If the temperature wraps around to negative, doesn't that imply a maximum value at the wrap point? Do we know the temperature value at which it wraps to negative? Is it literally "infinity Kelvin?"
For thinking about this point, it's much easier to talk about "thermodynamic beta" (sometimes called "coolness" or "coldness") which is just 1/T = partial S/partial E. The behavior of a spin system that admits negative temperatures can be described smoothly in terms of beta -- hotter systems have beta that is lower, and zero is not particularly special.
Now, any real system is coupled to the rest of the environment, so can't be in equilibrium at a negative temperature, as it would continuously leak heat until it cooled down enough to have some positive temperature. But if its internal equilibration proceeds much faster, then it's still useful to talk about its temperature as a quasi-equilibrium case.
