  We use rigorous non-equilibrium thermodynamic arguments to prove (i) the
residual entropy of any system is bounded below by the experimentally
(calorimetrically) determined absolute temperature entropy, which itself is
bounded below by the entropy of the corresponding equilibrium (metastable
supercooled liquid) state, and (ii) the instantaneous entropy cannot drop below
that of the equilibrium state. The theorems follow from the second law and the
existence of internal equilibrium and refer to the thermodynamic entropy. They
go beyond the calorimetric observations by Johari and Khouri [J. Chem. Phys.
134, 034515 (2011)] and others by extending them to all non-equilibrium systems
regardless of how far they are from their equilibrium states. We also discuss
the statistical interpretation of the thermodynamic entropy and show that the
conventional Gibbs or Boltzmann interpretation gives the correct thermodynamic
entropy even for a single sample regardless of the duration of measurements.
