The Sandier-Serfaty conjecture: the triangular lattice minimizes planar Coulomb renormalized energy

Theorem 1.1: for every admissible curl-free field with unit background, $W(E)\ge W(E_\triangle)$, and $\inf W=W(E_\triangle)$ is finite, with no separation, periodicity or Voronoi assumption on the competitor. Theorem 1.2 gives the finite version: on every square torus with $n\ge2$ points the renormalized energy is at least $nW(E_\triangle)$. The proof builds one trial potential from Voronoi sectors on a minimizing torus configuration, with an exact identity leaving a nonnegative square, plus interval-arithmetic estimates. It does not classify minimizers (compact changes keep the energy per area) and does not treat the Riesz or higher-dimensional cases.

Theorem 1.1: for every admissible curl-free field with unit background, $W(E)\ge W(E_\triangle)$, and $\inf W=W(E_\triangle)$ is finite, with no separation, periodicity or Voronoi assumption on the competitor. Theorem 1.2 gives the finite version: on every square torus with $n\ge2$ points the renormalized energy is at least $nW(E_\triangle)$. The proof builds one trial potential from Voronoi sectors on a minimizing torus configuration, with an exact identity leaving a nonnegative square, plus interval-arithmetic estimates. It does not classify minimizers (compact changes keep the energy per area) and does not treat the Riesz or higher-dimensional cases.