The isoperimetric profile of the cubic three-torus

What OpenAI claims

Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are 4π/81 and 1/π, with exactly the adjacent two types minimizing at each transition. (Lean)

Paper

1. The Isoperimetric Conjecture for the Cubic Flat Three-Torus

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove the isoperimetric conjecture for the cubic flat three-torus and classify all minimizers, including the equality cases at the transition volumes 4π/81 and 1/π. The minimizing regions are balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.