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Of course that’s basically the same as the “you travel backwards in time if you move faster than light” in the sense that these are things the math does, if you put in these totally unphysical values, while the same set of rules that give this predictions say you can’t reach these values.
Not really, as negative temperature has been experimentally measured.
Ah, yes. You’re referring to these gases with an inverted energy distribution. I guess that’s a question of definition, since they don’t exist in the thermodynamic limit.