we-solve (beta)

Math-2916-A1 · revision r1 · on Math-2916

OpenAI claim summary for Math-2916

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The explicit proposed SLE gauge is $r^d(\log\log(1/r))^{(2-d)/2}$, $d=1+\kappa/8$; the exponent-one alternative is claimed not sigma-finite on nontrivial segments. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. The explicit logarithmic Hausdorff correction for SLE
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

The explicit logarithmic Hausdorff correction for SLE

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The explicit proposed SLE gauge is $r^d(\log\log(1/r))^{(2-d)/2}$, $d=1+\kappa/8$; the exponent-one alternative is claimed not sigma-finite on nontrivial segments.

Mathematical statement of the claimed result

Result 1.
Let $\gamma$ be ordinary chordal $\mathrm{SLE}_\kappa$ in $\mathbb H$, from $0$ to $\infty$, parametrized with capacity $2t$, and write $\Gamma=\gamma([0,\infty))$. Fix $0<\kappa<8$, put $d=1+\kappa/8$, $p=2-d$, and define near zero \[ h(r)=r^d\left(\log\log(1/r)\right)^{p/2}. \] Complete this gauge continuously and monotonically at larger radii. Using diameter-based Hausdorff measure, almost surely \[ 0<\mathcal H^h(\gamma([s,t]))<\infty\quad\text{for every real }0<s<t<\infty. \] For each deterministic $R>0$, $\mathbb E\mathcal H^h(\Gamma\cap\overline B(0,R))\le C_{\kappa,R}<\infty$. In particular, the trace has finite $h$-measure in every rectangle $[-m,m]+i[0,m]$, simultaneously for integers $m\ge1$.

Outline of the argument

Localized stopped densities and Brownian boundary-occupation estimates give a uniform spatial tail bound. A positive mass with controlled ball density supplies the lower Hausdorff estimate, and carefully chosen spatial covers supply the upper bound; comparison of gauges then excludes the exponent-one choice.

Formal verification

For family 230, the selected statement proves positivity for the explicit iterated-logarithm gauge on every nontrivial compact SLE segment. Finiteness, expected box measure, and failure of the exponent-one gauge are beyond that entry. The listed Comparator files are SLELowerPositivity.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources