An exact Hausdorff gauge for SLE

Source and verification

This article explains family 230 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Hausdorff dimension identifies a critical power of length, but that power can assign zero measure to the entire curve. Family 230 claims an exact Hausdorff gauge for SLE, first by a deterministic construction and then through a regular logarithmic formula.

🏷️ Preliminaries: dimension, gauges, and random curves

Chordal ${\mathrm{SLE}}_{\kappa }$ in the upper half-plane is the curve generated by Loewner evolution driven by $\sqrt{\kappa }{B}_t$. For $0<\kappa <8$, its dimension is $d=1+\kappa /8$.

A Hausdorff gauge is a continuous nondecreasing function $h$ with $h(0)=0$ and $h(r)>0$ for $r>0$. Its measure optimizes the cost of arbitrarily fine countable covers:

$$\begin{array}{c}{\mathcal{H}}^h(E)\\ =\underset{\delta \downarrow 0}{\lim }\underset{\begin{array}{c}E\subseteq {\bigcup }_j{U}_j\\ \mathrm{diam}{U}_j\le \delta \end{array}}{\inf }\sum _{j}h(\mathrm{diam}{U}_j).\end{array}$$

The usual $d$-dimensional measure uses $h(r)=r^d$. A dimension theorem does not guarantee that this measure is positive and finite. The sources begin from the established fact that ordinary SLE has zero $d$-dimensional Hausdorff measure.

Hausdorff covers, Minkowski neighborhoods, and ordered partitions of a parametrized curve are different constructions of geometric size. Their correction factors need not agree.

🏷️ The two claimed exact gauges

The first companion constructs a deterministic gauge from moment integrals and finite batches of spatial scales. It claims positive finite measure simultaneously for every compact positive-time segment of the trace, together with finite expected measure of the entire trace in bounded boxes. (OpenAI, 2026)

The second companion claims the explicit small-radius formula

$$\begin{array}{c}h(r)=r^d(\log \log (1/r){)}^{(2-d)/2},\\ d=1+\kappa /8.\end{array}$$

One completes it continuously and monotonically at larger radii. For each fixed $0<\kappa <8$, on one probability-one event,

$$\begin{array}{c}0<{\mathcal{H}}^h(\gamma ([s,t]))<\infty \\ \text{for every }0<s<t<\infty .\end{array}$$

The paper also claims finite expected measure of the entire trace in every fixed bounded disk, including its boundary contacts. It does not obtain that spatial statement merely by adding bounds over time intervals. The relation between its regular gauge and the first paper’s constructed gauge remains open in the source. (OpenAI, 2026b)

🏷️ A consequence for a larger logarithmic correction

The source contrasts its formula with $g(r)=r^d\log \log (1/r)$. Since $1<d<2$, one has $h(r)/g(r)\to 0$. For any $\epsilon >0$, sufficiently small cover elements satisfy $h(r)\le \epsilon g(r)$.

If a set $A$ has finite ${\mathcal{H}}^g$-measure, applying this comparison to fine covers gives ${\mathcal{H}}^h(A)=0$. A positive-${\mathcal{H}}^h$ segment therefore cannot be a countable union of sets with finite ${\mathcal{H}}^g$-measure. Conditional on the claimed exact-gauge theorem, the larger-gauge measure is not sigma-finite on that segment.

This elementary argument explains why specifying the logarithmic power is substantive. A dimension statement cannot distinguish these gauges.

🏷️ Lower measure bounds from uniform tails

For a lower Hausdorff bound, one seeks a positive measure supported on a trace piece with mass in every small ball bounded by a constant times $h(r)$. The mass-distribution argument then compares any cover with that measure and forces positive total cost.

The explicit-gauge manuscript constructs approximate mass using stopped Green densities in a surviving domain after part of the curve has been explored. Write $M$ for this integrated stopped mass in unit-scale coordinates. With $p=2-d$, a central estimate is

$$\mathbb{P}\{M>u\}\le C\exp (-c{u}^{2/p}).$$

The constants must remain uniform in the permitted stopped history and final approximation mesh. Choosing $u$ of order $(\log j{)}^{p/2}$ at geometric spatial scale $r\asymp e^{-j}$ makes these exceptional probabilities summable after the appropriate localization. Since $\log j$ is comparable to $\log \log (1/r)$, the tail exponent explains the candidate correction.

🏷️ Upper covers and simultaneous statements

An upper bound needs a different construction: spatial balls carrying sufficiently high mass can cover the trace efficiently, while a fine residual cover controls points not captured by those balls. The source uses dense-visit estimates, finite batches, and actual stopped-domain restart identities to justify this procedure.

Uniform control near the real boundary is needed when $\kappa >4$, and a spatial estimate must include the whole trace rather than only an initial time segment. The proof also places all rational time intervals on one event before extending to arbitrary positive endpoints. Neither a favorable cover along one subsequence nor a lower mass estimate alone establishes an exact gauge.

