  The configurational states as introduced by Goldstein represent the system's
basins and are characterized by their free energies $\varphi(T,V)$ as we show
here. We find that the energies of some of the special points (termed basin
identifiers here) like the basin minima, maxima, lowest energy barriers, etc.
cannot be used to characterize the configurational states of the system in all
cases due to their possible non-monotonic behavior as we explain. The
complexity $\mathcal{S(}\varphi,T,V\mathcal{)},$ represents the configurational
state entropy. We prove that $S(T,V) \equiv S(\varphi_b,T,V) + S_b(T,V)$, where
$S_{\text{b}},$ and $\varphi_{\text{b}}$ are the basin entropy and free energy,
respectively. We further prove that all basins at equilibrium have the same
equilibrium basin energy $E(T,V)$ and entropy $S_{\text{b}% }(T,V).$ Here,
$\varphi$\ and $E$ are measured with respect to the zero of the potential
energy. The Boltzmann equiprobability principle is shown to apply to the basins
in that each equilibrium basin has an equal probability
$\mathcal{P=}\exp(-\mathcal{S})$ to be explored. This principle allow us to
draw some useful conclusions about the time-dependence in the system. We
discuss the percolation due to basin connectivity and its relevance for the
dynamic transition. Our analysis validates modified Goldstein's conjectures.
All the above results are shown to be valid at all temperatures, and not just
low temperatures as originally propsed by Goldstein.
