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Math-2933 · mathematics · Probability

Critical center magnetization in fixed-boundary XY boxes

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Assuming the explicitly scoped height, renormalization and spin-field companion inputs, is the center magnetization $a_n$ in the critical planar XY fixed-boundary box asymptotic to A $n^{-1/8}(\log $ $n)^{1/16}$, for a positive model-dependent A?

Short English statement

Assuming the explicitly scoped height, renormalization and spin-field companion inputs, is the center magnetization $a_n$ in the critical planar XY fixed-boundary box asymptotic to A $n^{-1/8}(\log $ $n)^{1/16}$, for a positive model-dependent A?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2933-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2933

For aligned square-boundary XY spins, the proposed center magnetization is $A n^{-1/8}(\log n)^{1/16}(1+o(1))$, conditional on the listed companion inputs. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Critical center magnetization in fixed-boundary XY boxes

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For aligned square-boundary XY spins, the proposed center magnetization is $A n^{-1/8}(\log n)^{1/16}(1+o(1))$, conditional on the listed companion inputs.

Mathematical statement of the claimed result

Result 1.
For angles $\theta_x\in\mathbb R/(2\pi\mathbb Z)$ on $\Lambda_R=[-R,R]^2\cap\mathbb Z^2$, use the free-boundary cosine law proportional to $\exp[b\sum_{\{x,y\}\subset\Lambda_R,\,|x-y|=1}\cos(\theta_x-\theta_y)]$, relative to product uniform angles. Count each edge once and omit exterior edges. Define \[ C_b(r)=\lim_{R\to\infty}\mathbb E_{R,b}^{\mathrm{free}}\cos(\theta_0-\theta_{re_1}),\quad m(b)=\lim_{r\to\infty}-r^{-1}\log C_b(r),\quad b_c=\inf\{b>0:m(b)=0\}. \] The thermodynamic limit precedes the separation limit. In $\Lambda_n$, now fix every boundary angle to zero and retain all internal bonds, including those incident to the boundary. Put $a_n=\mathbb E_{n,b_c}^{0}\cos\theta_0$. Using the companion height and spin-field constructions at critical height coefficient $a(b_c)=8\pi$, together with their finite Gaussian comparison, separated-pin restoration and local partition-map results, the theorem supplies $A_{\mathrm{XY}}>0$ with \[ a_n=A_{\mathrm{XY}}n^{-1/8}(\log n)^{1/16}(1+o(1)). \] The normalization is the actual centre magnetization, with inverse temperature fixed at the mass-defined $b_c$.

Outline of the argument

Critical height, local renormalization, and spin-field estimates compare the boundary-pinned observable across scales. The Green-function normalization separates the universal power and marginal logarithm from the positive square-domain amplitude.

Formal verification

Family 216 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

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Sources

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