  Observed well-developed $\alpha$ cluster states in $^{16}$O, located above
the four $\alpha$ threshold, are investigated from the viewpoint of
Bose-Einstein condensation of $\alpha$ clusters by using a field-theoretical
superfluid cluster model in which the order parameter is defined. The
experimental energy levels are reproduced well for the first time by
calculation. In particular, the observed 16.7 MeV $0_7^+$ and 18.8 MeV $0_8^+$
states with low-excitation energies from the threshold are found to be
understood as a manifestation of the states of the Nambu-Goldstone zero-mode
operators, associated with the spontaneous symmetry breaking of the global
phase, which is caused by the Bose-Einstein condensation of the vacuum 15.1 MeV
$0^+_6$ state with a dilute well-developed $\alpha$ cluster structure just
above the threshold. This gives evidence of the existence of the Bose-Einstein
condensate of $\alpha$ clusters in $^{16}$O. It is found that the emergence of
the energy level structure with a well-developed $\alpha$ cluster structure
above the threshold is robust, almost independently of the condensation rate of
$\alpha$ clusters under significant condensation rate. The finding of the
mechanism why the level structure that is similar to $^{12}$C emerges above the
four $\alpha$ threshold in $^{16}$O reinforces the concept of Bose-Einstein
condensation of $\alpha$ clusters in addition to $^{12}$C.
