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The question of whether the hyper-dissipative (HD) Napier-Stokes (NS) system can exhibit spontaneous formation of singularities in the super-critical regime--the hyper-diffusion being generated by a fractional power of the Laplacian, say $β$, 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, an evidence of criticality of the Laplacian is presented, more precisely, a class of plausible blow-up scenarios is ruled out as soon as $β$ is greater than one. While the framework 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 recently introduced by the authors, a major novelty in the current work is classification of the HD flows near a potential spatiotemporal singularity in two main categories, homogeneous (the case consistent with a near-steady behavior) and non-homogenous (the case consistent with the formation and decay of turbulence). The main theorem states that in the non-homogeneous case any $β$ greater than one prevents a singularity. In order to illustrate the impact of this result in a methodology-free setting, a two-parameter family of dynamically rescaled blow-up profiles is considered, and it is shown that as soon as $β$ is greater than one, a new region in the parameter space is ruled out. More importantly, the region is a neighborhood (in the parameter space) of the self-similar profile, i.e., the approximately self-similar blow-up, a prime suspect in possible singularity formation, is ruled out for all HD NS models.