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Schramm's Hausdorff-measure problem for SLE: an exact gauge function for the chordal SLE trace, 0 < kappa < 8

For 0<κ<80<\kappa<8 the chordal SLEκ\mathrm{SLE}_\kappa trace has Hausdorff dimension d=1+κ/8d=1+\kappa/8 (Beffara 2008), and Rezaei (2018) proved its dd-dimensional Hausdorff measure is zero. For a gauge hh (continuous, nondecreasing, h(0)=0h(0)=0) let Hh\mathcal H^h be the corresponding Hausdorff measure. Schramm, in his 2006 ICM collection of problems (Problem 7.1), asked whether the SLE trace has a sigma-finite Hausdorff measure and suggested rdlog⁡log⁡(1/r)r^d\log\log(1/r) as a possible gauge. Is there a deterministic gauge hh such that almost surely 0<Hh(γ([s,t]))<∞0<\mathcal H^h(\gamma([s,t]))<\infty for nontrivial trace segments, and if so, which?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Schramm-Loewner evolution; fractal geometry of random curves
Posed by
Oded Schramm, Conformally invariant scaling limits: an overview and a collection of problems (ICM 2006), Problem 7.1
Year posed
2006
Years open
20y
Solved
2026-09-26
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
24 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For fixed 0<κ<80<\kappa<8, d=1+κ/8d=1+\kappa/8, the gauge h(r)=rd(log⁡log⁡(1/r))(2−d)/2h(r)=r^d(\log\log(1/r))^{(2-d)/2} satisfies almost surely 0<Hh(γ([s,t]))<∞0<\mathcal H^h(\gamma([s,t]))<\infty for all 0<s<t<∞0<s<t<\infty, and EHh(Γ∩B‾(0,R))<∞\mathbb E\mathcal H^h(\Gamma\cap\overline B(0,R))<\infty. Corollary 1.2: Schramm's suggested gauge rdlog⁡log⁡(1/r)r^d\log\log(1/r) is not sigma-finite on any such segment, so the exponent he proposed is wrong. The September 25 companion first proved existence of an exact (non-closed-form) gauge. Not addressed: κ≥8\kappa\ge8, whether the two gauges are equivalent, and the relation of Hh\mathcal H^h to natural parametrization or Minkowski content.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts: 'An exact Hausdorff gauge for SLE' (September 25, 2026) answered the existence question with an implicitly defined moment-integral gauge; the principal manuscript (September 26, 2026) gives an explicit closed-form gauge with an independent proof and shows Schramm's suggested gauge is not sigma-finite.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against Schramm's Problem 7.1. The release has a Lean challenge (lean/ComparatorChallenges/SLELowerPositivity.json, declaration OAI.SLEExactGauge.sourceLowerMain_proved, solution module present at the pinned commit), not in the formalization catalogue. Its statement, read here, gives only the lower half: almost surely every gauge agreeing with hh near 0 gives positive measure to every positive-time segment. Finiteness, the expected bound in disks, and the companion's gauge are not formalized, so the headline is not covered and the tier stays unreviewed; not rebuilt here. Each κ\kappa is fixed; no uniformity in κ\kappa and no identification with natural parametrization is claimed.

Sources

Changelog1 change

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