Result 230, Probability and statistical mechanics

Exact Hausdorff gauges for SLE

Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, 0<κ<80\lt \kappa\lt 8. The explicit gauge rd(log⁡log⁡(1/r))(2−d)/2r^d(\log\log(1/r))^{(2-d)/2}, d=1+κ/8d=1+\kappa/8, gives almost surely positive finite measure to every trace segment γ([s,t])\gamma([s,t]) with 0<s<t<∞0\lt s\lt t\lt \infty, and finite expected measure to the trace in every bounded disk.

Lean formalization Proof

The bigger picture

Why it matters

SLE describes random planar curves whose irregularity makes ordinary length inadequate. The manuscripts report a precise way to measure their fractal size, capturing information that dimension alone does not provide.

What changes?

For chordal SLE, which joins boundary points, fix kappa strictly between 0 and 8 and set d = 1 + kappa/8. A Hausdorff gauge specifies the cost of covering a curve with small pieces. The reported gauge is r^d times (log(log(1/r)))^((2-d)/2) for sufficiently small diameters r. Almost surely, it assigns positive finite measure to every trace segment between times 0 < s < t < infinity. The entire trace has finite expected measure inside every bounded disk.

What does that help mathematicians do?

The explicit manuscript also reports that Schramm's suggested gauge, with iterated-logarithm exponent one, is not sigma-finite on any such segment: the segment cannot be covered by countably many sets of finite measure. This distinguishes a usable size measure from an excessively large one. Researchers therefore gain both a precise normalization for measuring these curves and a concrete reason to reject the proposed alternative.

Are there practical applications?

The immediate value is foundational in probability and statistical mechanics: a calibrated notion of size for these random curves, rather than a demonstrated technology. Finite expected measure in bounded disks allows researchers to average this size and bound the probability of unusually large values. These are mathematical consequences, not evidence of a practical simulation method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

An exact Hausdorff gauge for SLE

September 25, 2026 37 pages

For each 0<κ<80\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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-exact-Hausdorff-gauge-for-SLE-September-25-2026,
  author = {{OpenAI}},
  title = {{An exact Hausdorff gauge for SLE: A moment-integral and finite-batch construction}},
  howpublished = {OpenAI Math Release preprint
                  \href{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}{OAI:An-exact-Hausdorff-gauge-for-SLE-September-25-2026}},
  year = {2026}
}

An explicit exact Hausdorff gauge for SLE

September 26, 2026 49 pages

For each fixed 0<κ<80\lt \kappa\lt 8, let d=1+κ/8d=1+\kappa/8. The gauge h(r)=rd(log⁡log⁡(1/r))(2−d)/2h(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 SLEκ. 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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026,
  author = {{OpenAI}},
  title = {{An explicit exact Hausdorff gauge for SLE: A regular formula from quantitative tails and dense visits}},
  howpublished = {OpenAI Math Release preprint
                  \href{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}{OAI:An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/230.md.

Exact Hausdorff gauges for SLE

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

For 0<κ<80<\kappa<8, put d=1+κ/8d=1+\kappa/8. The formalization proves that every continuous nondecreasing Hausdorff gauge agreeing with h(r)=rd(log⁡log⁡(1/r))(2−d)/2h(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 SLEκ\mathrm{SLE}_\kappa.

Comparator links

Result Comparator statement
Positivity of the explicit gauge on every positive-time segment SLELowerPositivity.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.