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Math-2905-A1 · revision r1 · on Math-2905

OpenAI claim summary for Math-2905

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.

Uniform real triangular-lattice Lipschitz heights are claimed to give a Gaussian free field, and at a tuned two-arc amplitude their zero contour approaches chordal $\mathrm{SLE}_4$. 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. Interface and field scaling limits of triangular real Lipschitz surfaces
  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

Interface and field scaling limits of triangular real Lipschitz surfaces

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Uniform real triangular-lattice Lipschitz heights are claimed to give a Gaussian free field, and at a tuned two-arc amplitude their zero contour approaches chordal $\mathrm{SLE}_4$.

Mathematical statement of the claimed result

Result 1.
Let $D$ be bounded, smooth and simply connected, with two distinct marked boundary points, and let triangular-lattice polygons $D_\delta$ have uniformly converging marked boundaries. Sample real vertex heights with normalized Lebesgue measure on the polytope \[ h_\delta=\lambda\text{ on }A_{+,\delta},\quad h_\delta=-\lambda\text{ on }A_{-,\delta},\quad |h_\delta(x)-h_\delta(y)|\le1\ (x\sim y). \] Affine interpolation produces a zero-height chord $\gamma_\delta$ between the marks. There exist universal $A,\sigma>0$ and $\lambda\in(0,1/2]$ such that \[ \gamma_\delta\Longrightarrow\mathrm{SLE}_4(D;a,b),\qquad (h_\delta-\mathbb Eh_\delta)/\sigma\Longrightarrow\Phi_D. \] Curves use uniform distance modulo increasing reparametrization, and fields use distributional convergence with $\operatorname{Cov}\Phi_D=(-\Delta_D)^{-1}$. Writing $v=2/\sqrt3$, the constants satisfy \[ \sigma^2=(vA)^{-1},\qquad \lambda^2=\pi/(8vA),\qquad m_D(z)=\lambda[\omega_D(z,A_+)-\omega_D(z,A_-)], \] where $m_D$ is the limiting mean and $\omega_D$ denotes harmonic measure. The coefficient $A$ is determined implicitly by stationary cell flux, rather than by a claimed elementary value.

Outline of the argument

Local slack estimates and reflected stationary comparisons establish a tangent-flux cell problem. Its coefficient determines the bulk variance and tuned boundary amplitude; varying-continuum trace and conditional Dirichlet covariance estimates then identify both field and interface laws.

Formal verification

Family 232 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