  Fractional Navier-Stokes equations -- featuring a fractional Laplacian --
provide a `bridge' between the Euler equations (zero diffusion) and the
Navier-Stokes equations (full diffusion). The problem of whether an initially
smooth flow can spontaneously develop a singularity is a fundamental problem in
mathematical physics, open for the full range of models -- from Euler to
Navier-Stokes. The purpose of this work is to present a hybrid,
geometric-analytic regularity criterion for solutions to the 3D fractional
Navier-Stokes equations expressed as a balance -- in the average sense --
between the vorticity direction and the vorticity magnitude, key geometric and
analytic descriptors of the flow, respectively.
