Lean statement

Counterexample to harmonic dimension comparison

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.HarmonicCounterexample.main

Lean statement file: HarmonicGrowth.leanShow on the mapFamily 361. 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 MainClaim : Prop :=  ∃ (n k : ℕ) (g : SmoothMetric n), Even n ∧ 8 ≤ n ∧ 2 ≤ k ∧    Complete g ∧ RicciNonnegative g ∧ EuclideanNearOrigin g ∧ PositiveSubunitAVR g ∧    ∃ u : Fin (euclideanDimension n k + 1) → (Space n → ℝ),      LinearIndependent ℝ u ∧ ∀ i, HarmonicGrowth g k (u i)⋮axiom mainStatement : MainClaim⋮theorem main : MainClaim := mainStatement

A Lean statement states the theorem and leaves its proof open. This file writes axiom mainStatement where others write sorry; OpenAI’s solution module proves main directly, 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.HarmonicCounterexample.Space line 11
```
abbrev Space (n : ℕ) := Fin n → ℝ
```

abbreviation OAI.HarmonicCounterexample.Mat line 12

abbrev Mat (n : ℕ) := Matrix (Fin n) (Fin n) ℝ

structure OAI.HarmonicCounterexample.SmoothMetric line 15

structure SmoothMetric (n : ℕ) where  coeff : Space n → Mat n  smooth : ∀ i j, ContDiff ℝ ∞ (fun x => coeff x i j)  positive : ∀ x, (coeff x).PosDef

definition OAI.HarmonicCounterexample.coordDeriv line 21

def coordDeriv {n : ℕ} (i : Fin n) (f : Space n → ℝ) (x : Space n) : ℝ :=  fderiv ℝ f x (Pi.single i 1)

definition OAI.HarmonicCounterexample.christoffel line 25

def christoffel {n : ℕ} (g : SmoothMetric n) (x : Space n)    (k i j : Fin n) : ℝ :=  (1 / 2 : ℝ) * ∑ l, (g.coeff x)⁻¹ k l *    (coordDeriv i (fun y => g.coeff y j l) x +     coordDeriv j (fun y => g.coeff y i l) x -     coordDeriv l (fun y => g.coeff y i j) x)

definition OAI.HarmonicCounterexample.ricci line 33

def ricci {n : ℕ} (g : SmoothMetric n) (x : Space n) (i j : Fin n) : ℝ :=  (∑ k, (coordDeriv k (fun y => christoffel g y k i j) x -    coordDeriv j (fun y => christoffel g y k i k) x)) +  ∑ k, ∑ l, (christoffel g x k i j * christoffel g x l k l -    christoffel g x l i k * christoffel g x k j l)

definition OAI.HarmonicCounterexample.RicciNonnegative line 39

def RicciNonnegative {n : ℕ} (g : SmoothMetric n) : Prop :=  ∀ (x v : Space n), 0 ≤ ∑ i, ∑ j, ricci g x i j * v i * v j

definition OAI.HarmonicCounterexample.laplaceBeltrami line 43

def laplaceBeltrami {n : ℕ} (g : SmoothMetric n) (u : Space n → ℝ)    (x : Space n) : ℝ :=  ∑ i, ∑ j, (g.coeff x)⁻¹ i j *    (coordDeriv i (coordDeriv j u) x - ∑ k, christoffel g x k i j * coordDeriv k u x)

definition OAI.HarmonicCounterexample.pathLength line 49

def pathLength {n : ℕ} (g : SmoothMetric n) (γ : ℝ → Space n) : ℝ :=  ∫ t in (0 : ℝ)..1, Real.sqrt    (∑ i, ∑ j, g.coeff (γ t) i j * deriv γ t i * deriv γ t j)

definition OAI.HarmonicCounterexample.distance line 54

def distance {n : ℕ} (g : SmoothMetric n) (x y : Space n) : ℝ :=  sInf {L : ℝ | ∃ γ : ℝ → Space n,    ContDiff ℝ 1 γ ∧ γ 0 = x ∧ γ 1 = y ∧ L = pathLength g γ}

