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Math-2932 · mathematics · Probability

GFF limits of weighted integer Lipschitz surfaces

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the source-defined integer Lipschitz height model and each fixed $x\in [1/\operatorname{sqrt}(2),1]$, is there a positive normalization sigma(x), independent of the admissible bounded planar domain and its inside lattice approximation, such that the height distribution divided by sigma(x) converges to the domain GFF along the full mesh sequence, with every joint smooth-test moment?

Short English statement

For the source-defined integer Lipschitz height model and each fixed $x\in [1/\operatorname{sqrt}(2),1]$, is there a positive normalization sigma(x), independent of the admissible bounded planar domain and its inside lattice approximation, such that the height distribution divided by sigma(x) converges to the domain GFF along the full mesh sequence, with every joint smooth-test moment?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2932-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2932

Weighted triangular integer Lipschitz heights are claimed to have a Dirichlet Gaussian field limit for every fixed edge-change weight $x\in[1/\sqrt2,1]$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

GFF limits of weighted integer Lipschitz surfaces

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Weighted triangular integer Lipschitz heights are claimed to have a Dirichlet Gaussian field limit for every fixed edge-change weight $x\in[1/\sqrt2,1]$.

Mathematical statement of the claimed result

Result 1.
Let $D$ be bounded and simply connected with $C^2$ Jordan boundary. Approximate it from inside by regions bounded by simple honeycomb cycles, containing every enclosed vertex and edge, with uniformly converging boundary parametrizations and eventual containment of every compact subset of $D$. On enclosed faces, sample integer heights with boundary value zero and $|h(u)-h(v)|\le1$ for adjacent faces, using weight \[ \mathbb P_{\delta,x}(h)=Z_{\delta,x}^{-1}x^{N_\delta(h)},\qquad N_\delta(h)=\sum_{\{u,v\}:u\sim v}\mathbf1_{\{h(u)\ne h(v)\}}. \] Each unordered adjacency is counted once. Extend $h_\delta$ constantly over faces and by zero elsewhere, so $h_\delta(f)=\int_Dh_\delta f$. For each fixed $x\in[1/\sqrt2,1]$, there is a domain-independent $\sigma(x)>0$ such that \[ h_\delta/\sigma(x)\Longrightarrow\Phi_D\quad\text{in }\mathcal D'(D), \] where the zero-Dirichlet GFF has covariance $(-\Delta_D)^{-1}$. All finite joint test-function distributions and mixed moments converge.

Outline of the argument

Conditional comparisons across exact cuts and shell-count estimates bound smeared moments. Reflected plane normalization fixes the variance, while separated-pin transfer and boundary-attachment estimates remove boundary errors and identify Gaussian moments, including the critical endpoint.

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.

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