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An explicit exact Hausdorff gauge for SLE
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Abstract
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 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.
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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). Taking this paper's gauge $r^d(\log\log(1/r))^{(2-d)/2}$ and its exactness and finite-expectation theorem (Theorem 1.1) as input, proves that the resulting Hausdorff measure is a deterministic positive multiple of the Lawler–Rezaei occupation measure for $0<\kappa<8$. The author discloses GPT 6 assistance and personal checking of the manuscript.
Formalization
- OpenAI, Exact Hausdorff gauges for SLE — Lean scope note (OpenAI Math release, 2026-10-06). The documented formalization covers only the lower bound: almost surely, every gauge agreeing with $h$ at small radii gives positive measure to every nontrivial compact positive-time segment, for fixed $0<\kappa<8$; it does not cover finiteness or the expected-measure bound. [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 result. [family-level]
Related prior literature mentioned by others
- Gregory F. Lawler and Mohammad A. Rezaei, Minkowski content and natural parameterization for the Schramm–Loewner evolution (Annals of Probability 43(3), 2015; arXiv:1211.4146). Zhan's follow-up uses this paper's occupation measure and natural-parametrization identities to identify the Hausdorff measure of this paper's gauge up to a deterministic positive constant.
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 gauges with iterated-logarithm corrections for another random fractal, the Brownian continuum random tree; a possible point of comparison for how this paper derives its fractional $\log\log$ exponent from mass tails.
- Mohammad A. Rezaei and Dapeng Zhan, Green's function for chordal SLE curves (arXiv:1607.03840, PTRF 2018). Proves that multipoint Green's functions exist, with up-to-constant bounds, possibly relevant to the multipoint and stopped-Green-density estimates behind the tail $e^{-cu^{2/p}}$; this paper cites the earlier Rezaei–Zhan moments paper but not this one.
- Zixuan He, Exact Hausdorff measure of SLE boundary contacts (Zenodo preprint, doi:10.5281/zenodo.23233357, 8 October 2026). Gives an exact gauge of the form $r^\delta(\log\log(1/r))^\beta$ for the boundary-contact set of chordal SLE with $4<\kappa<8$, a boundary analogue of this paper's interior result. 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
- VibeMathed: Problem record
- Emergent Mind: family 230
Independent sites; their content is not reviewed here.
Cite
@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 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:
- An exact Hausdorff gauge for SLE · September 25, 2026