Reformulation and Interpretation of the Regularity Criterion for 3D NSE Based on Finitely Many Observations

We revisit and sharpen a recent observable regularity criterion for the three-dimensional Navier-Stokes equations on the periodic cube by requiring only finitely many measurements of the flow on a given time interval. Two data models are treated: (i) modal observations (a finite set of low Fourier modes), and (ii) nodal observations, i.e. values of the velocity field sampled at finitely many points on a uniform grid. The key upgrade is a piecewise linear interpolant built on a fixed five-tetrahedra subdivision of each grid cube, which removes the mollification step used previously and yields an explicit control of the derivatives of the interpolation operator purely in terms of the measured data. The criterion is also shown to be both necessary and sufficient for regularity.