An explicit exact Hausdorff gauge for SLE

OpenAI

For each fixed 0<κ<8, let d=1+κ/8. The September 26 manuscript claims that the explicit small-radius gauge r^d(log log(1/r))^((2−d)/2) gives positive finite Hausdorff measure to every compact positive-time segment of ordinary chordal SLEκ almost surely, with finite expected measure of the full trace inside bounded disks.

Sources for this record

What was already known

A published OpenAI record covers the September 25 existence result using a deterministic moment-integral gauge. Earlier work established dimension d and zero measure for the critical power gauge r^d.

AI involvement AI-led

OpenAI credits its internal model with both manuscripts. No named human checker or explainer of the full explicit-gauge theorem was documented.

OpenAI authorship and the cited production disclosure establish central internal-model work; autonomy of this individual result is not documented.

Importance Major

The later manuscript supplies a simple explicit formula for the gauge, beyond the published existence result.

Limitations

Lean checks a lower-positivity target for the explicit gauge, not the finite upper-measure or expectation claims. The two papers do not show their gauges are asymptotically equivalent.

Technical verification Partial

The cited Lean target proves a lower-positivity proposition conditional on its definitions of ordinary chordal SLE and the explicit gauge. It does not state the finite-measure upper bound, expected bounded-disk measure, or the moment-defined companion gauge. No completed check of the full theorem is documented.

Formal checking

Lean proof of lower positivity for the explicit gauge

Checker
Lean 4 kernel, as reported by the OpenAI repository
Scope
Positive Hausdorff gauge measure of each positive-time chordal SLE segment for the explicit formula under the formalized definitions
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Few

No named human explains the dense-visit, stopped-law and arbitrary-cover mechanism; Few is the evidence default.

Correction and publication history

2026-10-18 · publication

The later manuscript supplies a simple explicit formula for the gauge, beyond the published existence result.

Read this published revision