definition OAI.HarmonicCounterexample.Complete line 59

def Complete {n : ℕ} (g : SmoothMetric n) : Prop :=  ∀ z : ℕ → Space n,    (∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ p ≥ N, ∀ q ≥ N, distance g (z p) (z q) < ε) →    ∃ x : Space n, ∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ p ≥ N, distance g (z p) x < ε

definition OAI.HarmonicCounterexample.EuclideanNearOrigin line 64

def EuclideanNearOrigin {n : ℕ} (g : SmoothMetric n) : Prop :=  ∃ ε : ℝ, 0 < ε ∧ ∀ x : Space n, (∑ i, x i ^ 2) < ε ^ 2 → g.coeff x = 1

definition OAI.HarmonicCounterexample.ballVolume line 68

def ballVolume {n : ℕ} (g : SmoothMetric n) (R : ℝ) : ℝ≥0∞ :=  ∫⁻ x in {x : Space n | distance g 0 x < R},    ENNReal.ofReal (Real.sqrt (g.coeff x).det) ∂volume

definition OAI.HarmonicCounterexample.omega line 73

def omega (n : ℕ) : ℝ≥0∞ :=  volume {x : Space n | (∑ i, x i ^ 2) < 1}

definition OAI.HarmonicCounterexample.PositiveSubunitAVR line 77

def PositiveSubunitAVR {n : ℕ} (g : SmoothMetric n) : Prop :=  ∃ a : ℝ, 0 < a ∧ a < 1 ∧    Tendsto (fun R : ℝ => ballVolume g R / (omega n * ENNReal.ofReal (R ^ n)))      atTop (𝓝 (ENNReal.ofReal a))

definition OAI.HarmonicCounterexample.HarmonicGrowth line 83

def HarmonicGrowth {n : ℕ} (g : SmoothMetric n) (k : ℕ) (u : Space n → ℝ) : Prop :=  ContDiff ℝ ∞ u ∧ (∀ x, laplaceBeltrami g u x = 0) ∧    ∃ C : ℝ, 0 ≤ C ∧ ∀ x, |u x| ≤ C * (1 + distance g 0 x) ^ k

definition OAI.HarmonicCounterexample.euclideanDimension line 88

def euclideanDimension (n k : ℕ) : ℕ :=  (n + k - 1).choose k + (n + k - 2).choose (k - 1)

A note on how this file states its claim

Line 100 of this statement file writes axiom mainStatement : MainClaim where other statement files write sorry. It stands in for the proof on the statement side only. OpenAI’s solution proves main directly, in the solution file. A successful Comparator replay checks the solution against the allowed axioms: propext, Quot.sound, Classical.choice.

Possible paper match, with no recorded review

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

Words of the Lean statement, its names split into words, that also appear in the theorem: even, unit, omega, ball, ratio, volume, euclidean, origin and six more. Found in the paper’s title, which its main theorem states: dimension and harmonic. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: asymptotic.

Statements in A counterexample to integer-degree harmonic dimension comparison come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of A counterexample to integer-degree harmonic dimension comparison

Theorem

There are an even integer n\geq8, an integer k\geq2, and a complete smooth metric g on \mathbb R^n such that \operatorname{Ric}_g\geq0 and

h_k(\mathbb R^n,g)>h_k(\mathbb R^n,g_{\mathrm E}).

The metric is Euclidean near the origin, and its asymptotic volume ratio satisfies

0<\lim_{R\to\infty} \frac{\operatorname{vol}_g B_g(0,R)}{\omega_nR^n}<1,

where \omega_n is the volume of the Euclidean unit ball.

Section 1, Introduction Read it in the paper TeX source, line 23

Show the TeX

There are an even integer $n\geq8$, an integer $k\geq2$, and a complete
smooth metric $g$ on $\mathbb R^n$ such that $\operatorname{Ric}_g\geq0$
and
\[
 h_k(\mathbb R^n,g)>h_k(\mathbb R^n,g_{\mathrm E}).
\]
The metric is Euclidean near the origin, and its asymptotic volume ratio
satisfies
\[
 0<\lim_{R\to\infty}
 \frac{\operatorname{vol}_g B_g(0,R)}{\omega_nR^n}<1,
\]
where $\omega_n$ is the volume of the Euclidean unit ball.

