The isoperimetric conjecture for the cubic flat three-torus
From OpenAI Aftermath
The isoperimetric conjecture for the cubic flat three-torus predicts the least perimeter of a region of prescribed volume in the torus R3/Z3. The predicted minimizers are balls, solid tubes about shortest closed geodesics, slabs between coordinate tori, and their complements. A manuscript dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims to prove the conjecture and to classify all minimizers. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- The ball–cylinder–slab prediction appears in the periodic isoperimetric problem of Hauswirth, Pérez, Romon and Ros, and in Ros's account of the cubic torus and the cube.
- Reflection relates the periodic problem to relative perimeter in a cube. The canonical regions then become an eighth ball about a vertex, a quarter cylinder about an edge, and a region cut off by a coordinate plane.
- Hadwiger proved that a coordinate cut minimizes at half volume in the cube, and Barthe and Maurey recovered this by Gaussian comparison. Morgan and Johnson's small-volume comparison implies that balls minimize for sufficiently small volumes. Acerbi, Fusco and Morini proved that coordinate slabs are the only minimizers near half volume. These results came without effective bounds reaching the predicted transitions.
- E. Milman obtained effective ranges for the canonical phases. Using the method of Hauswirth, Pérez, Romon and Ros, he made explicit the reduction of the profile conjecture to the two transition volumes 4π/81 and 1/π.
The original result
- Claim. The manuscript The Isoperimetric Conjecture for the Cubic Flat Three-Torus states that, for a volume V in (0, 1) and v = min(V, 1 − V), the least perimeter among finite-perimeter subsets of R3/Z3 of volume V is min{(36π)1/3v2/3, 2√(πv), 2}.
- For V at most 1/2, every minimizing region is, up to an isometry and a null set, a ball when V ≤ 4π/81, a solid circular tube about a shortest closed geodesic when 4π/81 ≤ V ≤ 1/π, and a slab between coordinate tori when 1/π ≤ V ≤ 1/2.
- Both adjacent types occur at each transition volume, and it states that there are no further equality cases. For V above 1/2 the minimizers are the complements.
- Method, as described: Milman's reduction leaves endpoint exclusions at the two transition volumes, which the manuscript establishes directly. After Hsiang reflection and Ritoré's coordinate graph property, a nonstandard minimizer reduces to two vertex patterns. Flux, projection and coarea identities for planar sections give scalar inequalities that exclude most configurations. The main new ingredient is a quantitative use of nested planar sections at the cylinder–slab transition, with certified one-variable estimates in an appendix.
- Source. At commit
1581d4c: The Isoperimetric Conjecture for the Cubic Flat Three-Torus. - Formalization, as described by the repository. The scope note lean/docs/354.md states that the Lean formalization determines the profile and all minimizers in the unit cubic flat three-torus, including all equality cases at the transition volumes. The Comparator challenge is
CubicTorus.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 22 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | B. Deng, J. Li, Y. Liu, Relative isoperimetry in the three-dimensional cube: classification and quantitative stability (arXiv:2610.08644v1) | Parallel (does not cite the manuscript) | Determines the relative isoperimetric profile of the unit cube and classifies all minimizing sets, including the equality cases at the transition volumes 4π/81 and 1/π: corner eighth-balls, edge quarter-cylinders and coordinate slabs. It proves that the perimeter deficit controls the squared L1 distance to the minimizing family, with optimal exponent two. As consequences it states Milman's Conjecture 1.1 for every cuboid (0, β) × (0, 1)2 with 0 < β ≤ 1 and the three-dimensional case of Brezis's Open Problem 10.1. Its declaration of AI use states that the authors used ChatGPT for language editing, exploration of some proof strategies and calculations. | arXiv preprint, version 1. |
| 2026-10-10 | cert-machine, openai/math lane K: run 38040214623 for CubicTorus |
Verification (two kernels) | Records that Lean's default kernel and the nanoda kernel accept the solution, in 2,420 s. | Repository. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_260 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap and lists no issues after reading the whole manuscript, including both appendices, step by step. | Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The manuscript itself notes that reflection relates the torus problem to relative perimeter in the cube, the setting of the paper by Deng, Li and Liu.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
