Sharp singular-set bounds for stationary integral varifolds

What OpenAI claims

Proves that every stationary integral m-varifold in a Euclidean open set has singular set of Hausdorff dimension at most m-1, a sharp bound in every positive dimension and codimension. On round spheres, the family also proves almost-everywhere regularity: the singular set has zero m-dimensional Hausdorff measure.

Papers (2)

1. A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold

   5 October 2026 · Preprint, not peer reviewed · PDF

   The singular set of every stationary integral m-varifold in an open Euclidean set has Hausdorff dimension at most m-1, in every positive dimension and codimension. This sharp bound proves the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis.

2. Almost-everywhere regularity of stationary integral varifolds

   23 September 2026 · Preprint, not peer reviewed · PDF

   We resolve the almost-everywhere regularity conjecture of Brena, Decio, and De Lellis for stationary integral varifolds of arbitrary positive dimension and codimension in Euclidean open sets. The singular set has zero measure in the dimension of the varifold: near almost every support point, the varifold is a constant positive integer multiple of a smooth embedded minimal submanifold. The corresponding statement also holds on round spheres.

Status

Preprint. The papers have 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.