  In the theory of flat chains with coefficients in a normed abelian group, we
give a simple necessary and sufficient condition on a group element $g$ in
order for the following fundamental regularity principle to hold: if a
mass-minimizing chain is, in a ball disjoint from the boundary, sufficiently
weakly close to a multiplicity $g$ disk, then, in a smaller ball, it is a
$C^{1,\alpha}$ perturbation with multiplicity $g$ of that disk.
