Before revision:  The third is the assumption that a non-negative function whose gradient vanishes at some point necessarily has a minimum at that point and that its value at this minimum is zero. However, if the function is constant, which is the case here, its gradient vanishes everywhere, it has no minimum at any point, and its value can be arbitrarily large .
After revision:  The third is the claim that the Free Energy Lemma, when it does hold, implies equality of variational density and ergodic conditional density .