Lean statement

Positivity of the explicit gauge on every positive-time segment

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.SLEExactGauge.sourceLowerMain_proved

Lean statement file: SLELowerPositivity.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 230 link it.

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 SourceLowerMainTarget : Prop :=  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →      0 < hausdorffGauge h (segment (γ ω) s t)⋮theorem sourceLowerMain_proved : SourceLowerMainTarget := 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.

definition OAI.SLEExactGauge.dimension line 13

noncomputable def dimension (κ : ℝ) : ℝ := 1 + κ / 8

definition OAI.SLEExactGauge.codimension line 15

noncomputable def codimension (κ : ℝ) : ℝ := ((2 : ℕ) : ℝ) - dimension κ

definition OAI.SLEExactGauge.upperHalfPlane line 17

def upperHalfPlane : Set ℂ := {z | 0 < z.im}

definition OAI.SLEExactGauge.survivingDomain line 19

def survivingDomain (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : Set ℂ :=  {z | z ∈ upperHalfPlane \ (γ '' Icc 0 t) ∧    ¬ Bornology.IsBounded (connectedComponentIn (upperHalfPlane \ (γ '' Icc 0 t)) z)}

definition OAI.SLEExactGauge.IsCapacityTwoWitness line 23

def IsCapacityTwoWitness (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)    (G F : ℝ≥0 → ℂ → ℂ) : Prop :=

Show the remaining 15 lines

  Continuous γ ∧ γ 0 = 0 ∧ (∀ t, 0 ≤ (γ t).im) ∧    (∀ t, DifferentiableOn ℂ (G t) (survivingDomain γ t)) ∧    (∀ t, DifferentiableOn ℂ (F t) upperHalfPlane) ∧    (∀ t, MapsTo (G t) (survivingDomain γ t) upperHalfPlane) ∧    (∀ t, MapsTo (F t) upperHalfPlane (survivingDomain γ t)) ∧    (∀ t z, z ∈ survivingDomain γ t → F t (G t z) = z) ∧    (∀ t z, z ∈ upperHalfPlane → G t (F t z) = z) ∧    (∀ z ∈ upperHalfPlane, G 0 z = z) ∧    (∀ t z, z ∈ survivingDomain γ t →      HasDerivWithinAt (fun u : ℝ => G (Real.toNNReal u) z)        (2 / (G t z - (U t : ℂ))) (Ici 0) (t : ℝ)) ∧    (∀ t, Tendsto (fun z : ℂ => z * (G t z - z))      (cocompact ℂ ⊓ 𝓟 (survivingDomain γ t)) (𝓝 (2 * (t : ℂ)))) ∧    (∀ t, Tendsto (fun y : ℝ => F t ((U t : ℂ) + (y : ℂ) * Complex.I))      (𝓝[>] 0) (𝓝 (γ t)))

definition OAI.SLEExactGauge.IsCapacityTwoTrace line 41

def IsCapacityTwoTrace (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) : Prop :=  ∃ G F : ℝ≥0 → ℂ → ℂ, IsCapacityTwoWitness U γ G F

definition OAI.SLEExactGauge.IsOrdinaryChordalSLE line 44

def IsOrdinaryChordalSLE {Ω : Type*} [MeasurableSpace Ω]    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (P : Measure Ω) : Prop :=  (∀ t, Measurable (fun ω => γ ω t)) ∧  ∃ B : ℝ≥0 → Ω → ℝ, ProbabilityTheory.IsBrownianReal B P ∧    ∀ᵐ ω ∂P, IsCapacityTwoTrace (fun t => Real.sqrt κ * B t ω) (γ ω)

structure OAI.SLEExactGauge.IsGauge line 50

structure IsGauge (h : ℝ → ℝ) : Prop where  continuousOn : ContinuousOn h (Ici 0)  monotoneOn : MonotoneOn h (Ici 0)  zero : h 0 = 0  positive : ∀ r, 0 < r → 0 < h r

definition OAI.SLEExactGauge.hFormula line 56

noncomputable def hFormula (κ r : ℝ) : ℝ :=  r ^ dimension κ * (Real.log (Real.log (1 / r))) ^ (codimension κ / ((2 : ℕ) : ℝ))

definition OAI.SLEExactGauge.HasSmallRadiusFormula line 59

def HasSmallRadiusFormula (h f : ℝ → ℝ) : Prop :=  h =ᶠ[𝓝[>] 0] f

definition OAI.SLEExactGauge.extendedGauge line 62

noncomputable def extendedGauge (h : ℝ → ℝ) (r : ℝ≥0∞) : ℝ≥0∞ :=  ENNReal.ofReal (h r.toReal)

definition OAI.SLEExactGauge.hausdorffGauge line 65

noncomputable def hausdorffGauge (h : ℝ → ℝ) : Measure ℂ :=  Measure.mkMetric (extendedGauge h)

definition OAI.SLEExactGauge.segment line 68

def segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t

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: real, capacity, plane, half, probability, ordinary, sle and chordal. Found in the paper’s title, which its main theorem states: gauge and hausdorff. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: kappa.

