  The question of whether the hyper-dissipative Navier-Stokes (NS) system can
exhibit spontaneous formation of singularities in the super-critical
regime--the hyper-dissipation being generated by a fractional power of the
Laplacian confined to interval $\bigl(1, \frac{5}{4}\bigr)$--has been a major
open problem in the mathematical fluid dynamics since the foundational work of
J.L. Lions in 1960s. In this work, a mathematical evidence of the criticality
of the Laplacian is presented. While the framework for the proof is based on
the `scale of sparseness' of the super-level sets of the positive and negative
parts of the components of the higher-order derivatives of the velocity or
vorticity fields recently introduced by the authors, a major novelty in the
current work is the classification of the hyper-dissipative flows near a
potential spatiotemporal singularity in two main categories, that is,
`homogeneous' (the flows exhibiting a near-steady behavior, possibly after a
coordinate transformation) and `non-homogenous' (a generic case consistent with
the formation and decay of turbulence). The main result states that in the
non-homogeneous case, any power of the Laplacian greater than 1 yields a
contradiction, preventing a blow-up.