Other possible paper matches, closest wording first

Proposition 5.1 of A counterexample to integer-degree harmonic dimension comparison. Words of the Lean statement, its names split into words, that also appear in the theorem: distance, nonnegative, parameters, ricci, volume, euclidean and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: function and curvature.
Proposition 6.5 of A counterexample to integer-degree harmonic dimension comparison. Words of the Lean statement, its names split into words, that also appear in the theorem: growth, value, choose, parameters, finite, bounds and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: admitted and counting.
Theorem 1.1 of A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison. Words of the Lean statement, its names split into words, that also appear in the theorem: avr, ricci, pointed, euclidean, origin, complete and six more. Found in the paper’s title, which its main theorem states: dimension and harmonic. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: tangent and curvature.
Proposition 7.1 of A counterexample to integer-degree harmonic dimension comparison. Words of the Lean statement, its names split into words, that also appear in the theorem: nonnegative, harmonic, parameters, pointed, integer and metric. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constructed, cone, tangent and comparison.
Proposition 4.1 of A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison. Words of the Lean statement, its names split into words, that also appear in the theorem: 1/2, strictly, natural, round, independent, finite and nine more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: including, zero, dimensional, function and curvature.
Proposition 5.2 of A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison. Words of the Lean statement, its names split into words, that also appear in the theorem: operator, spaces, ball, one, matrix, complete and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: countable and function.

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 1096 s; peak memory 7.0 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 361’s Lean statements express the advertised result?

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

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

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
Their note: MainClaim is ∃ n (even, ≥ 8), k ≥ 2 and one complete smooth metric on ℝⁿ with Ric ≥ 0, Euclidean near 0, asymptotic volume ratio in (0,1), and euclideanDimension n k + 1 independent harmonic functions of growth ≤ k: a single higher-dimensional counterexample, as the docs say. The summary's claim (complete metrics on ℝ³ with ≥ (k+2)² harmonic functions of growth ≤ k for every sufficiently large integer k) is not implied. The challenge file also contains axiom mainStatement : MainClaim and theorem main := mainStatement instead of sorry, a non-standard stub shape; no axiom exists in the solution cone.
Definitions the group asks readers to compare with the paper: axiom mainStatement : MainClaim in the challenge file (the only challenge here with an axiom and no sorry) ; Hand-written Riemannian geometry on ℝⁿ: ricci, laplaceBeltrami, distance := sInf of C¹ path lengths, Complete, PositiveSubunitAVR (ball volume/(ω_n R^n) → a ∈ (0,1)), HarmonicGrowth (|u| ≤ C(1+d(0,x))^k) and euclideanDimension n k := C(n+k-1,k) + C(n+k-2,k-1) (dimension of Euclidean harmonic polynomials of degree ≤ k)
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: The three-dimensional statement — on R³, for every sufficiently large k, ≥ (k+2)² independent harmonic functions of growth ≤ k — is not stated. The Lean states a single (n, k) with n even ≥ 8 and k ≥ 2, with euclideanDimension(n,k) + 1 functions.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“It constructs one complete smooth metric with nonnegative Ricci curvature on an even-dimensional Euclidean space, Euclidean near the origin and with asymptotic volume ratio strictly between zero and one, admitting more independent harmonic functions of the prescribed growth degree than the Euclidean count. The construction uses dimension $16$ and degree $50000$. Separate tangent-cone and cone-spectrum conclusions are not included.”
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

108 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: 64 to its own declarations, 27 to Mathlib’s documentation. 18 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main 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 18 of the file’s 19 other declarations; with the theorem itself they span 57 lines. 1 of them is named in the statement itself.

