OpenAI · 230-02 · 2026-09-26 · Probability and statistical mechanics
An explicit exact Hausdorff gauge for SLE
preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026
Family 230. Exact Hausdorff measure for SLE.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Resolves Schramm’s Hausdorff-measure question for chordal , . The explicit gauge , , gives almost surely positive finite measure to every trace segment with , and finite expected measure to the trace in every bounded disk.
For each fixed , let . The gauge 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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 230: Exact Hausdorff gauges for SLE. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.