Exact Hausdorff gauges for SLE

What OpenAI claims

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

Papers (2)

1. An exact Hausdorff gauge for SLE

   25 September 2026 · Preprint, not peer reviewed · PDF

   For each 0 < κ < 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.

2. An explicit exact Hausdorff gauge for SLE

   26 September 2026 · Preprint, not peer reviewed · PDF

   For each fixed 0 < κ < 8, let d=1+κ/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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.