : Fin 3) : ℂ :=  ∑ e, (g p)⁻¹ e k * dz (fun q ↦ g q c e) p adef GeodesicallyComplete (M : Set Ambient) (g : Ambient → CMatrix) : Prop :=  ∀ p ∈ M, ∀ v : Ambient, ∃ γ : ℝ → Ambient,    ContDiff ℝ ∞ γ ∧ (∀ t, γ t ∈ M) ∧ γ 0 = p ∧ deriv γ 0 = v ∧      ∀ t k, coordinates (deriv (deriv γ) t) k +        ∑ a, ∑ c, christoffel g (γ t) k a c * coordinates (deriv γ t) a *          coordinates (deriv γ t) c = 0def IsNegativelyPinched (M : Set Ambient) (g : Ambient → CMatrix) (A B : ℝ) : Prop :=  0 < A ∧ A ≤ B ∧ ∀ p ∈ M, ∀ u v : Ambient, LinearIndependent ℝ ![u, v] →    -B ≤ sectional g p u v ∧ sectional g p u v ≤ -Adef NoBoundedCoordinates (M : Set Ambient) : Prop :=  ∀ F : Ambient → Target, AnalyticOnNhd ℂ F M →    (∃ B : ℝ, ∀ p ∈ M, ∀ i, ‖F p i‖ ≤ B) →      ¬ ∀ p ∈ M, (jacobian F p).det ≠ 0def BiholomorphicToBoundedDomain (M : Set Ambient) : Prop :=  ∃ N : Set Ambient, IsOpen N ∧ Bornology.IsBounded N ∧    ∃ F G : Ambient → Ambient, AnalyticOnNhd ℂ F M ∧ AnalyticOnNhd ℂ G N ∧      MapsTo F M N ∧ MapsTo G N M ∧ (∀ p ∈ M, G (F p) = p) ∧ (∀ q ∈ N, F (G q) = q)def MainTheorem : Prop :=  ∃ (M : Set Ambient) (g : Ambient → CMatrix) (A B : ℝ),    IsOpen M ∧ M.Nonempty ∧ ContractibleSpace M ∧ IsKahlerMetric M g ∧      GeodesicallyComplete M g ∧ IsNegativelyPinched M g A B ∧      NoBoundedCoordinates M ∧ ¬ BiholomorphicToBoundedDomain Mdef psi (z : Base) : ℝ := -Real.log (1 - ‖z‖ ^ 2)def metricPotential (φ : Base → ℝ) (lam : ℝ) (p : Ambient) : ℝ :=  lam * psi p.1 - Real.log (1 - Real.exp (φ p.1) * ‖p.2‖ ^ 2)def complexHessian (f : Ambient → ℝ) (p : Ambient) : CMatrix :=  fun i j ↦ dz (fun q ↦ dbar (fun r ↦ (f r : ℂ)) q j) p idef hartogsMetric (φ : Base → ℝ) (lam : ℝ) : Ambient → CMatrix :=  complexHessian (metricPotential φ lam)def dzBase (f : Base → ℂ) (z : Base) (i : Fin 2) : ℂ :=  (fderiv ℝ f z (EuclideanSpace.single i 1) -    Complex.I * fderiv ℝ f z (Complex.I • EuclideanSpace.single i 1)) / 2def dbarBase (f : Base → ℂ) (z : Base) (i : Fin 2) : ℂ :=  (fderiv ℝ f z (EuclideanSpace.single i 1) +    Complex.I * fderiv ℝ f z (Complex.I • EuclideanSpace.single i 1)) / 2def baseHessian (φ : Base → ℝ) (z : Base) : Matrix (Fin 2) (Fin 2) ℂ :=  fun i j ↦ dzBase (fun q ↦ dbarBase (fun r ↦ (φ r : ℂ)) q j) z idef centeredChart (r : ℝ) (U : Base ≃ₗᵢ[ℂ] Base) (ζ : Base) : Base :=  U (WithLp.toLp 2 ![((r : ℂ) + ζ 0) / (1 + (r : ℂ) * ζ 0),    ((Real.sqrt (1 - r ^ 2) : ℝ) : ℂ) * ζ 1 / (1 + (r : ℂ) * ζ 0)])def ChartControl (φ : Base → ℝ) (c C : ℝ) : Prop :=  ∀ r : ℝ, 0 ≤ r → r < 1 → ∀ U : Base ≃ₗᵢ[ℂ] Base, ∀ v : Base,    c * ‖v‖ ^ 2 ≤ (∑ i, ∑ j, baseHessian (φ ∘ centeredChart r U) 0 i j * v i * star (v j)).re ∧    (∑ i, ∑ j, baseHessian (φ ∘ centeredChart r U) 0 i j * v i * star (v j)).re ≤ C * ‖v‖ ^ 2def ChartJets (φ : Base → ℝ) (C : ℝ) : Prop :=  ∀ r : ℝ, 0 ≤ r → r < 1 → ∀ U : Base ≃ₗᵢ[ℂ] Base, ∀ i j k l : Fin 2,    ‖dzBase (fun z ↦ baseHessian (φ ∘ centeredChart r U) z i j) 0 k‖ +    ‖dbarBase (fun z ↦ baseHessian (φ ∘ centeredChart r U) z i j) 0 l‖ +    ‖dzBase (fun z ↦ dbarBase (fun q ↦ baseHessian (φ ∘ centeredChart r U) q i j) z l) 0 k‖ ≤ Cdef ControlledPotential : Prop :=  ∃ φ : Base → ℝ, ∃ C : ℝ, ContDiffOn ℝ ∞ φ ball ∧    ChartControl φ 1 C ∧ ChartJets φ C ∧ noMinorant φdef MetricTransfer : Prop :=  ∀ c C : ℝ, 0 < c → c ≤ C → ∃ lam0 : ℝ, 0 < lam0 ∧    ∀ φ : Base → ℝ, ContDiffOn ℝ ∞ φ ball → ChartControl φ c C → ChartJets φ C →      ∀ lam : ℝ, lam0 ≤ lam → ContractibleSpace (hartogs φ) ∧        IsKahlerMetric (hartogs φ) (hartogsMetric φ lam) ∧        GeodesicallyComplete (hartogs φ) (hartogsMetric φ lam) ∧        ∃ A B : ℝ, IsNegativelyPinched (hartogs φ) (hartogsMetric φ lam) A Btheorem main_theorem : MainTheorem := by  sorrytheorem controlled_potential : ControlledPotential := by  sorrytheorem metric_transfer : MetricTransfer := by  sorryend PinchedHartogsendend 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.

A negatively pinched Kähler threefold without bounded holomorphic coordinates

Theorem

There exist a contractible domain M\subset\C^3, a smooth complete Kähler metric g on M, and finite constants 0<A\le B such that

-B\le K_g(\sigma)\le -A<0

Section 1, Introduction Read it in the paper TeX source, line 39

Show the TeX

There exist a contractible domain $M\subset\C^3$, a smooth complete
K\"ahler metric $g$ on $M$, and finite constants $0<A\le B$ such that
\[
 -B\le K_g(\sigma)\le -A<0
\]
for every real two-plane $\sigma\subset T_pM$ and every $p\in M$.
No bounded holomorphic map $F:M\to\C^3$ has a nowhere-vanishing Jacobian
determinant. In particular, $M$ is not biholomorphic to a bounded domain
in $\C^3$.

Configuration and file facts

File: lean/ComparatorChallenges/PinchedKahler.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Geometry/Kahler/Main.lean
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 37 local vocabulary declarations
Lines of Lean, without comments or blank lines: 105
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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