I've been reading the Postulates of Classical Thermodynamics, and I haven't found anywhere to be said that the Absolute Entropy of a system has to be a positive number.
The third one states that the Temperature, $dU/dS$, must be Positive, since $S$ is monotone increasing with $U$, and that as $U(S)$ approaches flatness, $S$ approaches zero. But this doesn't mean that $S$ can't be negative. Consider for instance the function $U = kS^3$, which is monotone increasing, and where $S$ approaches zero as $dU/dS$ approaches zero from both sides, meaning $S$ could be negative.
Im pretty sure $S$ can't be negative, but I'd like to see the actual proof or Postulate that says so in CLASSICAL (not Statistical) Thermodynamics. Maybe it's included in the definition of entropy and I missed it?