Integral optimal flat-norm decompositions in codimension one

The real flat norm decomposes a current into a residual and the boundary of a filling, minimizing their total mass. We prove that every compactly supported integral $d$-current in $\R^{d+1}$ has an integral optimal decomposition, including when its boundary is nonzero. The filling is a scalar function of bounded variation. Rounding its values with one common shift preserves the mean filling mass and does not increase the mean residual mass. The integer multiplicities of the input are essential to the latter calculation. A compactness argument then gives an integral optimum.