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

abbrev Spaceline 11library
abbrev Matline 12library
structure SmoothMetricline 15library
def coordDerivline 21library
def christoffelline 25library
def ricciline 33library
def RicciNonnegativeline 39library
def laplaceBeltramiline 43library
def pathLengthline 49library
def distanceline 54library
def Completeline 59library
def EuclideanNearOriginline 64library
def ballVolumeline 68library
def omegaline 73library
def PositiveSubunitAVRline 77library
def HarmonicGrowthline 83library
def euclideanDimensionline 88library
def MainClaimline 92in the statementlibrary
theorem mainline 102selected theorem

Other Lean rules that may affect the statement 1

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

axiom mainStatement line 100

import Mathlibnamespace OAInoncomputable sectionopen Filter MeasureTheoryopen scoped BigOperators Topology ENNReal ContDiffnamespace HarmonicCounterexampleabbrev Space (n : ℕ) := Fin n → ℝabbrev Mat (n : ℕ) := Matrix (Fin n) (Fin n) ℝ/-- A genuine smooth positive definite covariant metric tensor on R^n. -/structure SmoothMetric (n : ℕ) where  coeff : Space n → Mat n  smooth : ∀ i j, ContDiff ℝ ∞ (fun x => coeff x i j)  positive : ∀ x, (coeff x).PosDef/-- The coordinate directional derivative. -/def coordDeriv {n : ℕ} (i : Fin n) (f : Space n → ℝ) (x : Space n) : ℝ :=  fderiv ℝ f x (Pi.single i 1)/-- Levi-Civita connection in the single global Cartesian chart. -/def christoffel {n : ℕ} (g : SmoothMetric n) (x : Space n)    (k i j : Fin n) : ℝ :=  (1 / 2 : ℝ) * ∑ l, (g.coeff x)⁻¹ k l *    (coordDeriv i (fun y => g.coeff y j l) x +     coordDeriv j (fun y => g.coeff y i l) x -     coordDeriv l (fun y => g.coeff y i j) x)/-- Ricci tensor, with the convention in which the round sphere has positive Ricci. -/def ricci {n : ℕ} (g : SmoothMetric n) (x : Space n) (i j : Fin n) : ℝ :=  (∑ k, (coordDeriv k (fun y => christoffel g y k i j) x -    coordDeriv j (fun y => christoffel g y k i k) x)) +  ∑ k, ∑ l, (christoffel g x k i j * christoffel g x l k l -    christoffel g x l i k * christoffel g x k j l)def RicciNonnegative {n : ℕ} (g : SmoothMetric n) : Prop :=  ∀ (x v : Space n), 0 ≤ ∑ i, ∑ j, ricci g x i j * v i * v j/-- Trace of the covariant Hessian, i.e. the Laplace-Beltrami operator. -/def laplaceBeltrami {n : ℕ} (g : SmoothMetric n) (u : Space n → ℝ)    (x : Space n) : ℝ :=  ∑ i, ∑ j, (g.coeff x)⁻¹ i j *    (coordDeriv i (coordDeriv j u) x - ∑ k, christoffel g x k i j * coordDeriv k u x)/-- Riemannian length of a path on the unit parameter interval. -/def pathLength {n : ℕ} (g : SmoothMetric n) (γ : ℝ → Space n) : ℝ :=  ∫ t in (0 : ℝ)..1, Real.sqrt    (∑ i, ∑ j, g.coeff (γ t) i j * deriv γ t i * deriv γ t j)/-- Infimum of smooth path lengths, the intrinsic Riemannian distance. -/def distance {n : ℕ} (g : SmoothMetric n) (x y : Space n) : ℝ :=  sInf {L : ℝ | ∃ γ : ℝ → Space n,    ContDiff ℝ 1 γ ∧ γ 0 = x ∧ γ 1 = y ∧ L = pathLength g γ}/-- Sequential metric completeness, for the intrinsic distance, not the background one. -/def Complete {n : ℕ} (g : SmoothMetric n) : Prop :=  ∀ z : ℕ → Space n,    (∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ p ≥ N, ∀ q ≥ N, distance g (z p) (z q) < ε) →    ∃ x : Space n, ∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ p ≥ N, distance g (z p) x < εdef EuclideanNearOrigin {n : ℕ} (g : SmoothMetric n) : Prop :=  ∃ ε : ℝ, 0 < ε ∧ ∀ x : Space n, (∑ i, x i ^ 2) < ε ^ 2 → g.coeff x = 1/-- Riemannian measure of the intrinsic open ball, retaining infinite values. -/def ballVolume {n : ℕ} (g : SmoothMetric n) (R : ℝ) : ℝ≥0∞ :=  ∫⁻ x in {x : Space n | distance g 0 x < R},    ENNReal.ofReal (Real.sqrt (g.coeff x).det) ∂volume/-- The ordinary Euclidean unit ball volume, in Cartesian Lebesgue measure. -/def omega (n : ℕ) : ℝ≥0∞ :=  volume {x : Space n | (∑ i, x i ^ 2) < 1}/-- The volume-ratio limit is finite, positive, and strictly below one. -/def PositiveSubunitAVR {n : ℕ} (g : SmoothMetric n) : Prop :=  ∃ a : ℝ, 0 < a ∧ a < 1 ∧    Tendsto (fun R : ℝ => ballVolume g R / (omega n * ENNReal.ofReal (R ^ n)))      atTop (𝓝 (ENNReal.ofReal a))/-- Membership in the actual space H_k, with the bound required at every point. -/def HarmonicGrowth {n : ℕ} (g : SmoothMetric n) (k : ℕ) (u : Space n → ℝ) : Prop :=  ContDiff ℝ ∞ u ∧ (∀ x, laplaceBeltrami g u x = 0) ∧    ∃ C : ℝ, 0 ≤ C ∧ ∀ x, |u x| ≤ C * (1 + distance g 0 x) ^ kdef euclideanDimension (n k : ℕ) : ℕ :=  (n + k - 1).choose k + (n + k - 2).choose (k - 1)/-- The manuscript's unconditional main existence claim. -/def MainClaim : Prop :=  ∃ (n k : ℕ) (g : SmoothMetric n), Even n ∧ 8 ≤ n ∧ 2 ≤ k ∧    Complete g ∧ RicciNonnegative g ∧ EuclideanNearOrigin g ∧ PositiveSubunitAVR g ∧    ∃ u : Fin (euclideanDimension n k + 1) → (Space n → ℝ),      LinearIndependent ℝ u ∧ ∀ i, HarmonicGrowth g k (u i)axiom mainStatement : MainClaimtheorem main : MainClaim := mainStatementend HarmonicCounterexampleendend 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 counterexample to integer-degree harmonic dimension comparison

