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Math-2932-A1 · revision r1 · on Math-2932

OpenAI claim summary for Math-2932

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.

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.

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

Manuscript

Contents
  1. GFF limits of weighted integer 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

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.

Sources