🏷️ Key steps and proof outlines

Put $p=2-d\in (0,1)$. The regular gauge is determined by a matched pair of estimates: normalized mass $m^p$ can be created at probability cost $e^{-C{m}^2}$, and larger stopped masses have tails bounded by $e^{-c{u}^{2/p}}$. The two estimates serve different sides of the Hausdorff comparison.

Finite stopped changes of law retain the actual domain

For an interior live point $z$, use the density

$$M_t(z)=R_{D_t}(z{)}^{-p}{\sin }^{a_0}{\theta }_t(z),a_0=8/\kappa -1,$$

where $R_{D_t}$ is the full conformal radius and ${\theta }_t$ is the angle in tip-target half-plane coordinates. Stop when the radius reaches a prescribed positive level. The stopped density has its original expectation, so it defines a finite change of law. Under this centered law, radius-time angle evolves by

$$d{\theta }_s=\frac{\sqrt{\kappa }}{2}d{W}_s+\cot {\theta }_{s}ds.$$

An inverse-angle moment bound supplies a positive probability of a uniformly good angle after one unit of radius time, regardless of the starting angle. Every restart retains the actual surviving-domain map. For $\kappa >4$, the separate physical-disk estimate says that after a disk hit, exiting a much larger disk while the marked point remains live has probability at most $C{L}^{-c}$. This distinguishes trace approach from mere swallowing and controls later mesh refinements.

Uniform tails require control all the way to the final mesh

Let $W_r(z)$ be the density frozen at the first live trace approach to distance $r$, and zero if no such approach occurs. In a collared unit disk the key assertion is

$$\mathbb{P}\left\{\int W_{r_f}(z)dA(z)>C{\delta }^{-p}\right\}\le C{e}^{-c{\delta }^{-2}},0<r_f\le \delta .$$

The estimate is uniform in the arbitrarily fine prescribed $r_f$. For $4<\kappa <8$, choose geometric radii $r_j=r_0{q}^j$ ending exactly at $r_f$. A local window ends at the next approach, a large-disk exit, or death of the marked point. Its outputs record successful approach mass and a live outer-exit failure. Conditional first moments come from the stopped density; the disk rule makes the bad-exit coefficient decay with the chosen outer radius.

Spatial window concentration controls integrated outputs with errors $b_j=(r_j/\delta {)}^p/(j+1{)}^2$. With outer multiplier $L_j=L_{\ast }{q}^{-ϵj}$, $0<ϵ<1-p$, the failure exponent is at least a constant times

$${\delta }^{-2}\frac{q^{-2(1-p-ϵ)j}}{(j+1{)}^4}.$$

These failure probabilities sum uniformly in the number of mesh levels. For integrated density $S_j$ and normalized approached area $A_j$, the successful estimates give

$$\begin{array}{rl}{S}_{j+1}& \le S_j+{\alpha }_j{A}_j+e_j,\\ A_{j+1}& \le C_1{S}_j+{\alpha }_j{A}_j+e_j,\end{array}$$

with summable ${\alpha }_j=O(L_{\ast }^{-c}{q}^{ϵcj})$ and $e_j=O({\delta }^{-p}(j+1{)}^{-2})$. Set $U_j=\max (S_j,A_j/\Lambda )$ with $\Lambda \ge \max \{1,2C_1\}$, then choose $L_{\ast }$ large. The recursion becomes $U_{j+1}\le (1+\Lambda {\alpha }_j)U_j+e_j$, so $U_j=O({\delta }^{-p})$ uniformly. Taking $u\asymp {\delta }^{-p}$ yields the stretched-exponential tail $C{e}^{-c{u}^{2/p}}$ (OpenAI, 2026b).

Delete excessive squares to retain a Frostman measure

Construct approximate stopped-density measures carried near a fixed compact trace piece. Their first moments stay positive and second moments stay bounded. At dyadic scale $2^{-k}$, delete all mass in squares exceeding a large multiple of

$$2^{-kd}(\log k{)}^{p/2}.$$

The uniform mass tail, combined with the one-point approach factor, makes expected removed mass summable over scales and squares. The cancellation $d+p=2$ is what compensates for the number of planar squares. Increase the threshold constant and then the initial scale so the expected loss is smaller than half the original mass. Paley–Zygmund gives a positive probability of retaining a fixed positive amount.

On that event, bounded measures on a fixed compact set have a weakly convergent subsequence. Its limit is supported on the trace. At each tested scale every square satisfies the mass bound; an open ball meets only a bounded number of comparable squares. Portmanteau gives $\nu (B(x,r))\le Ch(r)$ without requiring zero mass on grid boundaries. Any fine cover then has cost at least $\nu (\text{trace})/C$, proving positive Hausdorff measure with positive probability.

