  We are concerned with the critical threshold phenomena in the Restricted
Euler (RE) equations. Using the spectral and trace dynamics we identify the
critical thresholds for 3D and the 4D restricted Euler equations. It is well
known that the 3D RE solutions blow up. Projected on the 3-sphere, the set of
initial eigenvalues which give rise to bounded stable solutions is reduced to a
single point, which confirms that 3D RE blowup is generic. In contrast, we
identify a surprisingly rich set of the initial spectrum on the 4-sphere which
yields global smooth solutions; thus, 4D regularity is generic.
