Alright, let me see if I understand this:
Temperature is defined as: 1/T = del(S)/del(E).
Normally, one would expect Entropy to increase as Energy increases, thus positive Temperatures. However, as described in the paper, if you are apple to assemble a system in which High Energy States are more filled than Low Energy States, increasing Energy will decrease Entropy, causing negative Temperature.
While certainly interesting, this doesn't actually contradict any laws of Thermodynamics, specifically:
"The entropy of a perfect crystal at absolute zero is exactly equal to zero" is still true, and:
"It is impossible by any procedure, no matter how idealized, to reduce the temperature of any system to zero temperature in a finite number of finite operations" is still true.
Do I have this right?