Exact Hausdorff gauges for SLE

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Resolves Schramm’s Hausdorff-measure question for chordal SLE_κ, $0\lt \kappa\lt 8$. The explicit gauge $r^d(\log\log(1/r))^{(2-d)/2}$, $d=1+\kappa/8$, gives almost surely positive finite measure to every trace segment $\gamma([s,t])$ with $0\lt s\lt t\lt \infty$, and finite expected measure to the trace in every bounded disk. (OpenAI’s Lean note)

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The paper states

Theorem 1.1 (An exact moment gauge)

In An exact Hausdorff gauge for SLE

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Corollary 1.2 in An explicit exact Hausdorff gauge for SLE () and Theorem 1.1 in An explicit exact Hausdorff gauge for SLE ().

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.

No Lean statement in the release is paired with this theorem.

Reported proof verification: One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations: Two current papers, each readable here as a page beside its PDF.

Papers

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An exact Hausdorff gauge for SLE

Undated, current edition.

Read it here PDF

For each $0\lt \kappa\lt 8$, we construct a deterministic Hausdorff gauge that almost surely assigns positive finite measure to every nontrivial compact positive-time segment of chordal Schramm–Loewner evolution. This answers Schramm’s Hausdorff-measure existence problem in this parameter range. The entire trace has finite expected gauge measure in each bounded box.

An explicit exact Hausdorff gauge for SLE

Undated, current edition.

Read it here PDF

For each fixed $0\lt \kappa\lt 8$, let $d=1+\kappa/8$. The gauge $h(r)=r^d(\log\log(1/r))^{(2-d)/2}$ at sufficiently small radii almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLE_κ. This gives an explicit solution to Schramm’s Hausdorff-measure problem in this parameter range. The entire trace has finite expected measure in every bounded disk. The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.

One Lean statement in this family

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`SLELowerPositivity`

The paper states

Theorem 1.1

In An explicit exact Hausdorff gauge for SLE

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).

The words they share: real, capacity, plane, half, probability, ordinary, sle, chordal, gauge, hausdorff, kappa.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Positivity of the explicit gauge on every positive-time segment

SLELowerPositivity.lean, the release lists it as a supporting statement.

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

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).

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.

“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.

Their method: 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.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).

The gap they name: finiteness of the gauge measure on segments and finite expected measure in bounded disks (the upper half of the exact-gauge claim)

“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

For $0<\kappa<8$, put $d=1+\kappa/8$. 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 $\mathrm{SLE}_\kappa$.

AI-drafted comparisons, with no recorded human review

Positivity of the explicit gauge on every positive-time segment SLELowerPositivity
Unreviewed AI draft: states part of a paper’s theorem. States the positivity half of the explicit-gauge theorem: the gauge r^d(log log(1/r))^{(2−d)/2} gives positive measure to every positive-time segment of chordal SLE_κ. Finiteness of that measure and the expectation bound are not included.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Positivity of the explicit gauge on every positive-time segment `SLELowerPositivity`

Two reports, all at the release’s recorded version.

Reported by	Their word	Their account	Date	Original report
cert-machine	“CERTIFIED”	Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.	10 October 2026	Original report
MathVet	“comparator-sandboxed-pass”	Re-run through Comparator in a sandboxed virtual machine: accepted.	8 October 2026	Original report

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Author not stated. Source

grwtsk, [AUDIT.00] Audit the formal trust surface of the OpenAI math release (issue 2), 9 October 2026. Source
The SLE exact-gauge development mixes conditional and unconditional claims through published-input hypotheses; a child issue plans to separate them.

Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The Lean statement gives positivity only and does not make the quantity an exact gauge as the paper does.

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