An explicit dimension-free isoperimetric plateau in the cube

We prove that, in every dimension, the relative Euclidean isoperimetric profile of the unit cube equals one whenever the prescribed volume differs from one half by at most $6.25\times {10}^{-12}$. The only minimizers in this interval are coordinate slabs. This answers Glaudo’s question on the existence of a dimension-free central plateau. The proof combines a dimension-free local optimality estimate for slabs, established through BV slicing and a small-support estimate, with a quantitative Gaussian rigidity argument that forces the normal of an approximating half-space to be close to a coordinate direction.