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What does a negative heat capacity mean?
The definition of the heat capacity of a material is that it's the amount of energy you need to add to one gram of a material to heat it by one degree Celsius. In the case of water, you need to add 1 calorie or 4.184 Joules to heat it by one degree. In metals, you need to add less energy, which is why metals seem to heat faster than water. You get the idea.
Given this definition, there's no such thing as a negative heat capacity. If there was, it would mean that you'd actually have to remove energy from something for it to get colder. Or, to give an example of what that would look like, if you put a material like this in the freezer, it would eventually glow white hot.