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Math-2875-A1 · revision r1 · on Math-2875

OpenAI claim summary for Math-2875

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.

In the source setup the domain is the fixed unit square with its specified Dobrushin marks. The primary theorem gives the same q$\in$[1,4), c(q)n^(-d) occupation measure, ordered curve and same-sample Minkowski-content joint limit. This full classification uses that source-fixed domain, not an unstated every-domain extension. 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. Natural occupation measures of critical FK interfaces
  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

Natural occupation measures of critical FK interfaces

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For $1\leq q<4$, the rescaled edge-counting measure of the critical square FK interface is claimed to converge with the curve to its SLE Minkowski-content measure.

Mathematical statement of the claimed result

Result 1.
Fix $q\in[1,4)$, $p_q=\sqrt q/(1+\sqrt q)$, $\kappa=4\pi/\arccos(-\sqrt q/2)$, and $d=1+\kappa/8$. On mesh $1/n$ in $D=(0,1)^2$, use critical FK weights and wired/free boundary arcs between $(0,0)$ and $(1,1)$. Let $\eta_n$ be the oriented Dobrushin strand, and $z_{n,j}$ its medial-edge traversal midpoints, counting boundary traversals but not terminal half-edges. With $\Pi$ the coordinatewise projection onto $[0,1]^2$, the source claims a constant $c(q)>0$ such that \[ ([\eta_n],c(q)n^{-d}\sum_{j=1}^{N_n}\delta_{\Pi(z_{n,j})}) \Longrightarrow([\eta],\mu_\eta). \] Here $\eta$ is chordal $\operatorname{SLE}_\kappa$, curves retain order modulo increasing reparametrization, and measures converge weakly on $\overline D$. The limiting content measure is the weak limit of $r^{d-2}\Pi_*[\mathbf1_{\{\operatorname{dist}(z,\eta)<r\}}dA(z)]$, using whole-plane area. The two limit coordinates belong to the same random curve.

Outline of the argument

Row slicing and angular electric spectra determine Coulomb correlations with quantitative remainders. A two-arm outward coupling fixes the single-edge amplitude, and integrated correlation bounds identify the occupation measure and its total mass with one deterministic normalization.

Formal verification

Family 223 has no Lean scope page in this release. Its selected result is not presented as formalized. 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