Statements in An explicit exact Hausdorff gauge for SLE come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of An explicit exact Hausdorff gauge for SLE

Theorem

Let \gamma be ordinary chordal \mathrm{SLE}_\kappa from 0 to \infty in \mathbb H, parametrized by half-plane capacity 2t, and set \Gamma=\gamma([0,\infty)). For each fixed 0<\kappa<8, on one event of probability one,

0<\mathcal H^h(\gamma([s,t]))<\infty \qquad\text{for every real }0<s<t<\infty.

For every deterministic 0<R<\infty,

\mathbb E\mathcal H^h(\Gamma\cap\overline B(0,R)) \le C_{\kappa,R}<\infty.

In particular, on the same probability-one event,

\mathcal H^h\bigl(\Gamma\cap([-m,m]+i[0,m])\bigr)<\infty \qquad(m=1,2,\ldots).

Section 1, Introduction Read it in the paper TeX source, line 66

Show the TeX

Let $\gamma$ be ordinary chordal $\mathrm{SLE}_\kappa$ from $0$ to
$\infty$ in $\mathbb H$, parametrized by half-plane capacity $2t$,
and set $\Gamma=\gamma([0,\infty))$. For each fixed $0<\kappa<8$,
on one event of probability one,
\[
 0<\mathcal H^h(\gamma([s,t]))<\infty
 \qquad\text{for every real }0<s<t<\infty.
\]
For every deterministic $0<R<\infty$,
\[
 \mathbb E\mathcal H^h(\Gamma\cap\overline B(0,R))
       \le C_{\kappa,R}<\infty.
\]
In particular, on the same probability-one event,
\[
 \mathcal H^h\bigl(\Gamma\cap([-m,m]+i[0,m])\bigr)<\infty
 \qquad(m=1,2,\ldots).
\]

Other possible paper matches, closest wording first

Corollary 2.6 of An explicit exact Hausdorff gauge for SLE. Words of the Lean statement, its names split into words, that also appear in the theorem: target, trace, surviving, domain, half, plane and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: time.
Proposition 7.2 of An explicit exact Hausdorff gauge for SLE. Words of the Lean statement, its names split into words, that also appear in the theorem: segment, capacity, upper, probability, bound, hausdorff and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: compact and time.
Theorem 1.1 of An exact Hausdorff gauge for SLE. Words of the Lean statement, its names split into words, that also appear in the theorem: real, capacity, gauge, plane, half, upper and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: kappa, surely and almost.
Corollary 1.2 of An explicit exact Hausdorff gauge for SLE. Words of the Lean statement, its names split into words, that also appear in the theorem: log, small, continuous and measure. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nondecreasing, sufficiently and radii.
Proposition 3.7 of An explicit exact Hausdorff gauge for SLE. Words of the Lean statement, its names split into words, that also appear in the theorem: complex, trace, continuous, ordinary, sle and chordal. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: kappa.
Proposition 3.9 of An explicit exact Hausdorff gauge for SLE. Words of the Lean statement, its names split into words, that also appear in the theorem: surviving, domain, ordinary, sle and chordal. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: kappa and time.

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 230’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: Only positivity is stated: SourceLowerMainTarget : ... ∀ h, IsGauge h → HasSmallRadiusFormula h (hFormula κ) → ∀ᵐ ω ∂P, ∀ s t, 0 < s → s < t → 0 < hausdorffGauge h (segment (γ ω) s t) with hFormula κ r = r^d (log log (1/r))^((2-d)/2), d = 1 + κ/8, for 0 < κ < 8 (docs: 'almost surely assigns positive measure'). The summary's finiteness half, 'positive finite measure' for every segment and finite expected measure of the trace in every bounded disk, has no Lean statement, so 'exact gauge' is only half-established.
Definitions the group asks readers to compare with the paper: SLE is not a Mathlib object: IsOrdinaryChordalSLE κ γ P := (∀ t, Measurable ..) ∧ ∃ B, IsBrownianReal B P ∧ ∀ᵐ ω, IsCapacityTwoTrace (√κ * B · ω) (γ ω), with IsCapacityTwoTrace an existential Loewner-chain witness (conformal maps G_t, F_t, ODE HasDerivWithinAt (G · z) (2 / (G t z - U t)), hydrodynamic normalization, F t (U t + iy) → γ t); hausdorffGauge h := Measure.mkMetric (ofReal ∘ h ∘ toReal). A plausible Loewner-characterization but a hand-written definition of SLE.
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: finiteness of the gauge measure on segments and finite expected measure in bounded disks (the upper half of the exact-gauge claim)

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves that every continuous nondecreasing Hausdorff gauge agreeing with $h(r)=r^d(\log\log(1/r))^{(2-d)/2}$ at sufficiently small positive radii almost surely assigns positive measure to every nontrivial compact positive-time segment of ordinary chordal $\mathrm{SLE}_\kappa$.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states part of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

85 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: 26 to its own declarations, 37 to Mathlib’s documentation. 15 declared names also link to the same declaration in OpenAI’s library. 3 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names 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 sourceLowerMain_proved 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 14 other declarations of the file; with the theorem itself they span 53 lines. 1 of them is named in the statement itself.

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

def dimensionline 13library
def codimensionline 15library
def upperHalfPlaneline 17library
def survivingDomainline 19library
def IsCapacityTwoWitnessline 23library
def IsCapacityTwoTraceline 41library
def IsOrdinaryChordalSLEline 44library
structure IsGaugeline 50library
def hFormulaline 56library
def HasSmallRadiusFormulaline 59library
def extendedGaugeline 62library
def hausdorffGaugeline 65library
def segmentline 68library
def SourceLowerMainTargetline 70in the statementlibrary
theorem sourceLowerMain_provedline 78selected theoremlibrary

import Mathlibnamespace OAI/-! Positivity of the explicit Hausdorff gauge on all positive-time SLE segments. -/noncomputable sectionopen Set Filter MeasureTheoryopen scoped Topology ENNReal NNRealnamespace SLEExactGaugeuniverse v_lowernoncomputable def dimension (κ : ℝ) : ℝ := 1 + κ / 8noncomputable def codimension (κ : ℝ) : ℝ := ((2 : ℕ) : ℝ) - dimension κdef upperHalfPlane : Set ℂ := {z | 0 < z.im}def survivingDomain (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : Set ℂ :=  {z | z ∈ upperHalfPlane \ (γ '' Icc 0 t) ∧    ¬ Bornology.IsBounded (connectedComponentIn (upperHalfPlane \ (γ '' Icc 0 t)) z)}def IsCapacityTwoWitness (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)    (G F : ℝ≥0 → ℂ → ℂ) : Prop :=  Continuous γ ∧ γ 0 = 0 ∧ (∀ t, 0 ≤ (γ t).im) ∧    (∀ t, DifferentiableOn ℂ (G t) (survivingDomain γ t)) ∧    (∀ t, DifferentiableOn ℂ (F t) upperHalfPlane) ∧    (∀ t, MapsTo (G t) (survivingDomain γ t) upperHalfPlane) ∧    (∀ t, MapsTo (F t) upperHalfPlane (survivingDomain γ t)) ∧    (∀ t z, z ∈ survivingDomain γ t → F t (G t z) = z) ∧    (∀ t z, z ∈ upperHalfPlane → G t (F t z) = z) ∧    (∀ z ∈ upperHalfPlane, G 0 z = z) ∧    (∀ t z, z ∈ survivingDomain γ t →      HasDerivWithinAt (fun u : ℝ => G (Real.toNNReal u) z)        (2 / (G t z - (U t : ℂ))) (Ici 0) (t : ℝ)) ∧    (∀ t, Tendsto (fun z : ℂ => z * (G t z - z))      (cocompact ℂ ⊓ 𝓟 (survivingDomain γ t)) (𝓝 (2 * (t : ℂ)))) ∧    (∀ t, Tendsto (fun y : ℝ => F t ((U t : ℂ) + (y : ℂ) * Complex.I))      (𝓝[>] 0) (𝓝 (γ t)))def IsCapacityTwoTrace (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) : Prop :=  ∃ G F : ℝ≥0 → ℂ → ℂ, IsCapacityTwoWitness U γ G Fdef IsOrdinaryChordalSLE {Ω : Type*} [MeasurableSpace Ω]    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (P : Measure Ω) : Prop :=  (∀ t, Measurable (fun ω => γ ω t)) ∧  ∃ B : ℝ≥0 → Ω → ℝ, ProbabilityTheory.IsBrownianReal B P ∧    ∀ᵐ ω ∂P, IsCapacityTwoTrace (fun t => Real.sqrt κ * B t ω) (γ ω)structure IsGauge (h : ℝ → ℝ) : Prop where  continuousOn : ContinuousOn h (Ici 0)  monotoneOn : MonotoneOn h (Ici 0)  zero : h 0 = 0  positive : ∀ r, 0 < r → 0 < h rnoncomputable def hFormula (κ r : ℝ) : ℝ :=  r ^ dimension κ * (Real.log (Real.log (1 / r))) ^ (codimension κ / ((2 : ℕ) : ℝ))def HasSmallRadiusFormula (h f : ℝ → ℝ) : Prop :=  h =ᶠ[𝓝[>] 0] fnoncomputable def extendedGauge (h : ℝ → ℝ) (r : ℝ≥0∞) : ℝ≥0∞ :=  ENNReal.ofReal (h r.toReal)noncomputable def hausdorffGauge (h : ℝ → ℝ) : Measure ℂ :=  Measure.mkMetric (extendedGauge h)def segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s tdef SourceLowerMainTarget : Prop :=  ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →  ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) →  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →      0 < hausdorffGauge h (segment (γ ω) s t)theorem sourceLowerMain_proved : SourceLowerMainTarget := by  sorryend SLEExactGaugeendend 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.

