How negative temperatures can be possible has been treated on StackExchange before (several times in fact), but in light of some recent academic discussion, most of these answers seem to be possibly wrong or incomplete. The literature I am referring to is Dunkel & Hilbert, Nature Physics 10, 67 (2014) arXiv:1304.2066, where as I understand it, it is shown that negative temperatures are an artefact of choosing an incorrect definition of entropy. The Wikipedia article on the matter has also been amended to reflect this.
This was later challenged by similarly well-known scientists, in arXiv:1403.4299 where it was, among other things, pointed out that this argument is actually decades old (citing Berdichevsky et al., Phys. Rev. A 43, 2050 (1991)). The original authors quickly countered the arguments made in the comment, by what seems to be a rigorous treatment of the matter, arXiv:1403.6058. The first arXiv comment (arXiv:1403.4299) has been updated since and it still reads that "Obviously severe points of disagreement remain".
What I am asking, then, is whether someone on StackExchange might be able to shed some light on the matter as to how there can be a disagreement about something that seems should be a mathematical fact. I would also be interested in hearing whether changing the definition of entropy from that of Boltzmann to that due to Gibbs might potentially change any other results. Might for example the Wang-Landau algorithm be affected seeing that it does use the density of states and that you can never simulate infinite systems (although as I understand it, even in the present context with finite scaling you should be able to get consistent results)?
EDIT: An update on the matter for those who might care. arXiv:1407.4127 challenged the original paper and argued that negative temperatures ought to exist. They based their claims on their earlier experiments in Science 339, 52 (2013). A reply was offered in arXiv:1408.5392. More physicists keep joining in, arguing for arXiv:1410.4619 and against arXiv:1411.2425 negative temperatures.