  Lowest energy spectrum of the $^{12}$C nucleus is analyzed in the 3$\alpha$
cluster model with a deep $\alpha\alpha$-potential of Buck, Friedrich and
Wheatley with Pauli forbidden states in the $S$ and $D$ waves. The direct
orthogonalization method is applied for the elimination of the 3$\alpha$-Pauli
forbidden states. The effects of possible first order quantum phase transition
are shown in the lowest $^{12}$C($0_1^+)$ and $^{12}$C($2_1^+)$ states from
weakly bound phase to a deep phase. The ground and lowest $2^+$ states of the
$^{12}$C nucleus in the deep phase are created by the critical eigen states of
the Pauli projector for the $0^+$ and $2^+$ three-alpha functional spaces,
respectively.
