we-solve (beta)

Math-2649 · mathematics · Geometry

Triangular minimum of planar Coulomb renormalized energy

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Let $\Lambda\subseteq\mathbb R^2$ be locally finite, with no multiplicities and bounded centered-disk point density. Let $E$ be locally integrable, with $\operatorname{div}E=2\pi(\sum_{p\in\Lambda}\delta_p-1)$ and $\operatorname{curl}E=0$ in the distributional sense. For each $R$, choose a smooth cutoff $\chi_R$ supported on $[-R,R]^2$, equal to one on $[-R+1,R-1]^2$, taking values between zero and one, and with gradients uniformly bounded over $R$. Define \[W(E)=\limsup_{R\to\infty}\frac{1}{4R^2}\lim_{\eta\downarrow0}\left[\frac12\int_{\mathbb R^2\setminus\bigcup_{p\in\Lambda}B(p,\eta)}\chi_R|E|^2+\pi\log\eta\sum_{p\in\Lambda}\chi_R(p)\right].\] Must $W(E)$ be at least the finite value for the covolume-one triangular periodic field, for every such field and every such cutoff family?

Short English statement

Let $\Lambda\subseteq\mathbb R^2$ be locally finite, with no multiplicities and bounded centered-disk point density. Let $E$ be locally integrable, with $\operatorname{div}E=2\pi(\sum_{p\in\Lambda}\delta_p-1)$ and $\operatorname{curl}E=0$ in the distributional sense. For each $R$, choose a smooth cutoff $\chi_R$ supported on $[-R,R]^2$, equal to one on $[-R+1,R-1]^2$, taking values between zero and one, and with gradients uniformly bounded over $R$. Define \[W(E)=\limsup_{R\to\infty}\frac{1}{4R^2}\lim_{\eta\downarrow0}\left[\frac12\int_{\mathbb R^2\setminus\bigcup_{p\in\Lambda}B(p,\eta)}\chi_R|E|^2+\pi\log\eta\sum_{p\in\Lambda}\chi_R(p)\right].\] Must $W(E)$ be at least the finite value for the covolume-one triangular periodic field, for every such field and every such cutoff family?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2649-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2649

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 090; 2026-09-23).

Triangular minimum of planar Coulomb renormalized energy

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 090; 2026-09-23), the asserted result is that the triangular lattice minimizes planar Coulomb renormalized energy over the stated curl-free fields with unit background, with the spherical logarithmic-energy consequence.

Mathematical statement of the claimed result

Manuscript 1.
Let $\Lambda\subset\mathbb R^2$ be locally finite with no multiplicities and bounded density in centered disks. Admit locally integrable fields $E$ satisfying, distributionally, \[ \operatorname{div}E=2\pi(\sum_{p\in\Lambda}\delta_p-1),\qquad \operatorname{curl}E=0. \] For $K_R=[-R,R]^2$, take smooth $0\le\chi_R\le1$ supported in $K_R$, equal to one on $K_{R-1}$ and with uniformly bounded gradients. Define \[ W(E,\chi)=\lim_{\eta\downarrow0} \left[\frac12\int_{\mathbb R^2\setminus\bigcup_{p\in\Lambda}B(p,\eta)}\chi|E|^2 +\pi\log\eta\sum_{p\in\Lambda}\chi(p)\right], \qquad W(E)=\limsup_{R\to\infty}\frac{W(E,\chi_R)}{|K_R|}. \] For the unit-density triangular lattice \[ \Lambda_{\triangle}=\sqrt{2/\sqrt3}\,[\mathbb Z(1,0)+\mathbb Z(1/2,\sqrt3/2)], \] let $E_{\triangle}=-\nabla h$ with the periodic mean-zero solution $-\Delta h=2\pi(\delta_0-1)$. The claim, for every allowed field and every such cutoff family, is \[ W(E)\ge W(E_{\triangle})<\infty, \qquad\inf_E W(E)=W(E_{\triangle}). \] The cited spherical consequence concerns the ordered-pair logarithmic energy on the unit sphere: \[ \min_{y_1,\ldots,y_n\in S^2}-\sum_{i\ne j}\log\|y_i-y_j\| =\left(\frac12-\log2\right)n^2-\frac n2\log n+C n+o(n), \] \[ C=2\log2+\frac12\log(2/3) +3\log\left(\frac{\sqrt\pi}{\Gamma(1/3)}\right). \] The norm is Euclidean, collisions have infinite energy, and $n\to\infty$ through integers.

Outline of the argument

For manuscript 1 (family 090; 2026-09-23), a glued trial potential is built from Voronoi sectors with matching logarithmic singularities. An exact dual identity expresses the energy as sector contributions plus a nonnegative Dirichlet error; interval-certified scalar inequalities calibrate the sectors to the triangular cell.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 090; 2026-09-23).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

Go to contribution →