Measure 0 of the singular set for 2-valued stationary hypercurrents

The paper’s Theorem 1.1 exactly asserts that if T is an m-dimensional stationary integral current in U ⊂ R^{m+1} with density Θ_T ≤ 2, then T is smooth embedded off a closed set Sing(T) with H^m(Sing(T)) = 0; moreover, there exists an open O with ∥T∥(U\O) = 0 and dim(Sing(T) ∩ O) ≤ m−1 . The proof reduces this to a small-excess regime (Theorem 3.1) and uses a topological separation lemma and a Gehring-type higher-integrability estimate to build Almgren–De Lellis–Spadaro-type approximations and conclude the dimension bound on a set of full ∥T∥-measure, together with the measure-zero singular set globally . By contrast, the model’s outline incorrectly treats the global Hausdorff dimension bound dim_H(Sing) ≤ m−1 and H^m(Sing)=0 as classical consequences of stationarity plus Allard’s ε-regularity; in fact, the paper emphasizes that such conclusions require multiplicity-2 and orientability and new arguments, and it highlights Brakke-type constructions where large singular sets occur for varifolds without these hypotheses . The model also incorrectly claims that a density bound Θ ≤ 2 rules out triple junctions, which it does not in general for varifolds (e.g., Y × R^{m−1} has density 3/2).