The Hasse principle for size-minimizing flat chains

Let $[\Sigma]$ be an integral homology class, with reductions $[\Sigma]^{\mathbb{R}}$ and $[\Sigma]^{\mathbb{Z}/n\mathbb{Z}}$ in the corresponding coefficient systems. The Hasse principle for size-minimizing flat chains. $\mathbb{Z}$-size-minimizing representatives of $[\Sigma]$ exist and can be reconstructed from $\mathbb{R}$-size-minimizing representatives of $[\Sigma]^{\mathbb{R}}$ and $\mathbb{Z}/n\mathbb{Z}$-size-minimizing representatives of $[\Sigma]^{\mathbb{Z}/n\mathbb{Z}}$ for $n\in\mathbb{Z}_{\geq 2}$. Existence of integral size-minimizing representatives is described as a long-standing open problem because the trivial-norm flat-chain theory lacks compactness; finite-coefficient size minimizers do exist and have the regularity properties recalled immediately beforehand.