To obtain almost-sure positivity on a fixed time interval, divide it into $n$ deterministic capacity slots. Each restarted half-plane experiment succeeds with conditional probability at least the same $p_{\ast }>0$. Its compact interior witness is mapped back by a conformal map with bounded local distortion, preserving positive gauge measure. Simultaneous failure costs at most $(1-p_{\ast }{)}^n$. Let $n\to \infty$, then intersect over rational intervals. Hydrodynamic composition preserves these capacity increments; no unstated random time change is used.

A serpentine visit explains the matching large-mass cost

The dense-visit construction guides the trace through an embedded ribbon of width $\ell \asymp 1/m$ with order $m^2$ successive short passages. Each stopped passage has a fixed positive conditional probability and contributes mass in a disjoint middle strip. Thus the event costs at most an exponential in the number of steps, while its normalized sausage mass is at least

$$c{m}^2{\ell }^d\asymp m^{2-d}=m^p.$$

The guide keeps the marked point connected to the target through fixed-clearance routes. Its angle and radius therefore remain bounded below until completion, so the finite centered change of law retains a constant fraction of the ordinary success probability. The event is measurable at a finite radius-clock endpoint and works at one prescribed physical sausage mesh uniformly over the stopped state. These qualifications permit its repeated use in the cover construction.

One common sausage measure pays for a finite cover

At radii $r_j=e^{-T_{0}j}$, test a batch $n/3\le j\le 2n/3$ with $m_n=c_0\sqrt{\log n}$. For small fixed $c_0$, each adapted test succeeds with conditional probability at least $n^{-1/2}$ and generates mass at least $c{r}_j^d(\log n{)}^{p/2}$. Choose one deterministic fine sausage mesh satisfying all cutoffs in this finite batch. Since preceding tests finish before the next starts, all failures have probability at most $C{e}^{-c\sqrt{n}}$; independence is unnecessary.

Call a disk a candidate when the final sausage measure at that common mesh exceeds the required mass. Select disjoint candidate disks greedily in decreasing radius order. Their triples cover all discarded candidates, and their total gauge cost is bounded by a constant times that one sausage measure’s mass in a fixed larger region.

For a visited fine cell with no candidate, all adapted tests failed. Returning from the centered law at a finite stop adds the one-point factor $r_n^p$. With $O(r_n^{-2})$ cells and gauge cost $O(r_n^d(\log n{)}^{p/2})$, the expected cleanup cost is

$$O\left((\log n{)}^{p/2}{e}^{-c\sqrt{n}}\right)\to 0.$$

The whole-trace sausage charge has uniformly bounded expectation even near the real line, because its first-moment integral contains $y^{-p}$ and $p<1$. This gives the spatial expected upper bound, including boundary contacts, without assuming sausage measures converge.

The earlier gauge uses finite moment batches instead

The first companion only needs positive finite moments $m_n=\mathbb{E}{L}^n$ of its disk-model mass and increasing approximations to $L$. It repeats dyadic thresholds $2^k$ with multiplicities proportional to $4^n{2}^{kn}/m_n$, cut off at $4m_{n+1}/m_n$. Summing dyadic powers proves both a large total success probability within each batch and summable probabilities of exceeding sixteen times its thresholds.

For the resulting deterministic thresholds $b_i$ and radii $r_i$, the gauge is the quadratic envelope

$$h_{\kappa }(r)=\underset{i}{\inf }{b}_i{r}_i^d\max \{1,(r/r_i{)}^2\}.$$

Nonzero batch multiplicities and Jensen’s inequality give $b_i\ge \min \{1,m_1/4\}>0$ uniformly. Together with this lower bound, positivity at fixed $r>0$ uses $d<2$: terms with $r_i\to 0$ grow like $r_i^{d-2}$. Dedicated cleanup entries force continuity at zero, and $h_{\kappa }(Cr)\le C^2{h}_{\kappa }(r)$ controls changes of cover diameter. The finite-batch method balances success and excessive mass without yielding the later explicit regular formula. No equivalence between these two gauges is inferred (OpenAI, 2026a).

See Also

• Family 223 discusses natural occupation measures obtained through Minkowski content, which are distinct from Hausdorff covers.
• Family 237 studies dimension-type power exponents without claiming an exact Hausdorff gauge.

🏷️ Manuscripts and verification scope

• An exact Hausdorff gauge for SLE: A moment-integral and finite-batch construction — pinned manuscript and companion source.
• An explicit exact Hausdorff gauge for SLE: A regular formula from quantitative tails and dense visits — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented selection proves positivity for the regular logarithmic gauge on compact positive-time segments. It does not document finiteness, the spatial expectation bound, or the complete exact-gauge assertion. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. An exact Hausdorff gauge for SLE: A moment-integral and finite-batch construction.

🐻  OpenAI 2026b. An explicit exact Hausdorff gauge for SLE: A regular formula from quantitative tails and dense visits.