OpenAI · 354-01 · 2026-09-24 · Differential geometry
The Isoperimetric Conjecture for the Cubic Flat Three-Torus
preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026
Family 354. Isoperimetric regions in the cubic three-torus.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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 and , with exactly the adjacent two types minimizing at each transition.
We prove the isoperimetric conjecture for the cubic flat three-torus and classify all minimizers, including the equality cases at the transition volumes and . The minimizing regions are balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 354: The isoperimetric profile of the cubic three-torus. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.