An exact Hausdorff gauge for SLE

Theorem (An exact moment gauge)

Fix 0<\kappa<8, and let \gamma be chordal \SLE_\kappa from 0 to \infty in the upper half-plane \HH, parametrized by half-plane capacity 2t. There is a deterministic Hausdorff gauge h_\kappa, depending only on \kappa, such that almost surely, simultaneously for every real 0<s<t<\infty,

0<\Haus^{h_\kappa}(\gamma([s,t]))<\infty.

Section 1, Introduction Read it in the paper TeX source, line 38

Show the TeX

Fix $0<\kappa<8$, and let $\gamma$ be chordal $\SLE_\kappa$ from
$0$ to $\infty$ in the upper half-plane $\HH$, parametrized by half-plane capacity $2t$.
There is a deterministic Hausdorff gauge $h_\kappa$, depending only on
$\kappa$, such that almost surely, simultaneously for every real
$0<s<t<\infty$,
\[
 0<\Haus^{h_\kappa}(\gamma([s,t]))<\infty.
\]

An explicit exact Hausdorff gauge for SLE

Theorem

0<\mathcal H^h(\gamma([s,t]))<\infty \qquad\text{for every real }0<s<t<\infty.

For every deterministic 0<R<\infty,

\mathbb E\mathcal H^h(\Gamma\cap\overline B(0,R)) \le C_{\kappa,R}<\infty.

In particular, on the same probability-one event,

\mathcal H^h\bigl(\Gamma\cap([-m,m]+i[0,m])\bigr)<\infty \qquad(m=1,2,\ldots).

Section 1, Introduction Read it in the paper TeX source, line 66

Show the TeX

Let $\gamma$ be ordinary chordal $\mathrm{SLE}_\kappa$ from $0$ to
$\infty$ in $\mathbb H$, parametrized by half-plane capacity $2t$,
and set $\Gamma=\gamma([0,\infty))$. For each fixed $0<\kappa<8$,
on one event of probability one,
\[
 0<\mathcal H^h(\gamma([s,t]))<\infty
 \qquad\text{for every real }0<s<t<\infty.
\]
For every deterministic $0<R<\infty$,
\[
 \mathbb E\mathcal H^h(\Gamma\cap\overline B(0,R))
       \le C_{\kappa,R}<\infty.
\]
In particular, on the same probability-one event,
\[
 \mathcal H^h\bigl(\Gamma\cap([-m,m]+i[0,m])\bigr)<\infty
 \qquad(m=1,2,\ldots).
\]

Corollary

Let g(r)=r^d\log\log(1/r) at sufficiently small radii, with any continuous nondecreasing completion. On the event in Theorem thm:main, the restriction of \mathcal H^g to every \gamma([s,t]), 0<s<t<\infty, is not sigma-finite; in particular its total measure is infinite.

Section 1, Introduction Read it in the paper TeX source, line 92

Show the TeX

Let $g(r)=r^d\log\log(1/r)$ at sufficiently small radii, with any
continuous nondecreasing completion. On the event in
Theorem~\ref{thm:main}, the restriction of $\mathcal H^g$ to every
$\gamma([s,t])$, $0<s<t<\infty$, is not sigma-finite; in particular
its total measure is infinite.

Configuration and file facts

File: lean/ComparatorChallenges/SLELowerPositivity.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 230 link it.
Solution module: OAI.Probability.SLE.LowerPositivity
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: 14 local vocabulary declarations
Lines of Lean, without comments or blank lines: 53
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