# Exact Hausdorff gauges for SLE

> An explicit size rule gives every piece of an SLE random curve a positive, finite Hausdorff measure. Plain-English explainer, Lean status and sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-230/
Last updated: 2026-10-14

**An explicit size rule gives every piece of an SLE random curve a positive, finite Hausdorff measure.**

**Trust:** Formal proof listed · **Papers:** 2 · **First paper:** September 25, 2026 · **Field:** Probability and statistical mechanics · **Entry:** 230

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

SLE curves are random, very wiggly curves in the plane. Their dimension was known, but the matching power of distance gives them zero size. The papers claim a corrected size rule that gives every piece a positive, finite size, first implicitly and then as a formula. They also show that Schramm's own suggested rule is too large.

## At a glance

- **First step:** 2007 (Oded Schramm, Hausdorff-measure problem for SLE)
- **Significance:** Major. It answers Schramm's question with an explicit gauge and rules out his suggested candidate.
- **Trust:** Formal proof listed
- **Field:** Probability and statistical mechanics (2 papers)

## The explanation

To measure a smooth curve, cover it with small discs and add up their sizes. A curve of length 1 needs about 1/r discs of width r. Their widths add up to about 1, whatever r is. The power of r that makes the total settle down is the curve's dimension, here 1.

Some random curves are so wiggly that this fails. Schramm–Loewner evolution, or SLE, is a family of random curves in the plane. They describe boundaries in critical models of physics, such as percolation clusters. A parameter κ sets how wiggly the curve is. For κ between 0 and 8, Beffara showed that the dimension is d = 1 + κ/8[1]. So with κ = 4, d = 1.5.

One might then measure the curve with r^1.5 in place of r. Rezaei showed that this gives zero[2]. The pure power is slightly too small. It needs a tiny correction factor, and the full size rule is called a gauge. Schramm asked for a gauge that gives the curve a sensible, nonzero size[3]. He suggested r^d × log log(1/r).

The first paper claims that a suitable gauge exists for every κ between 0 and 8[4]. It gives every piece of the curve a positive, finite size. But the gauge is defined through integrals, and its shape need not be a neat formula[5].

The second paper names the gauge outright. It is h(r) = r^d × (log log(1/r))^((2−d)/2). It claims this gives every positive-time piece of the curve a positive, finite size[6]. The paper calls this an explicit solution to Schramm's problem[7]. The whole curve inside any bounded disc also has finite expected size[8].

Try κ = 4 and r = 10^(−6). Then log(1/r) is about 13.8, and its log is about 2.63. The correction is 2.63^0.25, about 1.27. Schramm's suggestion uses the full 2.63. The ratio of the two corrections is 2.63^(−0.75), about 0.48. At r = 10^(−100), it falls to about 0.28, and it keeps shrinking. So Schramm's gauge grows too fast in the limit. The paper claims it gives no sensible size on any piece of the curve[9].

The obstacle is that both directions need fine control at every scale. A lower bound needs a mass spread on the curve that is never too heavy in any small disc[10]. An upper bound needs covers whose total cost stays small on average[11]. For the lower bound, the paper deletes small squares that hold too much mass. The cost of reaching a square, its mass, and the number of squares balance out exactly[12]. With κ = 4, the exponents 0.5 and 1.5 add to 2, the dimension of the plane.

Other ways to measure SLE exist. The natural parametrization measures the curve along its own path. Holden and Yuan found a different log correction for ordered pieces of the curve. The gauge here belongs to arbitrary covers instead[13]. The argument needs no link to those other measures[14]. How the two exact gauges relate is still open[15].

How sure is this? Lean, a computer proof checker, has a formal version of one direction. For the explicit gauge, it states that every piece of the curve gets positive size, almost surely. The finite upper bound and the first paper's gauge are not in the scope note. No specialist review of these papers has been published.

**What it doesn't show.** This does not say how the explicit gauge relates to the first paper's gauge.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Major if the proofs hold. Lean has a formal version only of the positive lower bound for the explicit gauge; no specialist review is published.

## Words used

- **SLE:** Schramm–Loewner evolution: a family of random curves in the plane, set by a parameter κ.
- **Hausdorff dimension:** The power of size that makes the total of a fine cover settle to a nonzero, finite value, when one exists.
- **Gauge:** A size rule h(r) used in place of a pure power of the width r.
- **Hausdorff measure:** The smallest total gauge cost of covers by small sets, in the limit of finer covers.
- **Sigma-finite:** Splittable into countably many pieces of finite size.
- **Natural parametrization:** A way to measure an SLE curve along its own path, not by covers.

## The original papers (2)

- [An exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf) - September 25, 2026
- [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf) - September 26, 2026

- [Lean notes](https://github.com/openai/math/blob/main/lean/docs/230.md)

## Quoted lines from the paper (15)

[1] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> For ordinary chordal $\mathrm{SLE}_\kappa$, $0<\kappa<8$, Beffara identified the dimension $d=1+\kappa/8$; Rezaei proved that its $d$-dimensional Hausdorff measure is zero.

[2] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> For ordinary chordal $\mathrm{SLE}_\kappa$, $0<\kappa<8$, Beffara identified the dimension $d=1+\kappa/8$; Rezaei proved that its $d$-dimensional Hausdorff measure is zero.

[3] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> Schramm asked for a sigma-finite Hausdorff measure of the SLE trace and suggested the following candidate: \[ r^d\log\log(1/r). \]

[4] From [An exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf):

> This answers Schramm's Hausdorff-measure existence problem in this parameter range.

[5] From [An exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf):

> The gauge is explicit in terms of deterministic integrals, although its asymptotics need not have a regular form.

[6] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> 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 $\mathrm{SLE}_\kappa$.

[7] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range.

[8] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> The entire trace has finite expected measure in every bounded disk.

[9] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.

[10] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> For the lower Hausdorff bound we need a positive measure on a piece of the trace whose mass in every small ball is at most a constant times $h(r)$.

[11] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> For the upper bound we need spatial covers whose total $h$-cost has a bounded expectation.

[12] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> The powers cancel because $p+d=2$.

[13] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> The different logarithmic correction here belongs to arbitrary spatial covers.

[14] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> Our argument requires no identification of $\mathcal H^h$ with either natural parametrization or Minkowski content.

[15] From [An explicit exact Hausdorff gauge for SLE](https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf):

> The relation between the two exact gauges remains open.

## Other sources

- [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”
- [V. Beffara, “The dimension of the SLE curves”, Annals of Probability 36(4) (2008), 1421–1452](https://doi.org/10.1214/07-AOP364)

## Related

- [Gaussian fields and interfaces for triangular-lattice Lipschitz heights](https://closedform.nishachay.tech/e/openai-math-2026-232/) - Probability and statistical mechanics
- [Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling](https://closedform.nishachay.tech/e/openai-math-2026-223/) - Probability and statistical mechanics
- [Optimal logarithmic mixing of the Thorp shuffle](https://closedform.nishachay.tech/e/openai-math-2026-238/) - Probability and statistical mechanics
- [The Mézard–Parisi formula for diluted spin glasses](https://closedform.nishachay.tech/e/openai-math-2026-221/) - Probability and statistical mechanics
- [Sharp singularity rates for symmetric random sign matrices](https://closedform.nishachay.tech/e/openai-math-2026-239/) - Probability and statistical mechanics
- [Probability and statistical mechanics](https://closedform.nishachay.tech/science/mathematics/probability-and-statistical-mechanics/) - Field · 29 results
- [Open since the 2000s](https://closedform.nishachay.tech/open-since/2000s/) - Era · 48 results
- [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results
- [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20230%5D%20)