Theorem

There are an even integer n\geq8, an integer k\geq2, and a complete smooth metric g on \mathbb R^n such that \operatorname{Ric}_g\geq0 and

h_k(\mathbb R^n,g)>h_k(\mathbb R^n,g_{\mathrm E}).

The metric is Euclidean near the origin, and its asymptotic volume ratio satisfies

0<\lim_{R\to\infty} \frac{\operatorname{vol}_g B_g(0,R)}{\omega_nR^n}<1,

where \omega_n is the volume of the Euclidean unit ball.

Section 1, Introduction Read it in the paper TeX source, line 23

Show the TeX

There are an even integer $n\geq8$, an integer $k\geq2$, and a complete
smooth metric $g$ on $\mathbb R^n$ such that $\operatorname{Ric}_g\geq0$
and
\[
 h_k(\mathbb R^n,g)>h_k(\mathbb R^n,g_{\mathrm E}).
\]
The metric is Euclidean near the origin, and its asymptotic volume ratio
satisfies
\[
 0<\lim_{R\to\infty}
 \frac{\operatorname{vol}_g B_g(0,R)}{\omega_nR^n}<1,
\]
where $\omega_n$ is the volume of the Euclidean unit ball.

Configuration and file facts

File: lean/ComparatorChallenges/HarmonicGrowth.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Geometry/HarmonicGrowth/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: 19 local vocabulary declarations
Lines of Lean, without comments or blank lines: 58
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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

Plectis Atlas