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Math-2874-A1 · revision r1 · on Math-2874

OpenAI claim summary for Math-2874

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.

The stationary vertex walk on the same FK--Ising maps is claimed to converge in actual time to Brownian motion of the $\sqrt3$-Liouville sphere, with edge-count acceleration. 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. Liouville Brownian motion as the walk limit on FK–Ising maps
  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

Liouville Brownian motion as the walk limit on FK–Ising maps

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The stationary vertex walk on the same FK--Ising maps is claimed to converge in actual time to Brownian motion of the $\sqrt3$-Liouville sphere, with edge-count acceleration.

Mathematical statement of the claimed result

Result 1.
For FK--Ising spherical maps sampled with weight $2^{\ell/2}$, use all-edge distance $d_n$, stationary degree measure $\mu_n(v)=\deg(v)/(2n)$, and any deterministic spatial scale $a_n\to0$ yielding the metric-measure limit $(S,D_h,\mu_h)$ of the ordinary unit-area $\sqrt3$-LQG sphere. Start any fixed number $r$ of conditionally independent simple walks from $\mu_n$, interpolated along unit cable edges. The source claims \[ (\overline M_n,a_nd_n,\mu_n;\overline X^{n,j}_{nt},\ 1\le j\le r) \Longrightarrow(S,D_h,\mu_h;B_t^{h,j},\ 1\le j\le r). \] Conditioned on $h$, the limit walks are independent Liouville Brownian motions started from $\mu_h$, with Dirichlet energy $\tfrac12\int|\nabla f|^2$. Paths converge uniformly on compact time intervals in the common metric embeddings; time acceleration is exactly $n$.

Outline of the argument

The spectral companion supplies unit conductivity, local energy recovery, and the coupled surface. Dirichlet-form and inverse-operator estimates give tight path laws and identify the limiting Markov process, retaining conditional independence for any fixed number of walks on that surface.

Formal verification

For family 211, the selected Lean theorem is the $q>4$ Brownian tree metric-measure limit, with degree measure and the $n^{-1/2}$ distance scale. The subcritical, critical-Liouville, walk, and spectral companions are not included. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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