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An exact Hausdorff gauge for SLE

OpenAI · September 25, 2026

Lean

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Abstract

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.

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Notes

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Automatically collected notes (unreviewed)

work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

  • Dapeng Zhan, An Exact Hausdorff Measure as Natural Length for SLE (arXiv:2610.15066, 2026-10-10). Using the explicit-gauge companion's exactness and finite-expectation theorem, proves that its Hausdorff measure is a deterministic positive multiple of the Lawler–Rezaei natural length; it cites only that companion and does not identify this paper's moment-defined gauge. The author discloses GPT 6 assistance and personal checking of the manuscript. [family-level]

Formalization

  • OpenAI, Exact Hausdorff gauges for SLE — Lean scope note (OpenAI Math release, 2026-10-06). The documented formalization covers only almost-sure positivity of the explicit gauge $r^d(\log\log(1/r))^{(2-d)/2}$ on every nontrivial compact positive-time segment for fixed $0<\kappa<8$; it does not cover finiteness, finite expected measure, or the moment-defined gauge $h_\kappa$ of this paper. [family-level]
  • OpenAI math release, SLELowerPositivity.lean comparator statement (release comparator file). The Lean statement of the positivity theorem that the scope note lists as the comparator for this family. [family-level]

Possibly relevant literature (found by the agent, not checked)

  • Thomas Duquesne and Jean-François Le Gall, The Hausdorff measure of stable trees (arXiv:math/0509690, 2005; ALEA 2006). Finds exact Hausdorff measure functions for the Brownian continuum random tree, another random fractal; a possible point of comparison for this paper's construction of an exact gauge from moments of a local mass.
  • Zixuan He, Exact Hausdorff measure of SLE boundary contacts (Zenodo preprint, doi:10.5281/zenodo.23233357, 8 October 2026). Gives an exact iterated-logarithm gauge for the boundary-contact set of chordal SLE with $4<\kappa<8$, possibly relevant to this paper's bounded-box bound, which includes trace points on the real boundary. The record does not mention the OpenAI preprints.

Digests and expositions

  • Latent Seminar, An exact Hausdorff gauge for SLE (2026-10-06). This exposition contrasts the moment-defined and explicit gauges, outlines their mass and covering arguments, and states that it does not independently verify the complete proofs. [family-level]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@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}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/230.md. The Lean statements have not been checked by people against the paper's claims.

Family 230: Exact Hausdorff gauges for SLE

Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, $0\lt \kappa\lt 8$. 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.

Other papers in this family: