This paper completes the classification of the set S of all real parameter pairs (α, β) such that the dilated floor functions fα(x) = ⌊αx⌋, f β (x) = ⌊βx⌋ have a nonnegative commutator, i.e. [fα, f β ](x) = ⌊α⌊βx⌋⌋ − ⌊β⌊αx⌋⌋ ≥ 0 for all real x. It treats the negative dilation case, where both α, β < 0. This result is equivalent to classifying all positive α, β satisfying ⌊α⌈βx⌉⌋ − ⌊β⌈αx⌉⌋ ≥ 0 for all real x. The classification analysis for negative dilations is connected with the theory of Beatty sequences and with the Diophantine Frobenius problem in two dimensions.