The Isoperimetric Conjecture for the Cubic Flat Three-Torus

Formalization

• OpenAI math release, Lean scope notes for family 354: The isoperimetric profile of the cubic three-torus (lean/docs/354.md, October 2026). Describes the release's Lean formalization: existence of minimizers for every volume in (0,1), the profile, and the classification of minimizers including equality cases at the transition volumes 4π/81 and 1/π, with comparator statement CubicTorus.lean. [family-level]

Possibly relevant literature (found by the agent, not checked)

• F. Glaudo, On the Isoperimetric Profile of the Hypercube (arXiv:2305.20051, May 2023). Shows that subsets of the cube (0,1)^d with volume sufficiently close to 1/2 have relative perimeter at least 1; relates to the paper's discussion of near-half-volume results, which cites only Acerbi–Fusco–Morini, Hadwiger and Barthe–Maurey.
• L. Yu, An explicit dimension-free isoperimetric plateau in the cube (arXiv:2610.18351, October 2026). Proves that in every dimension the relative profile of the unit cube equals one, with coordinate slabs as the only minimizers, for volumes within 6.25×10⁻¹² of 1/2; relates to the paper's remark that earlier near-half-volume results came without effective bounds.
• L. Heller, S. Heller, M. Traizet, The Enclosed Volume for Periodic Constant Mean Curvature Surfaces (arXiv:2601.14935, January 2026). Gives a formula for the enclosed volume of lattice-periodic CMC surfaces and a counterexample concerning the isoperimetric problem in T²×ℝ; context for the periodic isoperimetric problem of Hauswirth–Pérez–Romon–Ros that the paper builds on.
• Emanuel Milman, Spectral and Isoperimetric Bounds on Flat Tori (arXiv:2608.13052, 2026-08-13). Its bounds for isoperimetric profiles and Cheeger constants on general flat tori provide context for the manuscript's claimed exact profile on the unit cubic three-torus.
• Ben Baker, Joseph Briggs, Manuel Fernandez V and Chris Wells, Isoperimetric Inequalities in the $\ell_{\infty}$ cube and torus (arXiv:2609.22456, 2026-09-18). Its exact isoperimetric inequality for cubes and tori in the $\ell_{\infty}$ metric provides a different-metric analogue of the manuscript's claimed Euclidean cubic-torus profile.

Digests and expositions

• Latent Seminar, Isoperimetry in the cubic three-torus: balls, tubes, and slabs (2026-10-07). This exposition traces the two-volume reduction, nested-section perimeter gain and finite arithmetic estimates, while explicitly distinguishing its selected-mechanism review from independent verification of the complete proof. [family-level]
• heyprof, Isoperimetry in the cubic torus (3D) (interactive exposition, undated). This interactive visualization illustrates the manuscript's ball–tube–slab profile, transition ties and complementary regions through periodic copies and a perimeter chart. [family-level]
• Gradient Damage, Sphere, Tube or Slab? The 3D Torus Geometry Problem (video, 2026-10-08). This short visualization of the reported result distinguishes physical boundaries from cuts through the displayed periodic cell and describes its shape transitions as illustrative. [family-level]