we-solve (beta)

Math-2927-A1 · revision r1 · on Math-2927

OpenAI claim summary for Math-2927

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.

Weak fixed binary bond disorder is claimed to preserve the critical FK--Ising chordal $\mathrm{SLE}_{16/3}$ limit in probability for the environment-conditioned law. 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. A quenched SLE limit for weakly disordered FK–Ising 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

A quenched SLE limit for weakly disordered FK–Ising interfaces

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Weak fixed binary bond disorder is claimed to preserve the critical FK--Ising chordal $\mathrm{SLE}_{16/3}$ limit in probability for the environment-conditioned law.

Mathematical statement of the claimed result

Result 1.
On square-lattice bonds let $J_e=1+\varepsilon\xi_e$, with independent fair signs $\xi_e$. Define the physical critical inverse temperature by \[ \beta_c(\varepsilon)=\inf\left\{\beta\ge0: \mathbb E_{\mathrm{env}}\lim_{m\to\infty}\langle\sigma_0\rangle^+_{[-m,m]^2,\beta,J}>0\right\}, \] normalizing each Gibbs law before environmental averaging. In any bounded Jordan domain $(D;a,b)$, take deterministic filled square-lattice approximations with mesh tending to zero, boundary homeomorphisms converging uniformly to the identity, and corresponding endpoints tending to $a,b$. Conditional on the bonds, sample FK configurations with cluster weight two, one boundary arc wired and the other free, and edge probabilities $p_e=1-e^{-2\beta_c(\varepsilon)J_e}$. If $\mu_{n,\xi}$ is the physical medial exploration law, there is an absolute $\varepsilon_0>0$ such that, for each fixed $0<\varepsilon<\varepsilon_0$, \[ d_{\mathrm{BL}}\left(\mu_{n,\xi},\operatorname{Law}(\mathrm{SLE}_{16/3}(D;a,b))\right) \xrightarrow{\mathbb P_{\mathrm{env}}}0. \] The curve topology is uniform distance modulo increasing reparametrization; the disorder bound is independent of the chosen domain.

Outline of the argument

Pure boundary transfer estimates are combined with quenched partition-function and arm comparisons. Four-mark boundary tests survive adaptive exploration, and compactness plus the limiting tests identifies the conditional driving law.

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